Ученого совета - rudnpde2018moscow.rudn.ru/abstracts.pdf · on partial di erential...
TRANSCRIPT
Moscow 2018
International Conferenceon Partial Differential Equations and Applications
in Memory of Professor B. Yu. SterninMoscow, Russia, November 6–9, 2018
ABSTRACTS
RUDN University,Moscow State University,
Ishlinsky Institute for Problems in Mechanics RAS
Sponsored by:
RUDN University, Program 5-100,Ishlinsky Institute for Problems in Mechanics RAS,RFBR (grant 18-01-20093)
УДК 517.9(063) ББК 22.161.6 М43
У т в е р ж д е н о РИС Ученого совета
Российского университета дружбы народов
Издание подготовлено в рамках Программы повышения конкурентоспособности РУДН «5-100»
проект М 2.3.2. П1.3
М43 Международная конференция по дифференциальным уравне-ниям с частными производными и приложениям, посвящённая памяти профессора Б.Ю. Стернина. Москва, Россия, 6–9 ноября 2018 г. : тезисы докладов = International Conference on Partial Differential Equations and Applications in Memory of Professor B.Yu. Sternin. Moscow, Russia, 6–9 November 2018 : abstracts. – Москва : РУДН, 2018. – 79 с.
Конференция посвящена памяти профессора Бориса Юрьевича Стернина (1939–2017). Научная программа включает доклады по разным разделам теории дифференциальных уравнений с частными производными и их приложениям, кото-рыми занимался проф. Б.Ю. Стернин. В работе конференции принимают участие ведущие российские и зарубежные специалисты, а также его коллеги, ученики и соавторы.
ISBN 978-5-209-08991-9 © Коллектив авторов, 2018 © Российский университет дружбы народов, 2018
Program Committee:S. Yu. Dobrokhotov, A. T. Fomenko, Yu. A. Kordyukov, V. M. Manuilov,V. P. Maslov (Chairman), A. S. Mishchenko (Vice-Chairman), V. E. Nazaikinskii,G. Rozenblum, E. Schrohe, B.-W. Schulze, J. Sjostrand.
Organizing Committee:A. V. Boltachev, V. I. Burenkov, V. M. Filippov (Co-Chairman), P. L. Gurevich,N. R. Izvarina, D. A. Loshchenova, E. I. Moiseev (Co-Chairman), A. Yu. Savin(Vice-Chairman), P. A. Sipailo, A. L. Skubachevskii, K. N. Zhuikov.
Contents
N. M. AbasovExtension of an orthogonally additive map dominated by a continuous operator 10
P. M. Akhmet’evA remark on the Mahowald elements and on iterated desuspension in the stablehomotopy groups of spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
A. V. AlekseevCohomology of n-categories and derivations in group algebras . . . . . . . . . . 11
Yu. A. Alkhutov, M. D. SurnachevRegularity of a boundary point for the p(x)-Laplacian . . . . . . . . . . . . . . . 12
A. I. AlliluevaParametrix and asymptotics of rapidly varying solutions for linearized equa-tions of gas dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
J. A. Alvarez Lopez, Yu. A. Kordyukov, E. LeichtnamA trace formula for foliated flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
I. Androulidakis, G. SkandalisA Baum–Connes conjecture for singular foliations and its use . . . . . . . . . . 14
A. Yu. Anikin, S. Yu. Dobrokhotov, A. I. Klevin, B. TirozziAsymptotic solutions of stationary problems for linearized plasma equations 15
A. AntonevichC∗-algebras generated by dynamical systems, and applications . . . . . . . . . 15
A. I. AristovExact solutions of a nonclassical nonlinear partial differential equation . . . . 16
A. A. ArutyunovDerivations on group algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
I. V. AstashovaOn atypicality of power-law solutions to highly superlinear Emden–Fowler typeequations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
J. BruningThe equivariant Atiyah–Patodi–Singer theorem . . . . . . . . . . . . . . . . . . . . 19
V. I. Burenkov, P. D. LambertiStability estimates for the eigenvalues of higher order elliptic operators upondomain variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
S. N. BurianMotion on manifolds with singularities . . . . . . . . . . . . . . . . . . . . . . . . . . 20
V. L. ChernyshevAsymptotics for random walks on metric graphs . . . . . . . . . . . . . . . . . . . 21
D. ChoudhuriSchrodinger–Poisson system involving a measure . . . . . . . . . . . . . . . . . . . 23
V. DanilovGeneralized exponential type solutions to the Kolmogorov–Feller equation: for-ward and backward in time motion, vanishing diffusion . . . . . . . . . . . . . . . 24
A. S. DemidovElliptic pseudo-differential boundary value problems and the inverse problemof magneto-electroencephalography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
M. V. DeminaFinding algebraic invariants and algebraically invariant solutions . . . . . . . . 26
5
S. Yu. Dobrokhotov, V. E. Nazaikinskii
Lagrangian manifolds related to Bessel functions, and their applications . . . 27
J. Ecalle
Singular & singularly perturbed differential systems and their multiple resur-gence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
A. V. Faminskii
Large-time decay of solutions to the Zakharov–Kuznetsov equation . . . . . . 28
A. A. Fedotov
Quasiclassical asymptotics of solutions to difference equations with two closeturning points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
A. T. Fomenko
Topological billiards and integrable Hamiltonian systems . . . . . . . . . . . . . 30
H. Gacki, R. Brodnicka
Asymptotic stability of an evolutionary nonlinear Boltzmann-type equation 31
V. Gasimov
Hochshild’s method for describing the Mackenzie obstruction to the construc-tion of a transitive Lie algebroid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
R. Gaydukov
Derivation of the Benjamin–Ono equation at construction of the triple-deckstructure in problems of a fluid flow along a plate with small irregularities onthe surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Yu. E. Gliklikh
On existence of time-global solutions for parabolic equations . . . . . . . . . . . 34
A. Gorokhovsky, H. Moscovici
Pairings for pseudodifferential symbols . . . . . . . . . . . . . . . . . . . . . . . . . . 35
B. Gramsch
Complex homotopy principle of Grauert and Gromov for algebras of pseudod-ifferential operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
D. Grieser, R. B. Melrose
Eigenvalues for adiabatic problems in the presence of conical singularities . 36
S. M. Gusein-Zade
Universal Euler characteristic of orbifolds . . . . . . . . . . . . . . . . . . . . . . . . 36
A. Ya. Helemskii
Multi-normed spaces based on non-discrete measures, and their tensor prod-ucts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
Yu. Ilyashenko
Global bifurcations on the two sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
V. Ivrii
Complete semiclassical spectral asymptotics for periodic and almost periodicperturbations of constant operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
V. S. Kalnitsky
Phase flow over 0-singularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
M. I. Katsnelson
Engineering of quantum Hamiltonians by high-frequency laser fields . . . . . 40
A. P. Kiselev
Simple localized solutions of the wave equation . . . . . . . . . . . . . . . . . . . . 41
6
Yu. Yu. KlevtsovaOn the rate of convergence as t→ +∞ of the distributions of solutions to thestationary measure for the stochastic system of the Lorenz model for a baro-clinic atmosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
V. N. KolokoltsovFractional PDEs: brief introduction and new perspectives . . . . . . . . . . . . . 43
Yu. A. KordyukovSpectral geometry of generalized smooth distributions . . . . . . . . . . . . . . . 44
L. M. KozhevnikovaEntropy and renormalized solutions of elliptic equations with variable expo-nents of nonlinearities and measure data . . . . . . . . . . . . . . . . . . . . . . . . . 45
M. V. KukushkinSome phenomena in the behavior of the eigenvalues of fractional differentiationoperators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
M. KumarNumerical solution of elliptic partial differential equations . . . . . . . . . . . . . 46
A. G. KyurkchanThe use of a priori analytic information to solve boundary-value problems ofdiffraction theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
A. Lazarev, J. Chuang, J. HolsteinStrong homotopies of differential graded algebras and theorems of Hilbert-Riemann type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
X. Li, A. S. MishchenkoThe homotopy classification of transitive Lie algebroids . . . . . . . . . . . . . . 49
V. LychaginOn classification of the second order differential operators and differential equa-tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
M. M. MalamudTo spectral theory of Schrodinger and Dirac operators with point interactions 50
D. MazlumPublishing Mathematics with Birkhauser . . . . . . . . . . . . . . . . . . . . . . . . . 51
V. Maz’yaSobolev inequalities in arbitrary domains . . . . . . . . . . . . . . . . . . . . . . . . . 51
G. A. MendozaBoundary values and problems for elliptic w-operators on compact manifoldswith fibered boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
D. V. MillionshchikovKlein–Gordon equation and growth of Lie algebras of vector fields . . . . . . . 52
D. S. MinenkovSemi-classical asymptotics for the two-dimensional radially symmetric Diracequation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
A. S. MishchenkoDerivations of group algebras and Hochschild cohomology . . . . . . . . . . . . . 53
E. I. Moiseev, D. A. GulyaevCompleteness of perturbed systems of trigonometric functions in the space ofLebesgue integrable functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
Q. Morales MelendezReduction of computation of the signature of a G-manifold in the simplest case 54
7
A. B. MuravnikOn absence of solutions of correlated-noise inequalities . . . . . . . . . . . . . . . 55
A. I. NazarovOn fractional Neumann Laplacians in the half-space . . . . . . . . . . . . . . . . . 55
N. N. NefedovPropagation and blowing-up of non-plane fronts in Burgers-type equationswith modular advection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
Nguyen Le AnhLie algebroids and Lie algebra bundles . . . . . . . . . . . . . . . . . . . . . . . . . . 56
J. R. Oliveira, A. S. MishchenkoWhitney-Sullivan constructions for transitive Lie algebroids: the polynomialcase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
A. Yu. PirkovskiiDense quasi-free subalgebras of the Toeplitz algebra . . . . . . . . . . . . . . . . . 57
M. A. PlievOn the sum of narrow and compact operators . . . . . . . . . . . . . . . . . . . . . 58
R. PongeCyclic homology of crossed-product algebras . . . . . . . . . . . . . . . . . . . . . . 59
Th. Yu. PopelenskyCombinatorial Ricci flow for degenerate metrics . . . . . . . . . . . . . . . . . . . . 60
K. Reijnders, M. KatsnelsonSemiclassical theory of electronic Veselago lensing in graphene . . . . . . . . . 61
L. Rossovskii, A. TovsultanovElliptic functional differential equation with contracted and shifted argument 62
M. Rouleux, H. LouatiQuasi-classical quantum maps of semi-hyperbolic type . . . . . . . . . . . . . . . 63
J. RowlettHow to hear the corners of a drum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
Yu. K. SabitovaDirichlet problem for mixed type equation with characteristic degeneration 64
T. SalnikovaSteady-state motion of charged dust particles under gravitational forces andforces of inertia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
E. Schrohe, A. Savin, B. SterninAnalytic and algebraic indices of elliptic operators associated with quantizedcanonical transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
S. Ya. Sekerzh–Zen’kovich, A. A. TolchennikovSome applications of asymptotic solutions of Boussinesq type equations for theapproximation of 2011 tsunami mareograms from DART stations . . . . . . . 67
A.G. SergeevQuantum Hall effect and noncommutative geometry . . . . . . . . . . . . . . . . . 67
S. A. SergeevAsymptotics of the solution explicit difference scheme for the wave equationwith localized initial data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
A. I. ShafarevichLaplacians and wave equations on two-dimensional polyhedra . . . . . . . . . . 69
T. ShaposhnikovaSobolev multipliers and their applications to differential and integral operators 69
8
G. SharyginBruhat order and the symmetric Toda flow on real Lie groups . . . . . . . . . . 69
A. A. ShkalikovOn perturbations of self-adjoint and normal operators: analytical aspects . . 70
B. K. Singh, U. GogoiStudy of some general classes of estimators for estimating population mean incompromised imputation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
A. L. SkubachevskiiStationary and nonstationary solutions of mixed problems for Vlasov–Poissonequations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
A. I. ShternLocally bounded finally precontinuous finite-dimensional quasirepresentationsof connected locally compact groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
G. SuLower bounds of Lipschitz constants on foliations . . . . . . . . . . . . . . . . . . . 72
I. TalibApproximate solution of fractional order partial differential equations and itscoupled systems using operational matrices approach . . . . . . . . . . . . . . . . 73
B. Tirozzi, P. Buratti, F. Alladio, P. MicozziSolution of the equations describing the plasma in a gas discharge . . . . . . . 74
A. A. TolchennikovAsymptotics of the solution of the Klein–Gordon equation with localized initialconditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
V. B. VasilyevPseudo-differential operators, equations, and elliptic boundary value problems 75
V. V. VedyushkinaThe non-trivial Liouville foliation of a three-dimensional torus . . . . . . . . . 76
B. VertmanThe resolvent expansion of geometric operators on stratified spaces . . . . . . 77
E. V. Vybornyi, M. V. KarasevAsymptotic of tunneling for Schrodinger equation with hyperbolic frequencyresonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
J. YuFried conjecture for Morse–Smale flow . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
W. ZhangPositive scalar curvature on foliations . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
C. Zhu, B. Booss-BavnbekThe Maslov index in symplectic Banach spaces . . . . . . . . . . . . . . . . . . . . 79
9
Extension of an orthogonally additive mapdominated by a continuous operator
N. M. AbasovMoscow Aviation Institute (National Research University), Moscow, Russia
Today, the theory of orthogonally additive operators in vector lattices is an activearea of functional analysis (see, for instance, [1, 2, 6, 7, 8])). In the recent years someconnections with the problems on convex geometry emerged [5, 9].
Let E and F be vector lattices.
Recently, in the forthcoming paper [3] it was proved that a dominated orthogonallyadditive operator T : V → W is laterally-to-order continuous if and only if so is itsexact dominant |T | : E → F . An element y of a lattice-normed space V is said to bea fragment of an element x ∈ V (in another terminology, a component) if y ⊥ (x−y).The set of all fragments of x is denoted by Fx. A subset D of a lattice-normed spaceV is called a lateral ideal if the following conditions hold:
• if x ∈ D, then y ∈ D for any y ∈ Fx;
• if x, y ∈ D, x ⊥ y, then x+ y ∈ D.
Theorem. Let (V,E) be a lattice-normed space, let (W,F ) be a Banach–Kantorovichspace over a Dedekind complete vector lattice F , let D be a lateral ideal in V , andlet T : D → W be an orthogonally additive map dominated by a laterally-to-ordercontinuous (σ-laterally-to-order continuous) positive orthogonally additive operatorS : E → F . Then there exists a dominated laterally-to-order continuous (σ-laterally-
to-order continuous) orthogonally additive operator TD : V →W such that TDx = Txfor any x ∈ D.
References[1] Abasov, N., Pliev, M., “On extensions of some nonlinear maps in vector lattices”,
J. Math. Anal. and Appl., 455, 2017, 516–527.[2] Abasov, N., Pliev, M., “Disjointness preserving orthogonally additive operators in
vector lattices”, Banach Journal of Math. Anal., 12:3, 2018, 730–750.[3] Abasov, N., Pliev, M., “Dominated orthogonally additive opera-
tors in lattice-normed spaces”, Advances in Operator Theory, 2018,doi:10.15352/aot.1804-1354.
[4] Kusraev, A. G., Dominated Operators, Kluwer Academic Publishers, 2000.[5] Ludwig, M., Reitzner, M., “A classification of SL(n) invariant valuations”, Ann.
of Math., 172:2, 2010, 1219–1267.[6] Pliev, M., Ramdane, K., “Order unbounded orthogonally additive operators in
vector lattices”, Mediterranean Journal of Math., 15:2, 2018.[7] Pliev, M., “Domination problem for narrow orthogonally additive operators”, Pos-
itivity, 21:1, 2017, 23–33.[8] Pliev, M. A., Weber, M. R., “Finite elements in some vector lattices of nonlinear
operators”, Positivity, 22:1, 2018, 245–260.[9] Tradacete, P., Villanueva, I., “Continuty and representation of valuations on star
bodies”, Adv. of Math., 329, 2018, 361–391.
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A remark on the Mahowald elementsand on iterated desuspension in the stable
homotopy groups of spheres
P. M. Akhmet’evHigher School of Economics, Moscow
A new geometrical approach toward a description of E2 term of the Adams spectralsequence is presented. This approach is based on skew-framed immersion cobordismgroup [1]. As an example, a geometrical description of the Mahowald elements [2] ofdimension 2i, i ≥ 3 is presented. The approach allows to compare the desuspensionproblem for stable homotopy groups Π128 and Π126. The Mahowald element in Π128
admits no 5-time desuspensions into the unstable range. Barratt–Jones–Mahowaldapproach relates the Snaith Conjecture [3] about Arf-invariant in dimension 126 withthe desuspension problem.
This is a part of joint results with Th. Yu. Popelenskii and O. D. Frolkina.
References[1] Akhmet’ev, P. M., Frolkina, O. D., “On properties of skew-framed im-
mersions cobordism groups”, Fundam. Prikl. Mat., 21:5, 2016, 19–46,arXiv:1712.00959v1.
[2] Mahowald, M., “A new infinite family in 2πS∗ ”, Topology, 16, 1977, 249–256.
[3] Snaith, V. P., Stable homotopy around the Arf-Kervaire invariant, BirkhauserProgress on Math. Series vol. 273, April 2009.
Cohomology of n-categories and derivationsin group algebras
A. V. AlekseevMoscow Institute of Physics and Technology, Moscow, Russia
This work represents the concept of an n-groupoid Γn and n-characters χn onn-groupoids as complex-valued maps from spaces of different classes of morphismssatisfying the condition χn(ψ kϕ) = χn(ψ)+χn(ϕ) for any possible compositions. Asequence of spaces of n-characters and morphisms between them is constructed andits exactness is shown. This construction has an important application for describingderivations in group algebras. In particular, this approach allows us to study theouter derivation algebra from a new point of view and also construct some interestingexamples. The research was supervised by Professor Arutyunov and is based on ideasdue to Professor Mishchenko.
References[1] Ehresmann, C., Categories et structures, Dunod, Paris, 1965.[2] Benabou, J., “Introduction to bicategories”, Reports of the Midwest Category
Seminar, Springer, Berlin, 1967, 1–77.[3] Mac Lane, S., Categories for the Working Mathematician, 2004, 352pp., ISBN
0-387-90036-5.
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[4] Arutyunov, A. A., Mishchenko, A. S., Shtern, A. I., “Derivations of Group Alge-bras”, arXiv:1708.05005.
Regularity of a boundary pointfor the p(x)-Laplacian
Yu. A. AlkhutovVladimir State University, Vladimir, Russia
M. D. SurnachevKeldysh Institute of Applied Mathematics, Moscow, Russia
In a bounded domain D ⊂ Rn, n ≥ 2 consider the generalized Dirichlet problem
Luf = div(|∇uf |p(x)−2∇uf
)= 0 in D, uf |∂D = f ∈ C(∂D).
with a measurable exponent p satisfying 1 < α ≤ p(·) ≤ β <∞ a.e. in D. A solutioncorresponding to a continuous boundary function is obtained as a limit of variationalsolutions with smooth boundary data, see [1] for details. Due to the Lavrentievphenomenon two different types of solutions may arise — H- and W -solutions [2].Thus the following definition can be considered in the H- and W -frameworks.
Definition 1. The boundary point x0 ∈ ∂D is called regular if ess limD3x→x0
uf (x) = f(x0)
for any function f ∈ C(∂D).
We assume that the exponent p is essentially continuous at the boundary pointx0 ∈ ∂D, and ess oscp, Bx0
r ≤ ω(r), where Bx0r is an open ball of radius r centered
at x0, and ω is a continuous, nondecreasing on [0, d) function, such that ω(0) = 0.We set p(x) = p(x0) for x ∈ Rn \D.
Definition 2. (see [1]). The H-capacity (W -capacity) of a compact set K ⊂ Bx0
R
with respect to the ball Bx0
R is the number Cp(K,Bx0
R ) = inf∫BR
|∇ϕ|p(x)(p(x))−1 dx,
where the infimum is taken over the set of function ϕ ∈ C∞0 (Bx0
R ) (ϕ ∈ W0(Bx0
R ))that are greater than or equal to 1 almost everywhere on K. For H- and W -solutionswe use H- and W -capacities, correspondingly.
Let the function θ(r) = r−ω(r) be nonincreasing on (0, d). We denote p0 = p(x0).
Theorem 3. Let∫0
exp (−c1θc2(t))
(Cp
(Bx0
t \D,Bx02t
)tp0−n
)1/(p0−1)
t−1 dt =∞,
where ci = ci(n, α, β) > 0, i = 1, 2. Then the boundary point x0 ∈ ∂D is regular.
For p = const this theorem was obtained in [3], and for ω(t) = C/(ln 1/t)−1 in [1].This work was supported by the Ministry of Education and Science of the Russian
Fedration (project No. 1.3270.2017/4.6).
12
References[1] Alkhutov, Yu. A., Surnachev, M. D., “Regularity of a boundary point for the p(x)-
Laplacian”, J. Math. Sci., 232:3, 2018, 206–231.[2] Zhikov, V. V., “On variational problems and nonlinear elliptic equations with
nonstandard growth conditions”, J. Math. Sci., 173:5, 2011, 463–570.[3] Maz’ya, V. G., “On continuity in a boundary point of solutions of quasilinear
elliptic equations”, Vestn. Leningr. Univ., Ser. Mat. Mekh. Astron., 25:13, 1970,42–55 [in Russian].
Parametrix and asymptotics of rapidly varyingsolutions for linearized equations of gas dynamics
A. I. AlliluevaIshlinsky Institute for Problems in Mechanics, RAS, Moscow, Russia
We study system of equations of gas dynamics with small viscosity linearized ona smooth external flow. The parabolic part of the system is degenerate; for thevanishing viscosity, we obtain nonstrictly hyperbolic system (containing simple aswell as multiple characteristics). For this system, we describe the expansion of theresolving operator, which is asymptotic with respect to smoothness and to smallviscosity simultaneously. The asymptotic series consists of summands of two differenttypes: they describe acoustic and hydrodynamic modes. For hydrodynamic modes,there are no focal points (characteristics lie in configuration space); the amplitude iscomputed from the system of ODE’s along the characteristics. For the acoustic modes,we use Maslov canonic operator to describe the summands of asymptotic series. Theamplitudes (functions on the corresponding Lagrangian manifolds) can be computedexplicitly. We apply this construction to the description of short-wave asymptotics.We show that for certain external flows the hydrodynamic modes can grow in time.
This work was supported by RFBR (grants 18-31-00273, 17-51-150006).
A trace formula for foliated flows
Jesus A. Alvarez LopezUniversity of Santiago de Compostela, Santiago de Compostela, Spain
Yuri A. KordyukovRussian Academy of Sciences, Ufa, Russia
Eric LeichtnamInstitut de Mathematiques de Jussieu, Paris, France
Let F be a smooth codimension one foliation on a compact manifold M . A flowφt on M is said to be foliated if it maps leaves to leaves. If moreover the closedorbits and preserved leaves are simple, then there are finitely many preserved leaves,which are compact, forming a compact subset M0, and a precise description of thetransverse structure of F can be given. A version of the reduced leafwise cohomology,
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HI(F), is defined by using distributional leafwise differential forms conormal to M0.The talk will be about our progress to define distributional traces of the inducedaction of φt on H
rI(F), for every degree r, and to prove a corresponding Lefschetz
trace formula involving the closed orbits and leaves preserved by φt. The formulaalso involves a version of the η-invariant of M0. This kind of distributional traceformula was conjectured by Christopher Deninger [3], and it was proved by the firsttwo authors when M0 = ∅ [1, 2].
References[1] Alvarez Lopez, J. A., Kordyukov, Yu. A., “Distributional Betti numbers of transi-
tive foliations of codimension one”. In: Foliations: geometry and dynamics. Pro-ceedings of the Euroworkshop, Warsaw, Poland, May 29–June 9, 2000 . World Sci.Publ., Singapore 2002, pp. 159–183.
[2] Alvarez Lopez, J. A., Kordyukov, Y. A., “Lefschetz distribution of Lie foliations”.In: C∗-algebras and elliptic theory II . Trends Math., Birkhauser, Basel, 2008,pp. 1–40.
[3] Deninger, C., “Analogies between analysis on foliated spaces and arithmetic ge-ometry”. In: Groups and analysis. London Math. Soc. Lecture Note Ser., vol. 354,Cambridge Univ. Press, Cambridge 2002, pp. 174–190.
A Baum–Connes conjecture for singular foliationsand its use
Iakovos AndroulidakisNational and Kapodistrian University of Athens, Athens, Greece
Georges SkandalisUniversite Paris 7, Denis Diderot, Paris, France
In several cases, a non-compact manifold can be understood as a dense leaf of asingular foliation. We will discuss how, in cases as such, spectral gaps of the Laplacianmay be detected using the K-theory of the foliation C∗-algebra. The “shape” of thisK-theory is described by a Baum–Connes assembly map for singular foliations.
References[1] Androulidakis, I., “Laplacians and spectrum for singular foliations”, Chin. Ann.
Math. Ser. B, 35:5, 2014, 679–690.[2] Androulidakis, I., Skandalis, G., “A Baum–Connes conjecture for singular folia-
tions”. Preprint: arXiv:1509.05862.
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Asymptotic solutions of stationary problemsfor linearized plasma equations
A. Yu. AnikinIshlinsky Institute for Problems in Mechanics of RAS, Moscow, Russia
S. Yu. DobrokhotovIshlinsky Institute for Problems in Mechanics of RAS, Moscow, Russia
A. I. KlevinIshlinsky Institute for Problems in Mechanics of RAS, Moscow, Russia
B. TirozziENEA Centro Ricerche Frascati, Roma, Italy
We propose a method described in [1] for calculating asymptotic solutions of sta-tionary problems for differential (or pseudodifferential) operators whose symbol is aself-adjoint matrix. We show that the problem of constructing asymptotic solutionscorresponding to a fixed eigenvalue (called an effective Hamiltonian, term, or mode)reduces to studying objects related only to the determinant of the principal matrixsymbol and the eigenvector corresponding to a given (numerical) value of this effectiveHamiltonian. We apply this method in a linearized 12× 12 system describing plasmamotion (e.g. in tokamak).
This work was supported by the RFBR grant 18-31-00273.
References[1] Anikin, A. Yu., Dobrokhotov, S. Yu., Klevin, A. I., Tirozzi, B., “Scalarization of
stationary semiclassical problems for systems of equations and its application inplasma physics”, Theoret. and Math. Phys., 193:3, 2017, 1761–1782.
C∗-algebras generated by dynamical systems,and applications
A. B. AntonevichBelarussian State University, Minsk, Belarus
The main problems we are dealing with in the present report originate in the studyof the operators acting in some function spaces over a set X of the form
Bu :=∑k
aku(αk(x)) = f(x), x ∈ X, (1)
where αk : X → X are certain transformations of the domain X and ak are operatorsfrom some operator’s algebra A, as a rule well-investigated. If X is a manifold andak = ak(x,D) are pseudodifferential operators (PDO) on X, then the operator (1) is anonlocal pseudodifferential operator. If ak = ak(x) are operators of multiplication byfunctions or matrix functions, then the operator (1) is a “pure” functional operator.An operator of the form bu = a(x)u(α(x)) is called a weighted shift operator.
15
The report contains a short survey of the results on this subject given in [1, 2], assoon as some new results from [1, 2].
We present the following results:1. General approach to investigate these operators based on the theory of C∗-algebrasgenerated by dynamical systems.2. Symbolic calculus for nonlocal pseudodifferential operators. The symbol σ(B) ofsuch operator B is a “pure” functional operator acting in a function space on thecotangent bundle of the manifold X.
Theorem. A nonlocal pseudodifferential operator B is Fredholm if and only if itssymbol σ(B) is invertible. The operator B is semi-Fredholm if and only if its symbolσ(B) is one-sided invertible.
3. The description of the spectral properties of weighted shift operator and in-vertibility conditions for a “pure” functional operator.4. One-sided invertibility conditions for weighted shift operators.
References[1] Antonevich, A. B., Linear Functional Equations. Operator Approach, Operator
Theory: Advances and Applications, Vol.83, Birkhauser, 1996.[2] Antonevich, A. B., Lebedev, A. V., Functional differential equations: I. C∗-theory,
Longman Scientific and Technical, 1994.[3] Antonevich, A. B., Ahmatova, A. A., Makowska, Ju., “Maps with separable dy-
namics and the spectral properties of the operators generated by them”, Sb. Math.,206:3, 2015, 341–369.
[4] Antonevich, A. B., Panteleeva, E. V., “Well-posed boundary value problems, righthyperbolicity, exponential dichotomy”, Mathematical Notes, 100:1, 2015, 11–23.
Exact solutions of a nonclassical nonlinear partialdifferential equation
A. I. AristovLomonosov Moscow State University, Moscow, Russia
We study the equation∂2u
∂t∂x+ up
∂u
∂x= uq. (1)
Here u(·) is a real function of the space variable x ∈ R and time t > 0. The parametersp and q are integer; moreover, p > 0, q > 1, p+ q > 2.
Earlier (see [1]), the similar equation
∂2u
∂t∂x+
∂
∂x
(|u|p−2u
)= |u|q
was studied. It describes a nonstationary process in a semiconductor with regardto heating. Sufficient conditions for the nonexistence of generalized solutions of theCauchy problem and some other problems for the equation on every finite time intervalwere obtained.
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Our aim is to construct exact solutions of (1) and to analyse their qualitativeproperties. Fourteen classes of exact solutions of (1) are constructed in the presentwork. Among them are both bounded solutions and solutions going to infinity whentime goes to a finite value.
We use separation of variables and the method of differential constraints. Also,travelling wave solutions, self-similarity solutions, and some special solutions are con-structed. Some computations are automated with the Maple system.
This work was supported by the Presidential Program for Support of Young Phi-losophy Doctors (project no. MK-1829.2018.1).
References[1] Korpusov, M. O., Lukyanenko, D. V., “Instantaneous blow-up versus local solv-
ability for one problem of propagation of nonlinear waves in semiconductors”,Journal of Mathematical Analysis and Applications, 459, 2018, 159–181.
Derivations on group algebras
Andronick ArutyunovMoscow Institute of physics and Technologies, Moscow, Russia
The main theme of the report is a combinatorial description of derivations on agroup algebra. The linear mapping d : C[G]→ C[G] is said to be the derivation if theLeibniz rule hold
d(uv) = d(u)v + ud(v),∀u, v ∈ C[G].
We can consider a gruppoid Γ in the following way.The set of objects is a set of elements Obj(Γ) = g ∈ G. The set of arrows is
a set of pairs Hom(Γ) := (u, v) | u, v ∈ G. An arrow φ = (u, v) has a sources(φ) = v−1u and a target t(φ) = uv−1. For arrows φ = (u1, v1) and ψ = (u2, v2) suchthat t(φ) = s(ψ) we will define a composition
ψ φ := (v2u1, v2v1).
We will call a character any mapping χ : Hom(Γ)→ C such that χ(ψ φ) = χ(ψ) +χ(φ) when t(φ) = s(ψ). We will call a charachter a locally finite character if ∀v ∈Gχ((u, v)) = 0, for all except of finite number of elements u ∈ G.
Theorem 1. For any derivation d exist a character χd such that ∀a ∈ G
d(a) = a
∑t∈[u]
χd(at, a)t
.
This theorem gives a possibility to calculate an algebra of derivations for differentgroup algebras. Let G be a central commutative extension group. Then we geta theorem. Let τ : G → C be a homomorphism on an additive group of complexnumbers. The space of such homomorphisms we will denote by T (G). And let z bean element of a group centre z ∈ Z(G). Then
dzτ (g) = τ(g)gz.
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Let G+ be a set of elements such that G/Z(u) has a nonfinite cyclic subgroup, butdoesn’t have a subgroup isomorphic to a a free abelian group of rank 2. Then
d+u (x) = z(t−1ut+ t−2ut2 + · · ·+ t−lutl).
Theorem 2. Any derivation d can be written in the following way
d = azjτi dτizj + bld+
ul+ . . .
where dτizj , zg ∈ Z(G), τi are generators of T (G), and dui for ui ∈ G+. By . . . we
mean inner derivations. Sure, almost all coefficients azjτi , b
l have to be equal to 0.
References[1] Arutyunov, A. A., Mishchenko, A. S., “Smooth Version of Johnson’s Problem Con-
cerning Derivations of Group Algebras”, arXiv:1801.03480.
On atypicality of power-law solutions to highlysuperlinear Emden–Fowler type equations
I. V. AstashovaLomonosov Moscow State University and Plekhanov Russian University of Economics,
Moscow, Russia
Consider the equation
y(n) = p(x, y, y′, . . . , y(n−1))|y|k sign y, n ≥ 2, k > 1. (1)
Results on asymptotic behavior of blow-up solutions to this equation are presented. Inparticular, for slightly superlinear equations we prove Kiguradze’s hypothesis (see [1],Problem 16.4) about power-law behavior of all blow-up solutions.
Theorem 1 ([2]). Suppose p ∈ C(Rn+1)⋂Lipy0,...,yn−1(Rn) and p → p0 > 0 as
x→ x∗, y0 →∞, . . . , yn−1 →∞. Then for any integer n > 4 there exists K > 1 suchthat for any real k ∈ (1,K), any solution to equation (1) tending to +∞ as x→ x∗−0has power-law asymptotic behavior, namely
y(x) = C(x∗ − x)−α(1 + o(1)), α =n
k − 1, Ck−1 =
1
p0
n−1∏j=0
(j + α) .
The existence of blow-up solutions with non-power-law behavior was also proved,which means that in general case this hypothesis is not true. (See [3] and [4]).
It appears that the asymptotic behavior of solutions to equation (1) depends onthe spectrum of a related linear operator, and in general case, even for equation (1)with p = p0 > 0, the power-law behavior of blow-up solutions is atypical.
Theorem 2. Suppose n ∈ N, α ∈ (0; +∞), and the algebraic equation∏n−1j=0 (λ +
j + α) =∏n−1j=0 (1 + j + α) has at least two roots with positive real part and no purely
imaginary root. Then solutions to the differential equation (1) with p = p0 > 0having the power-law asymptotic behavior y(x) ∼ C(x∗ − x)−β as x → x∗ − 0 withany C 6= 0, β > 0, x∗ < ∞ are atypical, i.e. at any point x0 the subset of allinitial data producing solutions with the above power-law asymptotic behavior has nullLebesgue measure in Rn.
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Corollary 1. Suppose n > 11 and the equation∏n−1j=0 (λ+ j) =
∏n−1j=0 (1 + j) has no
purely imaginary root. Then there exists Kn > 1 such that for any k > Kn solutionsto equation (1) with p = p0 > 0 with the power-law asymptotic behavior are atypical.
References[1] Kiguradze, I. T., Chanturia, T. A., Asymptotic Properties of Solutions of
Nonautonomous Ordinary Differential Equations, Kluver Academic Publishers,Dordreht–Boston–London, 1993.
[2] Astashova, I., “On Asymptotic Behavior of Blow-Up Solutions to Higher-OrderDifferential Equations with General Nonlinearity”, Springer Proceedings in Math-ematics and Statistics, 230, 2018, 1–12.
[3] Kozlov, V. A., “On Kneser solutions of higher order nonlinear ordinary differentialequations”, Ark. Mat., 37:2, 1999, 305–322.
[4] Astashova, I. V., “On power and non-power asymptotic behavior of positive so-lutions to Emden-Fowler type higher-order equations”, Advances in DifferenceEquations, 2013:220, doi:10.1186/1687-1847-2013-220.
The equivariant Atiyah–Patodi–Singer theorem
J. BruningInstitut fur Mathematik der Humboldt-Universitat zu Berlin, Germany
In 1975, Atiyah, Patodi and Singer proved a remarkable index theorem for a firstorder elliptic operator, D, acting on the smooth sections of a complex vector bundleover a compact manifold. The key point was that the manifold now was supposedto have a boundary, such that elliptic boundary conditions had to be imposed for D.The natural intuition that suitable local boundary conditions would work turned outto be wrong; what the authors brought in were spectral boundary conditions based onthe restriction of D to the boundary, that yielded an elliptic first order operator, A.Then the boundary condition was expressed in terms of the spectral projection P<(A).The index formula, therefore, was expected to involve interior as well as boundarycontributions. This could be verified and resulted in the local term, integrated overthe manifold, and the famous eta-invariant of the operator A.
The purpose of this talk is to present the equivariant version of this result. Weassume that a compact Lie group, G, acts on all data. This means that the actionof G is restricted to the (infinite dimensional) isotypical subspace formed by an irre-ducible representation; this is a subspace of the original space hence not the space ofsections of some smooth vector bundle over the manifold. Nevertheless, we can provea completely analogous index formula for the restricted operator that is independentof the complexity of the Thom–Mather space arising from the G-orbit spaces of theisotypical subspaces.
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Stability estimates for the eigenvalues of higherorder elliptic operators upon domain variation
V. I. BurenkovRUDN University, Moscow, Russia
P. D. LambertiPadova University
We consider the eigenvalue problem for the operator
Hu = (−1)m∑
|α|=|β|=m
Dα(Aαβ(x)Dβu
), x ∈ Ω,
subject to homogeneous Dirichlet or Neumann boundary conditions, where m ∈ N,Ω is a bounded open set in RN and the coefficients Aαβ are real-valued Lipschitzcontinuous functions satisfying Aαβ = Aβα and the uniform ellipticity condition∑
|α|=|β|=m
Aαβ(x)ξαξβ ≥ θ|ξ|2
for all x ∈ Ω and all ξα ∈ R, |α| = m, where θ > 0 is the ellipticity constant.We consider open sets Ω for which the spectrum is discrete and can be represented
by means of a non-decreasing sequence of non-negative eigenvalues
λ1[Ω] ≤ λ2[Ω] ≤ · · · ≤ λn[Ω] ≤ . . .
where each eigenvalue is repeated as many times as its multiplicity.We present estimates for the variation
|λn[Ω1]− λn[Ω2]|
of the eigenvalues corresponding to two open sets Ω1, Ω2. Our analysis comprehendsopen sets with arbitrarily strong degenerations.
Motion on manifolds with singularities
S. N. BurianSt. Petersburg State University, Saint Petersburg, Russia
The main problem of our research is the construction of some analog of differentialcalculus for manifolds with singularities. This problem arises in theoretical mechanicswhen a mechanism may have some branching points where it can change the typeof motion. It is important to answer the following question: what is an ODE onmanifolds with singularities? Classical techniques for deriving equations of motionsuch as Newton’s second law and the minimum action principle have difficulties.
We study the case when the model space X consists of two smooth curves γ1 andγ2 which are tangent at a point x. From general considerations we could predict thatmotion have to be smooth. But is there a possibility for system to change curve after
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passing through the singular point? The following geometric approach is native. Wecould study X as an object of S, some category of manifolds with singularities. Forthe case of manifolds M the equation of motion is a vector field on the tangent bundleTM . In the case of a space X ∈ S we generalize this scheme: it is required that forall Y ∈ S there exists a “tangent bundle” TY ∈ S and for all Y ∈ S the class of“smooth” vector fields VY and the class of “smooth” curves r : R→ Y are defined.
Calculations in this way when S is the category of Frolicher spaces [1, 2]) will beshown. The example of mechanical system with singularities could be found in [3].
References[1] Watts, J., “Diffeologies, differential spaces, and symplectic geometry”. Preprint:
arXiv:1208.3634 (2012).[2] Laubinger, M., “Differential geometry in Cartesian closed categories of smooth
spaces”, PhD thesis, Louisiana State University (2008).[3] Burian, S. N., Kalnitsky, V. S., “On the motion of one-dimensional double pendu-
lum”, AIP Conference Proceedings, 1959, 2018, 030004.
Asymptotics for random walks on metric graphs
V. L. ChernyshevNational Research University Higher School of Economics, Moscow, Russia
Let us consider a random walk on a finite compact metric graph (see, for example,[1]). Let one point move along the graph at the initial moment of time. The passagetime for each individual edge is fixed. In each inner vertex, the point with someprobability selects one of the edges for further movement. The reflection occurs inthe vertices of valence one. Backward turns on the edges are prohibited. The aimis to analyze the asymptotic behavior of the number N(T ) of possible endpoints ofsuch random walk on the metric graph as time T increases. We suppose that theprobability of choosing an edge is non-zero for all edges. It is a situation of a generalposition.
Such random walk is typical for evolution of wave packets, localized in a smallneighborhood of one point at the initial moment of time (see [2] and referencestherein).
In the case of linearly independent over the rationals lengths the problem is relatedto the problem of counting the number of lattice points in expanding simplexes withreal vertices. An asymptotic expansion for N(T ) using Barnes’ multiple Bernoullipolynomials [5, 7] (also known as Todd polynomials, see [6] for details) was found.Explicit formulas for the first two terms of the expansion for the counting function ofthe number of moving points are presented (see [4] for details). The leading term wasfound earlier (see [2]) and depends only on the number of vertices V , the number ofedges E and the sum and the product of lengths of the edges tj .
The second term of the asymptotics is more complicated. It is determined by thequadratic form (i.e. E by E symmetric matrix) of the lengths of the edges of themetric graph. And it, generally speaking, depends on the starting vertex.
Let G denote a finite connected subgraph of the graph Γ, containing the startingvertex s. For the subgraph G ⊂ Γ the vertex set is denoted by V (G) and the setof edges by E(G). For v ∈ V (G), we denote by ρ(G, v) the valency of the vertex
21
v in the subgraph G. The vertex v is said to be the end-vertex in the subgraph Gif ρ(G, v) = 1. Then the unique edge e ∈ E(G) is called the end-edge. The edgee ∈ E(G) is called the isthmus, if after deleting this edge the graph G splits into twoconnected components.
Theorem 1. Suppose that the finite graph Γ has edge lengths t1, . . . , tE, linearlyindependent over Q, and the starting vertex s is not an end-vertex, then the countingfunction has the decomposition N(T ) = N1T
E−1 +N2TE−2 + o(TE−2), where
N2 =1
2E−2(E − 2)!∏Ei=1 ti
−1
2
E∑j=1
E∑i=1,i6=j
tjtiγi,j2β1(Γ\ei) − 2β1(Γ)−1
(1)∑ej
t2j
+
(2)∑ei,ej
(4−m)2β1(G)titj +
(3)∑ei,ej
2β1(G)+δi,j titj −(4)∑ei,ej
2β1(G)titj
.Here γi,j = 1 if, after removing the edge ei, the edge ej and the vertex s lie in oneconnected component, and γi,j = 0 otherwise. Next, δi,j = 1 if ei and ej are cyclic
edges, and δi,j = 0 otherwise. Summation∑(1)
is taken over the non-end isthmuses
ej. The sum∑(2)
is taken over all unordered pairs of edges ei, ej, such that afterremoving these two edges, the graph G = Γ\ei, ej consists of m isolated vertices and
another connected component. The sum∑(3)
is taken over all unordered pairs of edgesei, ej, such that after removing isolated vertices from the graph G = Γ \ ei, ej, we
obtain two connected components. The sum∑(4)
is taken over all unordered pairsof edges ei, ej, such that they are incident to a vertex of valence 2 (where againG = Γ \ ei, ej).
Here β1(G) is the first Betti number of the graph G.
The second term of the asymptotic expansion is connected with the graph struc-ture. The graph can be recovered uniquely if the second term is known as a functionof lengths in the case of a tree (see [3]).
This is joint work with A. A. Tolchennikov. The research was partially financiallysupported by the grant 16-11-10069 of the Russian Science Foundation.
References[1] Berkolaiko, G., Kuchment, P., Introduction to Quantum Graphs, Mathematical
Surveys and Monographs, V. 186, AMS, 2014.[2] Chernyshev, V. L., Shafarevich, A. I., “Statistics of gaussian packets on metric and
decorated graphs”, Philosophical transactions of the Royal Society A., 372:2007,2014, 20130145.
[3] Chernyshev, V. L., Tolchennikov, A. A., “Correction to the leading term of asymp-totics in the problem of counting the number of points moving on a metric tree”,Russian Journal of Mathematical Physics, 24:3, 2017, 290–298.
[4] Chernyshev, V. L., Tolchennikov, A. A., “The Second Term in the Asymptoticsfor the Number of Points Moving Along a Metric Graph”, Regular and ChaoticDynamics, 22:8, 2017, 937–948.
[5] Spencer, D. C., “The Lattice Points of Tetrahedra”, Journal of Mathematics andPhysics, 21:1-4, 1942, 189–197, doi:10.1002/sapm1942211189.
[6] Barvinok, A., Integer points in polyhedra, European Mathematical Society, 2008,199 pages.
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[7] Barnes, E. W., “On the theory of the multiple gamma function”, Trans. CambridgePhilos. Soc., 19, 1904, 374–425.
Schrodinger–Poisson system involving a measure
D. ChoudhuriNational Institute of Technology Rourkela, Rourkela, India
We discuss the following system of PDEs
(−∆)su+ u+ l(x)φu+ w(x)|u|k−1u = µ in Ω,
(−∆)sφ = l(x)u2 in Ω,
u = φ = 0 in RN \ Ω,
(1)
where (−∆)s is the fractional Laplacian operator and Ω is a bounded domain in RNwith boundary ∂Ω. We guarantee the existence and uniqueness of a solution to (1)when the nonlinearity g(u) = u+ l(x)φu+w(x)|u|k−1u in the problem (1) satisfies a“subcritical integrability condition.”
The main results of this paper is the following.
Theorem. The problem
(−∆)su+ u+ l(x)φu+ w(x)|u|k−1u = µ in Ω,
(−∆)sφ = l(x)u2 in Ω,
u = φ = 0 in RN \ Ω
admits a unique very weak solution (u, φ) corresponding to µ ∈ m(Ω, ρβ). Further,
−G[µ−] ≤ u ≤ G[µ+] a.e. in Ω
where l, w are nonnegative bounded functions on Ω, 1 < k, 0 < s < 1, Ω ⊂ RN is abounded domain, µ is a nonnegative bounded Radon measure, g(x, t) = t + l(x)φt +w(x)|t|k−1t is a continuous, nondecreasing function satisfying rg(r) ≥ 0 ∀r ∈ R, andfor each x ∈ Ω
∫∞1
(g(x, s)− g(x,−s))s−1−ks,βds <∞. Here µ−, µ+ are the positiveand negative parts of the Jordan decomposition of µ, G[.] being the Green’s operatorcorresponding to the fractional Laplacian (−∆)s. The critical exponent is defined as
k(s, β) =
N
N−2α β ∈ [0, N−2αN α),
N+αN−2α+β β ∈ (N−2α
N α, α]
for N ≥ 2, 0 < s < 1, 0 < β < α.
References[1] Chen, H., Veron, L., “Semilinear fractional elliptic equations involving measures”,
J. Differetial Equations, 257, 2014, 1457-1486.[2] Brezis, H., Operateurs maximaux monotones et semi-groupes de contractions dans
les espaces de Hilbert, North-Holland Math. Stud., Vol. 5, North-Holland, Ams-terdam, 1973, Notas de Matematica Vol. 50.
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[3] Feyel, D., de la Pradelle, A., “Topologies fines et compactifications assiciees acertains espaces de Dirichlet”, Ann. Inst. Fourier (Grenoble), 27, 1977, 121–146.
Generalized exponential type solutionsto the Kolmogorov–Feller equation: forward
and backward in time motion, vanishing diffusion
Vladimir DanilovNational Research University Higher School of Economics, Moscow, Russia
WKB solutions were popular in the theory of linear hyperbolic PDE. There wasa lot of reasons for that, one of them is the possibility of parametrix construction.
At the same time, nonoscillating WKB-type solutions in the theory of linearparabolic equations are not so popular. The direct analog between these two the-ories has been done by Yu. Kifer (for small times) and by V. P. Maslov in the large.
The last theory is spiritually similar to Maslov’s canonical operator theory anduses integral transformation with the heat kernel instead of Fourier transform inhyperbolic theory.
But some special features of parabolic equations allow to avoid any integral trans-formations. The price of that is the using of nonsmooth solutions to Hamilton-Jacobiequations (viscosity solutions) and transport equations with singular coefficients. Thelast problem relates to delta-shock solutions in the theory of hyperbolic conservationlaws, which was developed in part by V. Shelkovich and the author.
Using this approach, we can construct WKB-type solutions by means of charac-teristics only. More that, this approach can be extended to construction solutions ofa Cauchy problem backward in time.
Another application is degenerated parabolic equations. As an example, we con-sider asymptotic of the fundamental solution to the equation
ut − εx2uxx = 0,
where ε→ +0 is small parameter.
References[1] Kifer, Y., “On the asymptotics of transition density of processes with small diffu-
sion”, Theory Probab. Appl., 21, 1976, 513–522.[2] Maslov, V. P., “Global exponential asymptotic behavior of solutions of the tunnel
equations and the problem of large deviations” In: International Conference onAnalytical Methods in Number Theory and Analysis (Moscow, 1981), Proc. SteklovInst. Math., 163, MAIK Nauka, Moscow, 1985, pp. 177–209.
[3] Danilov, V. G., “Global-in-time asymptotic solutions to Kolmogorov–Feller-type parabolic pseudodifferential equations with small parameter (forward- andbackward-in-time motion)”, Abstr. Appl. Anal., 2013, 941878.
[4] Danilov, V. G., “Nonsmooth Nonoscillating Exponential-type Asymptotics forLinear Parabolic PDE”, SIAM Journal on Mathematical Analysis, 49:5, 2017,3550–3572.
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Elliptic pseudo-differential boundary valueproblems and the inverse problemof magneto-electroencephalography
A. S. DemidovLomonosov Moscow State University and Moscow Institute of Physics and Technology,
Moscow, Russia
Contrary the already prevailing for several decades opinion about the incorrect-ness of the inverse-MEEG problems (see, for example: Sheltraw, D. and Coutsias, E.(2003) Journal of Applied Physics, 94 (8), 5307-5315), the report will show that thisproblem is absolutely correct. Namely: under the condition of reconstruction electro-magnetic field according to its measurement in the final set of points xk on the headof the patient the inverse MEEG problem has, and the only solution in a special classof functions (different from those considered by biophysicists). Moreover, the oper-ator of this problem realizes an isomorphism of the corresponding function spaces.
The solution has the form q = q0 + p0δ∣∣∣∂Y
, where q0 is an ordinary function defined
in the domain of the region Y occupied by the brain, and p0δ∣∣∣∂Y
is a δ-function
on the boundary of the domain Y with a certain density p0. The functions p0 andq0 are interrelated and explicitly depend on the reconstructed electromagnetic field.Its reconstruction reduces to a finite-dimensional problem of minimizing a quadraticfunctional and revealing “essentially” various minimizing elements. The latter ques-tion echoes the analogous problem for the inverse problem of an equilibrium plasmain a tokamak [1].
This result [2] was obtained due to the fact that: 1) Maxwell’s equations are takenas a basis; 2) a transition was made to the equations for the potentials of the magneticand electric fields; 3) the theory of boundary value problems for elliptic pseudodiffer-ential operators with an entire index of factorization is used. This allowed us to findthe correct functional class of solutions of the corresponding integral equation of thefirst kind: the solution has a singular boundary layer in the form of a delta function(with some density) at the boundary of the domain.
References[1] Demidov, A. S., Savelyev, V. V., “Essentially different distributions of current in
the inverse problem for the Grad-Shafranov equation”, Russian J. Math. Ph., 17:1,2010, 56–65.
[2] Demidov, A. S., “Inverse problems in magneto-electroscaning (in encephalo-graphiy, for magnetic microscopes, etc.)”, Journal of Applied Analysis and Com-putation, 8:3, 2018, 915–927.
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Finding algebraic invariants and algebraicallyinvariant solutions
M. V. DeminaNational Research Nuclear University “MEPhI” and National Research University Higher
School of Economics, Moscow, Russian Federation
Nonlinear partial and ordinary differential equations arise in a variety of processesand phenomena in physics, biology, chemistry, economics, etc. In this talk we shalldiscuss the concept of algebraic invariants and algebraically invariant solutions forautonomous ordinary differential equations and systems of autonomous ordinary dif-ferential equations. The motivation of introducing the notion of algebraic invariantslies in the fact that a variety of known exact solutions of autonomous ordinary dif-ferential equations and a great number of traveling wave solutions of famous partialdifferential equations are in fact algebraically invariant solutions.
Our aim is to present a novel method of finding algebraic invariants in explicitform. The basic idea of the method is to use Puiseux series satisfying a non–autonomous ordinary differential equation related to the original equation [2, 3]. Agreat advantage of our approach lies in the fact that the method under considerationis able to give not some but all algebraically invariant solutions.
As an example, we shall classify all algebraic invariants and algebraically invariantsolutions for traveling wave reductions of the modified Burgers–Kolmogorov equation
ut = Duxx + βu2ux + δu(1− u2),
the Newell – Whitehead – Segel equation
ut = Duxx + βu+ δum, m ∈ N, m ≥ 2,
and some other physically relevant partial differential equations.Further, using our approach we shall solve completely the problem of Liouvil-
lian integrability for dynamical systems related to the traveling wave reductions ofaforementioned partial differential equations [1, 2, 3].
References[1] Singer, M. F., “Liouvillian first integrals of differential equations”, Trans. Amer.
Math. Soc., 333, 1992, 673–688.[2] Demina, M. V., “Invariant algebraic curves for Lienard dynamical systems revis-
ited”, Appl. Math. Lett., 84, 2018, 42–48.[3] Demina, M. V., “Novel algebraic aspects of Liouvillian integrability for two–
dimensional polynomial dynamical systems”, Phys. Lett. A, 382:20, 2018, 1353–1360.
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Lagrangian manifolds related to Bessel functions,and their applications
S. Yu. DobrokhotovIshlinsky Institute for Problems in Mechanics, Moscow, Russia and Moscow Institute of
Physics and Technology, Dolgoprudny, Moscow Oblast, Russia
V. E. NazaikinskiiIshlinsky Institute for Problems in Mechanics, Moscow, Russia and Moscow Institute of
Physics and Technology, Dolgoprudny, Moscow Oblast, Russia
We discuss the question about the relationship between the Bessel functions andLagrangian manifolds in the phase space. We present such 2-D regular and 1-Dsingular Lagrangian manifolds Λ that the Maslov canonical operator Kh
ΛA acting toappropriate functions A on Λ gives the functions expressed via the Bessel functionsof the complex arguments. Also we discuss the new integral representations of theMaslov canonical operator near the Lagrangian singularities and the Fock quantizationof canonical transforms in the problems connected with the case of singular Lagrangianmanifolds. We discuss the application of constructed asymptotic formulas in the beampropagation theory and in problems for the wave equations with degenerated velocityappearing in the water wave theory.
Singular & singularly perturbed differential systemsand their multiple resurgence
Jean EcalleParis-Sud University, Orsay, France
The following may be taken as a model of a doubly singular differential system
0 = ε t2 ∂t yi + λi y
i + bi(t, ε, y1, . . . , yν) (1 ≤ i ≤ ν) (1)
since it is not only singular (in the time variable t) but also singularly perturbed (bythe small parameter ε). Under suitable assumptions on the coefficients bi(t, ε,y), it isalso doubly resurgent, meaning that the formal solutions of (1)), whether expanded inpowers of t =: 1/z or ε =: 1/x, lead to convergent analytic germs in the correspondingBorel planes ζ or ξ, with all the trade-mark features of resurgence: endless analyticcontinuation; isolated singularities; remarkable self-reproduction properties.
t ∼ 0 ⇒ z := 1t = critical variable ⇒ ζ-Borel plane
(∑αnz
−n 7→∑αn
ζn−1
(n−1)!
)ε ∼ 0 ⇒ x := 1
ε = critical param. ⇒ ξ-Borel plane(∑
βnx−n 7→
∑βn
ξn−1
(n−1)!
)But whereas the now classical resurgence in z (equational resurgence) is well-understoodand easily described, the loosely dual but incomparably more complex resurgence inx (co-equational resurgence) was long thought to defy a unified treatment. As weshall attempt to show, such pessimism is unwarranted. Nothing stands in the way ofa general approach: one can produce a complete system of resurgence equations thatsubsume all the properties of the x-expansions.
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Two features stand out: the centrality of combinatorics to the theory, and the
crucial part played by the so-called tessellation coefficients tes(u1v1,...,,...,
urvr ). The latter
are piece-wise constant functions on C2r, paradoxically defined as superpositions ofhyperlogarithms. These coefficients possess no end of remarkable properties. Mainly,they encapsulate a sort of universal geometry that rules co-equational resurgence, andplay there roughly the same role as the Stokes constants do in equational resurgence.
Large-time decay of solutionsto the Zakharov–Kuznetsov equation
A. V. FaminskiiRUDN University, Moscow, Russia
The two-dimensional Zakharov–Kuznetsov equation
ut + ux + uxxx + uxyy + uux = 0 (1)
is a model of nonlinear waves in dispersive media propagating in one preassigned(x) direction with deformations in the transverse (y) direction. In the case of theinitial value problem, the well-known conservation law in the space L2(R2) excludesthe large-time decay of solutions in this space. From the physical point of view,initial-boundary value problems in domains where the variable y is considered on abounded interval seem to be natural. Then certain internal dissipation can ensuresuch large-time decay.
Let I1 = R, I2 = R+, I3 = (0, R), and Σj = Ij × (0, L). Consider three initial-boundary value problems, posed on the domains Σj with the initial data u0(x, y),homogeneous Dirichlet boundary conditions u|y=0 = u|y=L = 0, homogeneous Dirich-let boundary condition on the left side u|x=0 = 0 for j = 2 and j = 3, and additionalboundary conditions on the right side u|x=R = ux|x=R = 0 for j = 3.
Theorem 1. For j = 1 and j = 2 there exists an L0 > 0 such that for any L ∈ (0, L0)there exist α0 > 0, ε0 > 0, and β > 0 such that if (1+eαx)u0 ∈ L2(Σj) for α ∈ (0, α0]and ‖u0‖L2(Σj) ≤ ε0, then there exists a weak solution of the corresponding problemfor equation (1) satisfying the inequality
‖eαxu(t, ·, ·)‖L2(Σj) ≤ e−αβt‖eαxu0‖L2(Σj) ∀t ≥ 0.
Theorem 2. Let u0 ∈ L2(Σ3), and let
π2( 3
R2+
1
L2
)> 1.
Then there exist ε0 > 0 and β > 0 such that if ‖u0‖L2(Σ3) ≤ ε0, then there exists aweak solution of the corresponding problem (1) satisfying the inequality
‖u(t, ·, ·)‖L2(Σ3) ≤√
1 +Re−βt‖u0‖L2(Σ3) ∀ t ≥ 0.
In the cases of j = 2 and j = 3 these weak solutions are unique in certain classes.This work was supported by RFBR grants 17-01-00849, 17-51-52022, 18-01-00590.
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References[1] Faminskii, A. V., “An initial-boundary value problem in a strip for two-
dimensional equations of Zakharov–Kuznetsov type”, Contemp. Math., 653, 2015,137–162.
[2] Faminskii, A. V., “Initial-boundary value problems in a half-strip for two-dimensional Zakharov–Kuznetsov equation”, Ann. I. H. Poincare – AN, 35, 2018,1235–1265.
[3] Faminskii, A. V., “Initial-boundary value problems in a rectangle for two-dimensional Zakharov–Kuznetsov equation”, J. Math. Anal. Appl., 463, 2018,760–793.
Quasiclassical asymptotics for solutions todifference equations with two close turning points
A. A. FedotovSt. Petersburg State University, St. Petersburg, Russia
In the complex plane we study analytic solutions Ψ to the difference equation
Ψ(z + h) = M(z)Ψ(z), z ∈ C, (1)
where h > 0 is a parameter, and M is a given SL (2,C)-valued analytic function.We focus on the asymptotics of Ψ as h→ 0. Formally, Ψ (z+h) = exp(h d
dz )Ψ (z),and h appears in front of a derivative. So, it can be regarded as a quasiclassicalparameter.
To study quasiclassical asymptotics of analytic solutions to ordinary differentialequations on the complex plane, for example, to the Schrodinger equation
−h2ψ′′(z) + v(z)ψ(z) = Eψ(z) (2)
where v is an analytic potential, and E ∈ C is the spectral parameter, one uses thewell-known method often called the complex WKB method, see, e.g., [2, 7]. A versionof the complex WKB method for difference equations was developed in [1, 4, 5, 6].
As h→ 0 solutions to (1) have a standard simple asymptotic behavior if TrM(z) 6=±2. We call the points where TrM(z) ∈ −2, +2 turning points as they play thesame role as the turning points for the differential equation (2).
A.Fedotov and F.Klopp studied solutions to the difference Schrodinger equation
ψ(z + h) + ψ(z − h) + v(z)ψ(z) = Eψ (3)
near simple turning points (defined by the equations E − v(z) = ±2), see [3].In the framework of the complex WKB method we study solutions to (1) in the
case of two coalescent turning points. A similar analysis was carried out by V.Buslaevand A.Fedotov for the Harper equation, i.e., equation (3) with v(z) = cos(z), but thisnever was published.
Our work was supported by RFBR under the grant No. 17-01-00668-a.
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References[1] Buslaev, V., Fedotov, A., “Complex WKB method for Harper equation”, St. Pe-
tersburg Math. J., 6:3, 1995, 495–517.[2] Fedoryuk, M. V., Asymptotic Analysis. Linear Ordinary Differential Equations,
Springer-Verlag, Berlin, Heidelberg, 2009.[3] Fedotov, A., Klopp, F., “Difference equations, uniform quasiclassical asymptotics
and Airy functions”, Days on diffraction 2018, IEEE, 2018.[4] Fedotov, A., Shchetka, E., “The complex WKB method for difference equations
in bounded domains”, J. of Math. Sciences (New York), 438, 2015, 236–254.[5] Fedotov, A., Shchetka, E., “Complex WKB method for the difference Schrodinger
equation with the potential being a trigonometric polynomial”, St. PetersburgMath. J., 29, 2018, 363–381
[6] Fedotov, A., Shchetka, E., “Berry phase for difference equations”, Days on Diffrac-tion 2017, IEEE, 2017.
[7] Wasow, W., Asymptotic expansions for ordinary differential equations, Dover Pub-lications, New York, 1987.
Topological billiards and integrable Hamiltoniansystems
A. T. FomenkoLomonosov Moscow State University, Moscow, Russia
When analyzing quite a few topological invariants (so-called marked molecules)calculated so far for various integrable billiards and other integrable systems with twodegrees of freedom, the author stated a conjecture consisting of five parts A, B, C,D, and E. The talk will also describe the proof of two parts of a conjecture obtainedjointly with V. V. Vedyushkina (Fokicheva) and I. S. Kharcheva.
Consider nondegenerate (Bott) integrable systems with two degrees of freedom,their three-dimensional isoenergetic surfaces, and the corresponding Liouville folia-tions.
Conjecture A. (Atoms). Any bifurcation of two-dimensional Liouville tori inan isoenergetic 3-manifold of any integrable nondegenerate system with two degreesof freedom can be modelled by integrable billiards. In other words, any orientable3-atom can be realized as an appropriate billiard.
Conjecture B. (Rough molecules). Any so-called rough molecule can be modelledby an integrable billiard.
Conjecture C. (Marked molecules). Any marked molecules (i.e., Fomenko–Zieschang invariants) can be modelled by integrable billiards. In other words, allLiouville foliations of nondegenerate integrable systems on isoenergetic 3-surfaces areLiouville equivalent to the corresponding foliations of some billiard.
Conjecture D. Any closed three-dimensional isoenergetic surface of any inte-grable nondegenerate system with two degrees of freedom can be realized as an isoen-ergetic surface of some integrable billiard. This conjecture is a special case of Con-jecture C, and so it is true provided that Conjecture C is. Recall that, according to atheorem due to A. V. Brailov and A. T. Fomenko, the class of isoenergetic 3-manifoldscoincides with the class of graph-manifolds (Waldhausen manifolds).
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Conjecture E. According to Conjecture C, there exists a large class of Liouvillefoliations that can be realized as integrable topological billiards. Then the Fomenko–Zieschang invariant for this class is “equivalent” to the corresponding billiard. Moreprecisely, given an integrable billiard, consider it as a two-dimensional CW-complexin which the boundaries of 2-cells are formed by edges of the billiard and segments offocal lines. Then there exists a one-to-one correspondence between marked molecules(up to a fiberwise equivalence of Liouville foliations) and integrable billiards (up tocellular homeomorphisms of two-dimensional CW-complexes).
References[1] Vedyushkina, V. V., Fomenko, A. T., Kharcheva, I. S., “Modeling nondegenerate
bifurcations of closures of solutions for integrable systems with two degrees offreedom by integrable topological billiards”, Dokl. Math., 97:2, 2018, 174–176.
Asymptotic stability of an evolutionary nonlinearBoltzmann-type equation
dr hab. prof. US Henryk GackiUniversity of Silesia, Katowice, Poland
mgr Roksana BrodnickaUniversity of Silesia, Katowice, Poland
Some problems of the mathematical physics can be written as differential equationsfor functions with values in the space of measures. The vector space of signed measuresdoesn’t have good analytical properties. For example, this space with the Fortet–Mourier metric is not complete. There is the method to overcome this problem. Wemay restrict our equations to some complete convex subsets of the vector space ofmeasures. This approach seems to be quite natural and it is related to the classicalresults concerning differential equations on convex subsets of Banach spaces (see [2]).
The main purpose of our lecture is to show that the Kantorovich–Rubinsteinmaximum principle combined with the LaSalle invariance principle allow us to findnew sufficient conditions for the asymptotic stability of solutions of the followingversion of the nonlinear Boltzmann-type equation.
dψ
dt+ ψ = P ψ for t ≥ 0 (1)
with the initial conditionψ (0) = ψ0,
where ψ0 ∈ M1(R+) and ψ : R+ →Msig(R+) is an unknown function. Moreover Pis the collision operator acting on the space of probability measures. The collisionoperator P is a convex combination of N operators P1, . . . , PN , where Pk for k ≥ 2describes the simultaneous collision of k particles and P1 the influence of externalforces.
We will show that if our equation has a stationary measure µ? such that suppµ? =R+, then this measure is asymptotically stable with respect to the Kantorovich–Wasserstein metric.
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The open problem related with characterization of the stationary measure µ? ofthe equation (1) will end the talk.
References[1] Brodnicka, R., Gacki, H., “Asymptotic stability of a linear Boltzmann-type equa-
tion”, Appl. Math., 41, 2014, 323–334.[2] Crandall, M. G., “Differential equations on convex sets”, J. Math. Soc. Japan, 22,
1970, 443–455.[3] Gacki, H., “Applications of the Kantorovich–Rubinstein maximum principle in
the theory of Markov semigroups”, Dissertationes Math., 448, 2007, 1–59.[4] Lasota, A., “Asymptotic stability of some nonlinear Boltzmann-type equations”,
J. Math. Anal. Appl., 268, 2002, 291–309.[5] Lasota, A., Traple, J., “Properties of stationary solutions of a generalized Tjon–
Wu equation”, J. Math. Anal. Appl., 335, 2007, 669–682.
Hochshild’s methodfor describing the Mackenzie obstruction
to the construction of a transitive Lie algebroid
Vagif GasimovBaku State University, Baku, Azerbaijan
The Mackensie obstruction is a three dimensional class of cohomologies ([1], 7.2.12,pp. 277) whose triviality is provided by the existence and construction of the transitiveLie algebroid on the manifold M if we are given the set a data:
1. The local trivial bundle L will typical fiber isomorphic to the finite-dimensionalLie algebra g and structural group of all automorphisms Aut (g) of algebra g,denoted by LAB.
2. Coupling between the tangent bundle TM and bundle LAB in the form ofhomomorphism
Γ∞ (Ξ) : Γ∞ (TM)→ Γ∞ (Dout (L))
of the cut space as finite-dimensional Lie algebras.
Following the paper [2], we formulate the problem of description of Mackenzieobstruction to construction of transitive Lie algebroid by the set of data: 1) LABbundle L on a smooth manifold M and 2) coupling Ξ between the bundle L andtangent bundle TM .
According to Hochschild [2] we consider the set of all bundles (LAB) equippedwith the coupling Ξ with tangent bundle TM , moreover with fixed modulus ZL overthe Lie algebra Γ∞ (TM) of vector fields. In the case of Lie algebras in Hochschild’swork ([2], pp. 698) it was shown that in such a set it is possible to set up the structureof linear space, and the mapping determined by the Mackenzie obstruction is a linearmonomophism.
Our task is to take the Hochschild construction to the case of transitive Lie al-gebroids and to prove the similar theorem: the set of all Mackenzie obstructions for
32
bundles (LAB) L with coupling Ξ with one and the same center ZL as a modulesover the algebra of vector fields Γ∞ (TM) generates a linear subspace in the groupH3 (M ;ZL).
The second task is to generalize Hochschild’s theorem that states that this spacecoincides with the group H3 (M ;ZL) ([2], pp. 716, Theorem 3.1, [3], pp. 777, Theo-rems 4,5).
References[1] Mackenzie, K. C. H., General Theory of Lie Groupoids and Lie Algebroids, Cam-
bridge University Press, 2005.[2] Hochschild, G., “Lie Algebra Kernels and Cohomology”, Amer. J. Math., 76:3,
1954, 698-716.[3] Hochschild, G., “Cohomology classes of Finite Tgpe and Finite Dimensional
Kernels for Lie Algebras”, Amer. J. Math., 76:4, 1954, 763–778.
Derivation of the Benjamin-Ono equationat construction of the triple-deck structure
in problems of a fluid flow along a platewith small irregularities on the surface
Roman GaydukovNational Research University Higher School of Economics, Moscow, Russia
We consider a viscous fluid flow problem along a semi-infinite plate with small pe-riodic irregularities on the surface for large Reynolds numbers. Depending on the scaleof irregularities, the asymptotic solution of the problem under study has a double-deckor a triple-deck boundary layer structure.
In both cases of the boundary layer structure, the flow in the near-wall region isdescribed by a Prandtl-type system of boundary layer equations with induced pressure(with some differences in the boundary conditions). A numerical simulation showsthat the behavior of the flow in the near-wall region is similar in the both cases (thedifference lies in different values of the critical amplitude at which the laminar flowbecomes a vortex flow).
It is of interest to study the equations describing the velocity oscillations in theboundary layers. In the double-deck case, it is a Rayleigh-type equation which isconsidered on the semi-infinite cylinder, and the lower boundary condition is thetrace at infinity of the velocity in the near-wall region. In the triple-deck case, it isthe Laplace equation on the semi-infinite cylinder, but the lower boundary conditionis a solution of a Benjamin–Ono-type equation. However, we show that the flowsatisfies the Benjamin–Ono-type equation automatically, and the “solution” of thisequation can be determined from the trace at infinity of the velocity in the near-wallregion.
References[1] Danilov, V. G., Gaydukov, R. K., “Vortices in the Prandtl boundary layer induced
by irregularities on a plate”, Russian Journal of Mathematical Physics, 22:2, 2015,161–173.
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[2] Gaydukov, R. K., Borisov, D. I., “Existence of the Stationary Solution of aRayleigh-Type Equation”, Mathematical Notes, 99:5, 2016, 636–642.
[3] Dobrokhotov, S. Yu., Krichever, I. M., “Multi-phase solutions of the Benjamin-Ono equation and their averaging”, Mathematical Notes, 49:6, 1991, 583–594.
[4] Gaydukov, R. K., Danilov, V. G., “Equations for velocity oscillations in problemsof a fluid flow along a plate with small periodic irregularities on the surface forlarge Reynolds numbers”, Proceedings of the International Conference “Days onDiffraction 2018” (in print), 6 pp.
On existence of time-global solutionsfor parabolic equations
Yuri E. GliklikhVoronezh State University, Voronezh, Russia
The symbol E denotes the expectation and P , the probability.Consider in Rn the parabolic equation
∂
∂tu = Au (1)
where A is an autonomous strictly elliptic operator with C∞ coefficients without con-
stant term. In coordinates the operator A is represented in the form ai ∂∂xi +αij ∂2
∂xi∂xj
where ai are coordinates of some vector a and the matrix (αij) is a symmetric n× nmatrix α of a certain (2, 0)-tensor field. Since A is strictly elliptic, α is non-degenerateand we can find a unique n×n matrix A such that α = AA∗ where A∗ is the conjugatematrix to A. Thus we can construct a stochastic differential equation in Ito form
dξ(t) = a(ξ(t))dt+A(ξ(t))dw(t) (2)
for which A plays the role of generator. Since the coefficients of (2) are smooth, ithas unique at least local solutions for any initial time and value. Denote by ξ(s)the stochastic flow generated by (2) and by ξt,x(s) its orbit, i.e., the solution of (2)with initial condition ξt,x(t) = x. Consider a smooth function u0(x) on Rn. It is awell-known fact (see, e.g. [1]), that if the flow exists up to T > 0, then the functionu(t, x) = Eu0(ξt,x(T )) is the solution of (1) on [t, T ] with condition u(T, x) = u0(x).Thus, if the flow ξ(s) is complete, i.e., all the orbits exist up to +∞, we can constructthe above solutions for any T > 0.
We say that the flow ξ(s) is continuous at infinity if for any t < T , any compactK ⊂ Rn and every sequence xi →∞ the relation limxi→∞ P (ξt,xi(T ) ∈ K) = 0 holds(see [2, 3]). Note that a sufficient condition for the flow to be continuous at infinityis the fact that it satisfies C0-property.
Construct the direct product Rn+ = [0,∞) × Rn and introduce on Rn+ the flow
ξ+(s) = (s, ξ(s)). Obviously the generator of ξ+(s) has the form ∂∂t +A.
Recall that on the topological space X the function ϕ : X → R is called proper ifthe preimage of every relatively compact set in R is relatively compact in X.
Theorem. Let ξ(s) on Rn be continuous at infinity. Then ξ(s) is complete if andonly if on Rn+ there exists a smooth positive proper function ϕ such that ( ∂∂t+A)ϕ < Cfor a certain constant C > 0.
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This work was supported by RFBR grant 18-01-00048.
References[1] Gikhman, I. I., Skorokhod, A. V., Introduction to the Theory of Random Processes,
Dover Publications, Inc., Mineola, New York, USA, 1980.[2] Schwartz, L., “Processus de Markov et desingration regulieres”, Annales de
l’Institut Fourier de l’Universite de Grenoble, 27:3, 1977, 211–277.[3] Schwartz, L., “Le semigroupe d’une diffusion en liaison avec les trajectories”,
Seminair de Probabilites XXIII / Lect. Notes Math, 1372, 1989, 326–342.
Pairings for pseudodifferential symbols
Alexander GorokhovskyUniversity of Colorado, Boulder, CO, USA
Henri MoscoviciOhio State University, Columbus, OH, USA
We give a uniform construction of the higher indices of elliptic operators associ-ated to Alexander–Spanier cocycles of either parity in terms of a pairing between theK-theory and the cyclic cohomology of the algebra of complete symbols of pseudod-ifferential operators. While the formula for the lowest index of an elliptic operatorD on a closed manifold M (which coincides with its Fredholm index) reproduces theAtiyah-Singer index theorem, our formula for the highest index of D yields an ex-tension to arbitrary manifolds of any dimension of the Helton–Howe formula for thetrace of multicommutators of classical Toeplitz operators on odd-dimensional spheres.In fact, the totality of higher analytic indices for an elliptic operator D amount toa representation of the Connes-Chern character of the K-homology cycle determinedby D in terms of expressions which extrapolate the Helton–Howe formula below thedimension of M .
Calculation of these higher indices can be obtained by comparison between variouscalculations of cyclic (co)homology of pseudodifferential symbols. We will also discusshow this comparison result leads construction of invariants of the algebraic K-theoryof the algebra of pseudodifferential symbols.
References[1] Gorokhovsky, A., Moscovici, H., “Index pairing with Alexander–Spanier cocycles”,
J. Geom. Phys., 133, 2018, 195–209.
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Complex homotopy principle of Grauertand Gromov for algebras of pseudodifferential
operators
Bernhard GramschGutenberg University Mainz, Germany
Let H := H(X;M) resp. C := C(X,M) be the set of holomorphic resp. contin-uous mappings from the Stein manifold X into the complex manifold M . For eachn = 0, 1, 2, . . . the homotopy groups of H and C are isomorphic by deformations if Mis finite dimensional, homogeneous or an elliptic bundle [2, 1]. We extend this resultto the case where M is the set of Fredholm operators of kernel dimension smaller thank in some classes of pseudodifferential operators (e.g. of Hormander) with spectralinvariance. For finite dimensional matrices this is due to Gromov [1]. If the dimensionof the kernel of the Fredholm operators is precisely k this is treated in [4]. Inducedby [3] we get connections to locally bounded algebras and their countable projectivelimits.
References[1] Gromov, M., “Oka’s principle for holomorphic sections of elliptic bundles”, J.
Amer. Math. Soc., 2, 1989, 851–897.[2] Forstneric, F., Stein manifolds and holomorphic mappings, Springer, Heidelberg,
2011.[3] Albrecht, E., “Small invariant function spaces on the unit ball”, Operator Theory,
Conf. Proc. Theta Series in Adv. Math, Bucharest 2018, 1–24.[4] Gramsch, B., “Oka’s principle for special Frechet-Lie groups and homogeneous
manifolds in topological algebras of the microlocal analysis”, Proc. Banach Alge-bras, De Gruyter, Berlin, 1998, 189–204.
Eigenvalues for adiabatic problems in the presenceof conical singularities
Daniel GrieserCarl von Ossietzky Universitat Oldenburg, Germany
Richard B. MelroseMIT, Cambridge, Massachusetts, USA
The expression “adiabatic limit” refers to a family of Riemannian metrics gh ona compact fibre bundle where lengths in the fibre direction are shrunk by a factor hcompared to a fixed reference metric. An elementary example is the Euclidean metricon a 1×h rectangle. It is well-known that, under certain assumptions, the eigenvaluesλk(h) of the Laplacian for gh have a complete asymptotic expansion as h → 0. Weanalyze the asymptotic behavior of the eigenvalues in certain non-smooth settingsinvolving conical singularities, specifically to families of triangles degenerating to aninterval.
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Universal Euler characteristic of orbifolds
S. M. Gusein-ZadeLomonosov Moscow State University, Moscow, Russia
The Euler characteristic is the only additive topological invariant for spaces ofcertain sort, in particular, for manifolds with some finiteness properties. A general-ization of the notion of a manifold is the notion of a (real) orbifold called sometimesa V -manifold. We discuss a universal additive topological invariant of orbifolds: theuniversal Euler characteristic. It takes values in the ring R freely generated (as aZ-module) by the isomorphism classes of finite groups. The ring R is the polyno-mial ring in the variables corresponding to the indecomposable finite groups. We alsoconsider the universal Euler characteristic on the class of locally closed equivariantunions of cells in equivariant CW-complexes. We show that it is a universal additiveinvariant satisfying a certain “induction relation”. We give Macdonald type equa-tions for the universal Euler characteristic for orbifolds and for cell complexes of thedescribed type.
The talk is based on a joint work with I. Luengo and A. Melle-Hernandez (Com-plutense University of Madrid). The work was supported by the RSF grant 16-11-10018.
Multi-normed spaces based on non-discretemeasures, and their tensor products
A. Ya. HelemskiiMoscow State University, Moscow, Russia
It was A. Lambert who discovered a new type of structures, situated, in a sense,between normed spaces and (abstract) operator spaces. His definition was basedon the notion of amplification a normed space by means of spaces `n2 . Afterwardsseveral mathematicians investigated more general structure, “p–multi-normed space”,introduced with the help of spaces `np ; 1 ≤ p ≤ ∞. In the present talk we pass from `pto Lp(X,µ) with an arbitrary measure. This happened to be possible in the frame-work of the non-coordinate (“index-free”) approach to the notion of amplification,equivalent in the case of a discrete counting measure to the approach in mentionedarticles.
Two categories arise. One consists of amplifications by means of an arbitrarynormed space, and another one consists p–convex amplifications by means of Lp(X,µ).Each of them has its own tensor product of its objects whose existence is proved by arespective explicit construction. As a final result, we show that the “p–convex” tensorproduct has especially transparent form for the so-called minimal Lp–amplificationsof Lq–spaces, where q is the conjugate of p. Namely, tensoring Lq(Y, ν) and Lq(Z, λ),we get Lq(Y × Z, ν × λ).
37
Global bifurcations on the two sphere
Yulij IlyashenkoIndependent University of Moscow and National Research University Higher School of
Economy, Moscow, Russia
Differential equations deal with the same matters as children do: pictures in theplane. If a picture related to a differential equation remains (topologically) the sameafter the equation is slightly perturbed, this equation is structurally stable. If it is not,abrupt changes of the corresponding picture may occur under a small perturbation.These abrupt changes are the subject of the bifurcation theory. This talk manifeststhe first steps of a new born branch of the bifurcation theory: global bifurcationson the two sphere. Bifurcations in generic one-parameter families were classiffed;the answer appeared to be quite unexpected. An important and non-trivial question“who bifurcates?” was answered. In all the previous works on the planar bifurcations,the result was described by a finite number of phase portraits that may occur underthe perturbations of degenerate vector fields. In the global theory, this is no morethe case. Even three-parameter families of vector fields on the two sphere may havenumeric invariants, and six-parameter families may have functional invariants. Theseare joint results of the speaker and his collaborators: N. Goncharuk, D. Filimonov,Yu. Kudryashov, N. Solodovnikov, I. Schurov and others.
The development of the bifurcation theory will be outlined from the very begin-ning. Some open problems will be stated.
Complete semiclassical spectral asymptoticsfor periodic and almost periodic perturbations
of constant operators
Victor IvriiUniversity of Toronto, Canada
Under certain assumptions we derive a complete semiclassical asymptotics
eh,ε(x, x, λ) ∼∑n≥0
κn(x, τ, ε)h−d+n
of the spectral function for a scalar operator
A(x, hD, h) = A0(hD) + εB(x, hD, h),
where A0 is an elliptic operator and B(x, hD, h) is a periodic or almost periodicperturbation.
In particular, a complete semiclassical asymptotics of the integrated density ofstates also holds. The proof (see [1] combines the “Gauge transformation” of [2, 3]and the “hyperbolic operator method”.
Further, we consider generalizations.
38
References[1] Ivrii, V., “Complete semiclassical spectral asymptotics for periodic and almost
periodic perturbations of constant operators”, arXiv:1808.01619, 24pp.[2] Parnovski, L., Shterenberg, R., “Complete asymptotic expansion of the integrated
density of states of multidimensional almost-periodic Schrodinger operators”, Ann.of Math., Second Series, 176:2, 2012, 1039–1096.
[3] Parnovski, L., Shterenberg, R., “Complete asymptotic expansion of the spectralfunction of multidimensional almost-periodic Schrodinger operators”, Duke Math.J., 165:3, 2016, 509–561.
Phase flow over 0-singularity
V. S. KalnitskySaint-Petersburg State University, Russia
The problem under discussion arose when the author offered the construction ofone-dimensional double pendulum with special type of constraint [1]. The configura-tion space of this pendulum consists of two smooth lines in tangency. This situationmeans a geometrical uncertainty for trajectories of the motion equation. The seriesof experiments showed that there is no dynamical uncertainty [2]. Trajectories ofmotion always cross each other. However, absolutely unexpected was the fact thatthe lines has non zero curvature at the tangency point and the real trajectories onthe state-space are not C2-smooth. So the adequate mathematical model is neededto explain this phenomenon.
We offer the geometrical model of special embedding of vector bundle over singularmanifold in R3. Consider the equation in R3
y2 −(zx+ sign(x)x2ez ln |x|
)2
= 0.
1. For any section x = const its components of connectedness are homeomorphic toR. Hence, this manifold has natural structure of one-dimensional vector bundle.
2. All sections z = const 6= 0 are two transversal lines.
3. The section z = 0 is the base of bundle and is the lines in tangency of 1-order.
So, even a trajectory is smooth its projection on the base is not C2-smooth. Nowwe can apply to this geometrical model different approaches to build the differentialcalculus over the base for modeling the above mechanical system.
References[1] Burian, S. N., Kalnitsky, V. S., “On the motion of one-dimensional double pendu-
lum”, AIP Conference Proceedings, 1959, 2018, 030004.[2] Burian, S. N., “Behaviour of the pendulum with a singular configuration space”,
Vestnik SPbSU. Mathematics. Mechanics. Astronomy, 4:62, 2017, 541–551.
39
Engineering of quantum Hamiltoniansby high-frequency laser fields
M. I. KatsnelsonRadboud University, Nijmegen, Netherlands
Asymptotic methods are widely used in theory of dynamical systems with param-eters quickly oscillating in time. In such cases, an effective perturbation theory in theinverse oscillation frequencies can be developed resulting in effective static Hamilto-nians describing dynamics of slow degrees of freedom. This approach was developedfirst within classical mechanics [1, 2] and further modified to describe also quantumcase [3, 4]. Physical realizations of the latter include ultracold gases in optical lat-tices [4] and solids under the effect of strong laser fields with photon energies muchlarger than the electron bandwidth [5, 6]. I will review applications of the pertur-bation theory to various quantum many-body systems including interacting electronsand phonons in a crystal [5] and spin systems at lattices [6]. The use of the strong laserfield opens a way to manipulate very effectively the properties of physical systems, inparticular, transforming antiferromagnetic exchange interaction to the ferromagneticone, or creating exotic spin textures such as nanoscyrmion mosaics [6, 7, 8].
This work was supported by ERC Advanced Grant No. 338957 and by NWO viaSpinoza Prize.
References[1] Bogoliubov, N., Asymptotic methods in the theory of non-linear oscillators, Gor-
don and Breach, Paris, 1961.[2] Arnold, V. I., Kozlov, V. V., Neishtadt, A. I., Mathematical aspects of classical and
celestial mechanics, Springer, Berlin, 2006.[3] Itin, A. P., Neishtadt, A. I., “Effective Hamiltonians for fastly driven tight-binding
chains”, Phys. Lett. A, 378, 2014, 822–825.[4] Itin, A. P., Katsnelson, M. I., “Effective Hamiltonians for rapidly driven many-
body lattice systems: Induced exchange interactions and density-dependent hop-pings”, Phys. Rev. Lett., 115, 2015, 075301.
[5] Dutreix, C., Katsnelson, M. I., “Dynamical control of electron-phonon interactionswith high-frequency light”, Phys. Rev. B, 95, 2017, 024306.
[6] Stepanov, E. A., Dutreix, C., Katsnelson, M. I., “Dynamical and reversible controlof topological spin textures”, Phys. Rev. Lett., 118, 2017, 157201.
[7] Stepanov, E. A., Nikolaev, S. A., Dutreix, C., Katsnelson, M. I., Mazurenko, V. V.,“Heisenberg-exchange-free nanoskyrmion mosaic”, arXiv:1710.03044.
[8] Badrtdinov, D. I., Nikolaev, S. A., Rudenko, A. N., Katsnelson, M. I., Mazurenko,V. V., “Nanoskyrmion engineering with sp-electron materials: Sn monolayer onSiC(0001) surface”, arXiv:1804.00440.
40
Simple localized solutions of the wave equation
Aleksei P. KiselevSt. Petersburg Department of Steklov Mathematical Institute, St.Petersburg State
University and Institute for Problems in Mechanical Engineering RAS, St.Petersburg,Russia
A review is given of simple explicit localized solutions of the wave equation
uxx + uyy + uzz −1
c2utt = 0, c = const > 0.
Two classes of such solutions are under consideration.First, we describe analytic solutions dependent on a certain parameter and be-
coming highly localized as it tends to infinity. These are the ones based on thecomplexified Bateman theory (see, e.g., [1, 2]) as well as those associated with theso-called “complex sources” [3, 4, 5, 6].
Second, we discuss simple solutions of the homogeneous wave equation, havinga singularity at a running point. Much attention is paid to a detailed analyticalinvestigation of the solution presented by Hormander [7]. We found that it is aspecification of the classical Bateman solution [8]. Also, we are concerned with acertain specialized complexified Bateman solution, having similar properties [9].
The author was supported by RFBR grant 17-01-00535.
References[1] Perel, M. V., Kiselev, A. P., “Highly localized solutions of the wave equation”, J.
Math. Phys., 41, 2000, 1934.[2] Kiselev, A. P., “Localized light waves: Paraxial and exact solutions of the wave
equation (a review)”, Opt. Spectr., 102, 2007, 603.[3] Izmest’ev, A. A., “One parameter wave beams in free space”, Radiophys. Quant.
Electron., 13, 1970, 1062.[4] Deschamps, G. A., “Gaussian beam as a bundle of complex rays”, Electron. Lett.,
7, 1971, 684.[5] Tagirdzhanov, A. M., Blagovestchenskii, A. S., Kiselev, A. P., ““Complex source”
wavefields: sources in real space”, J. Phys. A., 44, 2011, 425203.[6] Tagirdzhanov, A. M., Kiselev, A. P., “Complexified spherical waves and their
sources. A review”, Opt. Spectr., 119, 2015, 271.[7] Hormander, L., The Analysis of Linear Partial Differential Operators I. Distribu-
tion Theory and Fourier Analysis, Springer, Berlin, 1983.[8] Bateman, H., “The conformal transformations of a space of four dimensions and
their applications to geometrical optics”, Proc. Lond. Math. Soc., 2, 1909, 70.[9] Blagovestchenskii, A. S., Kiselev, A. P., Tagirdzhanov, A. M., “Simple solutions of
the wave equation with a singularity at a running point, based on the complexifiedBateman solution”, J. Math. Sci., 224, 2017, 47.
41
On the rate of convergence as t→ +∞of the distributions of solutions to the stationarymeasure for the stochastic system of the Lorenz
model for a baroclinic atmosphere
Yu. Yu. KlevtsovaFederal State Budgetary Institution “Siberian Regional Hydrometeorological ResearchInstitute”, Siberian State University of Telecommunications and Information Sciences,
Novosibirsk, Russia
We consider the system of equations for the quasi-solenoidal Lorenz model for abaroclinic atmosphere
∂
∂tA1u+ νA2u+A3u+B(u) = g, t > 0, (1)
on the two-dimensional unit sphere S centered at the origin of the spherical polarcoordinates (λ, ϕ), λ ∈ [0, 2π), ϕ ∈
[−π2 ,
π2
], µ = sinϕ. Here ν > 0 is the kinematic
viscosity, u(t, x, ω) = (u1(t, x, ω), u2(t, x, ω))T
is an unknown vector function and
g(t, x, ω) = (g1(t, x, ω), g2(t, x, ω))T
is a given vector function, x = (λ, µ), ω ∈ Ω,(Ω, P, F ) is a complete probability space,
A1 =
(−∆ 00 −∆ + γI
), A2 =
(∆2 00 ∆2
), A3 =
(−k0∆ 2k0∆k0∆ −(2k0 + k1 + νγ)∆ + ρI
),
B(u) =(J(∆u1 + 2µ, u1) + J(∆u2, u2), J(∆u2 − γu2, u1) + J(∆u1 + 2µ, u2)
)T.
Also, γ, ρ, k0, k1 ≥ 0 are numerical parameters, I is the identity operator, J(ψ, θ) =ψλθµ − ψµθλ is the Jacobi operator and ∆ψ = ((1− µ2)ψµ)µ + (1− µ2)−1ψλλ is theLaplace-Beltrami operator on the sphere S. A random vector function g = f + ηis taken as the right-hand side of (1); here f(x) = (f1(x), f2(x))T and η(t, x, ω) =(η1(t, x, ω), η2(t, x, ω))T is a white noise in t. For the existence of a unique stationarymeasure for the Markov semigroup defined by solutions of the Cauchy problem for (1)and for the exponential convergence of the distributions of solutions to the stationarymeasure as t → +∞, in [1] and the present work we obtain sufficient conditions onthe right-hand side of (1) and the parameters ν, γ, ρ, k0, k1:
k0 < mini=1,2,...,i∗
ς(i), ς(i) =2
(j(i)− γ)2
(3νj2(i)(j(i) + γ) + χ(j(i))
+
√(3νj2(i)(j(i) + γ) + χ(j(i)))2 + (j(i)− γ)2 (ν2j3(i)(j(i) + γ) + νj(i)χ(j(i)))
),
χ(y) = (k1 + νγ)(y2 + γy) + ρ(γ + y), j(y) = y(y + 1), y > 0,
i∗ =
[c∗2ν
(√1 +
c∗ν
+ 1
)−1]> 1, c∗ =
ς(1), if γ 6= 2,
ς(2), if γ = 2,[r] - the integer part of r.
A similar result is obtained for the equation of a barotropic atmosphere and thetwo-dimensional Navier–Stokes equations. A comparative analysis with some of theavailable related results is given for the latter.
The author was supported by the Russian Foundation for Basic Research (grantnumber 14-01-31110).
42
References[1] Klevtsova, Yu. Yu., “On the rate of convergence as t→ +∞ of the distributions of
solutions to the stationary measure for the stochastic system of the Lorenz modeldescribing a baroclinic atmosphere”, Sb. Math., 208:7, 2017, 929–976.
Fractional PDEs: brief introductionand new perspectives
V. N. KolokoltsovWarwick University, UK, and SPB State University, Petersburg, Russia
We shall discuss recent achievements in the analytic and probabilistic interpreta-tion and extensions of the fractional PDEs of Caputo and Riemann–Liouville typeand on the path integral representations for their solutions arising from this point ofview. The main advantage of these path integral representations is their universalityallowing to cover a variety of different problems in a concise unified way, and thepossibility to yield solutions in a compact form that is explicitly stable with respectto the initial data and key parameters and is directly amenable to numeric schemes(Monte-Carlo simulation). Some of the author’s papers on the subject are cited.
References[1] Kolokoltsov, V., “Generalized Continuous-Time Random Walks (CTRW), Subor-
dination by Hitting Times and Fractional Dynamics”, Theory of Probability andits Applications, 53:4, 2009, 594–609.
[2] Kolokoltsov, V. N., “On fully mixed and multidimensional extensions of theCaputo and Riemann-Liouville derivatives, related Markov processes and frac-tional differential equations”, Fract. Calc. Appl. Anal., 18:4, 2015, 1039–1073,arXiv:1501.03925.
[3] Kolokoltsov, V., Veretennikova, M., “Fractional Hamilton Jacobi Bellmanequations for scaled limits of controlled Continuous Time Random Walks”,Communications in Applied and Industrial Mathematics, 6:1, 2014, e-484,http://caim.simai.eu/index.php/caim, doi:10.1685/journal.caim.484
[4] Kolokoltsov, V., Veretennikova, M., “Well-posedness and regular-ity of the Cauchy problem for nonlinear fractional in time andspace equations”, Fractional Differential Calculus, 4:1, 2014, 1–30,http://files.ele-math.com/articles/fdc-04-01.pdf.
[5] Kolokoltsov, V., “Chronological operator-valued Feynman-Kac formulae for gen-eralized fractional evolutions”, arXiv:1705.08157 (2017).
[6] Hernandez-Hernandez, M. E., Kolokoltsov, V. N., “On the solution of two-sidedfractional ordinary differential equations of Caputo type”, Fract. Calc. Appl.Anal., 19:6, 2016, 1393–1413.
43
Spectral geometryof generalized smooth distributions
Yu. A. KordyukovInstitute of Mathematics, Ufa Federal Research Centre RAS, Ufa, Russia
We will discuss the Laplacians associated with a generalized smooth distributionon a smooth manifold M . By a generalized smooth distribution, we mean a locallyfinitely generated C∞(M)-submodule D of the C∞(M)-module Xc(M) of smoothcompactly supported vector fields on M . Given a distribution D, let Dx, x ∈ M , bea linear subspace of TxM , which is the image of the evaluation map evx : D → TxM ,X 7→ X(x). A family Dx : x ∈M can be considered as a usual distribution on M ,in general, of non-constant rank. If the dimension of Dx is constant, D is a smoothsubbundle of the tangent bundle TM and D is the C∞(M)-module of smooth sectionsof this bundle: D = C∞(M,D). In this case D is projective.
The fiber of D at x ∈M is the finite dimensional linear space Dx = D/IxD, whereIx = f ∈ C∞(M) : f(x) = 0. We define a Riemannian metric on D as a family ofinner products 〈 , 〉x on Dx, depending smoothly on x ∈M in some sense. We provethat such a Riemannian metric exists for an arbitrary distribution D.
Given a smooth distribution D on a smooth manifold M , a Riemannian metricon D and a positive density µ on M , we construct the associated (horizontal) Lapla-cian ∆D as follows. First, we define the horizontal differential to be the operatordD : C∞c (M) → C∞c (M,D∗) given by dD = ev∗ d, where d : C∞c (M) → Ω1
c(M) isthe de Rham differential and ev∗ : Ω1
c(M)→ C∞c (M,D∗) is induced by the evaluationmaps evx : Dx → TxM , x ∈ M . Then ∆D is the second order differential operator∆D = d∗D dD : C∞c (M)→ C∞c (M), where d∗D : C∞c (M,D∗)→ C∞c (M) is the adjointof dD with respect to natural inner products on C∞c (M) and C∞c (M,D∗) defined bythe Riemannian metric on D and the density µ. If M is compact, the horizontalLaplacian ∆D as an unbounded operator on the Hilbert space L2(M,µ) with domainC∞(M) is essentially self-adjoint.
A distribution D is called involutive if it is closed under Lie brackets. An involutivesmooth distribution is called a singular foliation. In [1], I. Androulidakis and G.Skandalis constructed a longitudinal pseudodifferential calculus and the correspondingscale of longitudinal Sobolev spaces for an arbitrary singular foliation on a compactmanifold.
Given a smooth distribution D on a compact manifold M , consider the smallestinvolutive C∞(M)-submodule F of X (M), which contains D. It is generated by theelements of D and their iterated Lie brackets [X1, . . . , [Xk−1, Xk]] such that Xi ∈ D,i = 1, . . . , k, for every k ∈ N. Assume that F is a singular foliation (that is, itis finitely generated). We prove that the horizontal Laplacian ∆D is longitudinallyhypoelliptic in the scale of longitudinal Sobolev spaces associated with F .
If the distribution has constant rank, similar results were obtained in [2].
This is joint work with I. Androulidakis. It was supported by the Russian Foun-dation of Basic Research, grant no. 16-01-00312.
References[1] Androulidakis, I., Skandalis, G., “Pseudodifferential calculus on a singular folia-
tion”, J. Noncommut. Geom., 5, 2011, 125–152.
44
[2] Kordyukov, Y. A., “Laplacians on smooth distributions”, Mat. Sb., 208:10, 2017,91–112.
Entropy and renormalized solutions of ellipticequations with variable exponents of nonlinearities
and measure data
L. M. KozhevnikovaSterlitamak Branch of the Bashkir State University, Sterlitamak, Russia
The paper focuses on the existence of entropy and renormalized solutions to Dirich-let problems for elliptic equations with variable exponents of nonlinearities of the form
div a(x, u,∇u) = |u|p0(x)−2u+ b(x, u,∇u) + µ, x ∈ Ω ( Rn, (1)
where µ is a bounded Radon measure.
The problem in bounded domains was studied by T. Ahmedatt, M.B. Benboubker,E. Azroul, H. Chrayteh, M. El Moumni, H. Hjiaj, A. Touzani, C. Zhang, S. Zhou,and others. In the case of p(x) = p and bounded domain Ω it was proved (see [1])that µ is diffuse with p-capacity if and only if µ ∈ L1(Ω) +W−1
p′ (Ω), i.e.
µ = f − div f + f0, f ∈ L1(Ω), f = (f1, . . . , fn) ∈ (Lp′(Ω))n, f0 ∈ Lp′0(Ω).
Here we consider Eq. (1) with µ = f −div f+f0, f ∈ L1(Ω), f = (f1, . . . , fn) ∈Lp′1(·)(Ω) × · · · × Lp′n(·)(Ω), f0 ∈ Lp′0(·)(Ω). It is assumed that the vector functiona(x, s0, s) in Eq. (1) obeys the coercivity condition of the form
a(x, s0, s) · s ≥ an∑i=1
|si|pi(x) − φ(x), φ ∈ L1(Ω), (2)
for all s0 ∈ R and s ∈ Rn and for almost all x ∈ Ω.
Set a(x, s0, s) = a(x, s0, s) + f(x). Then Eq. (1) takes the form
div a(x, u,∇u) = |u|p0(x)−2u+ b(x, u,∇u) + f(x) + f0(x)
with the function a(x, s0, s) satisfying the coercivity condition of the form (2). There-fore, it suffices to consider Eq. (1) with µ = f + f0 ∈ L1(Ω) + Lp′0(·)(Ω).
We prove the existence of an entropy solution to the Dirichlet problem for Eq. (1)in an arbitrary domain Ω. Previously, this was done only for bounded domains. Inaddition, we establish that the constructed solution is a renormalized solution of theproblem.
This work was supported by RFBR grant 18-01-00428-a.
References[1] Malusa, A., Porzio, M. M., “Renormalized solutions to elliptic equations with
measure data in unbounded domains”, Nonlinear Analysis, 67, 2007, 2370–2389.
45
Some phenomena in the behavior of the eigenvaluesof fractional differentiation operators
M. V. KukushkinInternational Committee Continental, Geleznovodsk, Russia
This work is devoted to an exploration of some class of non-selfadjoint operatorsacting in a complex separable Hilbert space. We deal with the pair of complex separa-ble Hilbert spaces H,H+, with assumptions H+ →→ H. We consider a perturbationof the non-selfadjoint operator T represented by an expression W = T + A withcertain assumptions relative to so-called main part - operator T and a lower term -operator A, which is also a non-selfadjoint operator. Both of these operators act in H.We suppose that there exists a linear manifold dense in H+ on which operators T,Aare well defined with their adjoint operators. Also we suppose that T,A are strictlyaccretive. In opposite to the approach that was used in [1], where spectral propertiesof perturbation of selfadjoint and normal operators were studied, our considerationsare founded on known spectral properties of the real component of non-selfadjointoperators. Having used the technic of sesquilinear forms theory we establish a com-pactness property of the resolvent, obtain asymptotic equivalence between the realcomponent of the resolvent and the resolvent of the real component of non-selfadjointoperators. We conduct a classification of non-selfadjoint operators by belonging oftheir resolvent to Schatten-von Neumann’s class and formulate a sufficient conditionfor a completeness of the root vectors. Finally we obtain an asymptotic formula forthe eigenvalues.
As an application of obtained theoretical results, we solve the eigenvalues problemfor second order differential operators with fractional derivatives in the lower terms.This question is still relevant and many papers devoted to one, for instance the pa-pers [2, 3, 4]. More precisely in [3] a spectrum problem for a second order differentialoperator with Riemman-Liuvile’s fractional derivative in the lower terms was con-sidered, it was proved that the resolvent of this operator belongs to Hilbert-Shmidtclass. We would like to research the multidimensional case which can be reduced tothe cases considered in the works listed above. As a resume, it should be noted thatresults obtained in the theoretical part of our work alow us: to conduct a classificationof second order differential operators with fractional derivatives in the lower terms bybelonging of their resolvent to Schatten-von Neumann’s class, formulate the sufficientconditions for completeness of the root functions, to obtain the asymptotic formulafor the eigenvalues.
References[1] Shkalikov, A. A., “Perturbations of selfadjoint and normal operators with discrete
spectrum”, Russian Mathematical Surveys, 71, 2016, 113–174.[2] Nakhushev, A. M., “Sturm-Liouville problem for an ordinary differential equation
second order with the fractional derivatives in the lower terms”, Proceedings ofthe Academy of Sciences. USSR, 234, 1977, 308–311.
[3] Aleroev, T. S., “Spectral analysis of one class of non-selfadjoint operators”, Dif-ferential Equations, 20, 1984, 171–172.
[4] Aleroev, T. S., Aleroev, B. I., “On eigenfunctions and eigenvalues of one non-selfadjoint operator”, Differential Equations, 25, 1989, 1996–1997.
46
Numerical solutionof elliptic partial differential equations
Dr. Manoj KumarDepartment of Mathematics Motilal Nehru National Institute of Technology
Allahabad-211 004 (U.P.), India
The present paper deals a fifth-order Newton-type iterative method for solvingnonlinear equations. The derivative term has been removed from the proposed methodusing divided differences and it was proved that the method is fifth-order convergentin both the cases with and without derivative term. Various numerical comparisonsare made in MATLAB to demonstrate the performance of the developed methods. Fi-nally, fifth-order Newton-type iterative method has been applied to solve the nonlinearsystem of equations in finite element solution of nonlinear elliptic partial differentialequations.
The use a priori analytic information to solveboundary-value problems of diffraction theory
A. G. KyurkchanMoscow Technical University of Communications and Informatics,
Moscow, Russian Federation
The talk is devoted to some application aspects of one problem on which Prof.B. Yu. Sternin was working: the theory of analytical continuation of solutions todifferential equations of elliptic type. More precisely, we discuss methods for findingsolutions to diffraction problems based on a priori information concerning analyticalproperties of solution. These methods include the method of auxiliary sources, thepattern equations method, the null field methods, the T-matrix method, and theextended boundary condition method. We also briefly discuss the problem of therecognition of mirror-like objects.
References[1] Sternin, B., Shatalov, V., Differential Equations on Complex Manifolds, Academic
Publ., Dodrecht, Boston, 1994.[2] Kyurkchan, A. G., Sternin, B. Yu., Shatalov, V. Ye., “Singularities of continuation
of the Maxwell equations solutions”, Journal of Communications Technology andElectronics, 37:5, 1992, 675–694.
[3] Kyurkchan, A. G., Sternin, B. Yu., Shatalov, V. Ye., “The singularities of the con-tinuation of wave fields”, Phys. Usp., 39:12, 1996, 1221–1242.
[4] Kyurkchan, A. G., Smirnova, N. I., Mathematical Modeling in Diffraction TheoryBased on a Priori Information on the Analytic Properties of the Solution, Elsevier,Amsterdam, 2016.
[5] Kyurkchan, A. G., “On the method of auxiliary currents and sources in problemsof wave diffraction”, Radio Eng. Electron. Phys., 29:10–11, 1984, 2129–2139.
[6] Kyurkchan, A. G., Minaev, S. A., Soloveichik, A. L., “A modification of the methodof discrete sources based on a priori information about the singularities of thediffracted field”, Journal of Communications Technology and Electronics, 46:6,2001, 615–621.
47
[7] Kyurkchan, A. G., “A new integral equation in the diffraction theory”, SovietPhysics Doklady, 37:7, 1992, 338–340.
[8] Kyurkchan, A. G., Anyutin, A. P., “The method of extended boundary conditionsand wavelets”, Doklady Mathematics, 66:1, 2002, 132–135.
[9] Kyurkchan, A. G., Smirnova, N. I., “Generalization of the T-matrix method toproblems of wave scattering by bodies with nonanalytic boundary”, Journal ofCommunications Technology and Electronics, 62:5, 2017, 476–480.
[10] Kyurkchan, A. G., “About recognition of mirror-like objects”, Physics-Uspekhi,60:10, 2017, 1018–1024.
Strong homotopies of differential graded algebrasand theorems of Hilbert-Riemann type
J. ChuangCity University, UK
J. HolsteinCambridge University, UK
A. LazarevLancaster University, UK
The simplest version of the Riemann-Hilbert correspondence is the statement,known for many decades, that the category of flat vector bundles on a smooth man-ifold M is equivalent to the category of representations of its fundamental groupπ1(M). Recently a higher generalization of this statement was developed, [1], wherethe category of representations of π1(M) was replaced by a differential graded cat-egory of infinity local systems on M and the category of flat vector bundles by adifferential graded (dg) category of certain modules, called cohesive modules, overΩ(M), the de Rham algebra of M . The correspondence is given by a certain A∞functor.
The proof in loc.cit. is technically complicated and our original motivation wasto understand it in simple terms, particularly keeping in mind that one side of theequivalence – the category of infinity local systems – is essentially the same as themore classical notion of a cohomologically locally constant (clc) dg sheaf, i.e. a dgsheaf whose cohomology forms an ordinary (graded) locally constant sheaf. Ourapproach is based on the observation that Ω(M) is the global sections of the sheafof de Rham algebras on M and the latter is a soft resolution of the constant sheafR. Similarly, a dg module N over Ω(M) could be sheafified and viewed as a moduleover the sheaf of de Rham algebras. Imposing suitable restrictions on M , we showthat the resulting sheaf of modules is quasi-isomorphic to a clc sheaf and that thisprocedure establishes an equivalence between the derived category of clc dg sheaves onM and a suitable homotopy subcategory of dg Ω(M)-modules (such as cohesive Ω(M)-modules). Our approach also generalizes to spaces more general than manifolds, e.g.simplicial complexes. Finally, we achieve a similar result working with the singularcochain complex of a topological space or a simplicial set, with values in rings otherthan R, e.g. Z.
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This work has appeared as a preprint [2]. It was supported by the EPSRC.
References[1] Block, J., Smith, A., “The higher Riemann–Hilbert correspondence”, Adv. Math.,
252, 1963, 382–405.[2] Chuang, J, Holstein, J., Lazarev, A., “Maurer–Cartan moduli and theorems of
Riemann–Hilbert type”, arXiv:1802.02549.
The homotopy classificationof transitive Lie algebroids
Xiaoyu LiNanjing University of Posts and Telecommunications, Nanjing, China
A. S. MishchenkoMoscow State University, Moscow, Russia
Transitive Lie algebroids have specific properties that allow to look at the tran-sitive Lie algebroid as an element of the object of a homotopy functor. Roughlyspeaking each transitive Lie algebroids can be described as a vector bundle over thetangent bundle of the manifold which is endowed with additional structures. There-fore transitive Lie algebroids admits a construction of inverse image generated by asmooth mapping of smooth manifolds. Then the non-abelian extension of transitiveLie algebroids due to K.Mackenzie (2005) can be used to construct a classifying space.The intention of my talk is to show the existence and classification of coupling betweenLi algebra bundles and tangent bundles which plays a crucial role on classification oftransitive lie algebroids.
References[1] Mackenzie, K., General theory of Lie Groupoids and Lie Algebroids, Cambridge
University Press, 2005.
On classification of the second order differentialoperators and differential equations
Valentin LychaginUniversity of Tromso, Norway
We discuss a local classification of second order linear differential operators andcorresponding the differential equations. Possibly Riemann [1] was apparently thefirst who analyzed this problem and found curvature as an obstruction to transformdifferential operators of the second order to operators with constant coefficients. Indimension two Laplace [2] found “Laplace invariants” which are relative invariantsof subgroup of rescaling transformations of unknown functions, and Ovsyannikov [3]found the corresponding invariants. All invariants for hyperbolic equations in dimen-sion two with respect to pseudogroup transformations, including also diffeomorphisms
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of the base manifold, were found by Ibragimov [4]. For the case of ordinary differentialoperators, this was done by Kamran and Olver [5], and for the case of linear ordinarydifferential equations of any order relative invariants were found by Wilczynski [6].We are going to consider the problem in all dimensions. The talk is based on jointwork with Valeriy Yumaguzhin [7].
References[1] Riemann, B., Gesammelte mathematische werke und avissenschaftlicher nachlass,
XXII, pp. 357-370, Leipzig, Teubner, 1876.[2] Laplace, P. S., “Recherches sur le calcul int egral aux di erences partielles”, in:
Memoires de l’Academie royale des Sciences de Paris (1773/77), pp. 341-402;reprinted from P. S. Laplace, Oeuvres Completes, Vol. 9, Gauthier-Villars, Paris,1893.
[3] Ovsyannikov, L. V., “Group properties of the Chaplygin equation”, J. Appl. Mech.Tech. Phys., 3, 1960, 126–145.
[4] Ibragimov, N. Kh., “Invariants of hyperbolic equations: solutions of the Laplaceproblem”, Journal of Applied Mechanics and Technical Physics, 45:2, 2004, 158–166.
[5] Kamran, N., Olver, P., “Equivalence of differential operators”, SIAM J. Math.Anal., 20:5, 1989, 1172–1185.
[6] Wilczynski, E. J., Projective differential geometry of curves and ruled surfaces,Teubner, Leipzig, 1905.
[7] Lychagin, V., Yumaguzhin, V., “Classification of the second order differentialoperators and differential equations”, Geometry and Physics, 130, 2018, 213–228.
To spectral theory of Schrodinger and Diracoperators with point interactions
M. M. MalamudPeoples Friendship University of Russia (RUDN University), Moscow, Russia
Schrodinger and Dirac operators with δ-interactions are studied without any as-sumptions on the lengths of intervals. A connection between spectral properties ofthese operators and certain Jacobi matrices will be discussed. In particular, we showthat Schrodinger operators and respective Jacobi operators are lower semibounded,non-negative, discrete, etc. only simultaneously. We will also discuss types (continu-ous, absolutely continuous, singular) of spectrum of these operators. A non-relativisticlimit will be discussed too. Similar results have recently been obtained for Schrodingeroperators on quantum graphs.
The talk is based on results of the papers [1, 2, 3] and recent publication [4]. Thelater publication was prepared with the support of the “RUDN University Program5-100”.
References[1] Kostenko, A., Malamud, M., “1–D Schrodinger operators with local point inter-
actions on a discrete set”, J. Differential Equations, 249, 2010, 253–304.[2] Malamud, M. M., Neidhardt, H., “Sturm–Liouville boundary value problems with
operator potentials and unitary equivalence”, J. Differential Equations, 252, 2012,5875–5922.
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[3] Carlone, R., Malamud, M., Posilicano, A., “On the spectral theory of Gesztezy–Seba realizations of 1-D Dirac operators with point interactions on discrete set”,J. Differential Equations, 254, 2013, 3835–3902.
[4] Kostenko, A., Malamud, M., Neidhardt, H., Exner, P., “Infinite quantum graphs”,Dokl. Acad. Nauk, 472:3, 2017, 1–6.
Publishing Mathematics with Birkhauser
Dorothy MazlumEditor Mathematics Birkhauser, Basel, Switzerland
In this presentation, we will give some background information on Birkhauser,highlight its mathematics program, and explain the publishing process from submis-sion to publication. Moreover, current developments in the publishing industry, suchas Open Access, will be discussed.
Sobolev inequalities in arbitrary domains
Vlaimir Maz’yaLinkoping University
A theory of Sobolev inequalities in arbitrary open sets in Rn is presented. Bound-ary regularity of domains is replaced with information on boundary traces of trialfunctions and of their derivatives up to some explicit minimal order. The relevantSobolev inequalities involve constants independent of the geometry of the domain,and exhibit the same critical exponents as in the classical inequalities on regular do-mains. Our approach relies upon new representation formulas for Sobolev functions,and on ensuing pointwise estimates which hold in any open set. This is a joint workwith A.Cianchi.
Boundary values and problems for ellipticw-operators on compact manifolds
with fibered boundary
Gerardo A. MendozaTemple University, Philadelphia
I will present an overview of joint work with T. Krainer concerning boundaryvalues of elliptic wedge operators on compact manifolds with fibered boundary, orcomplexes of cone operators (cone operators in the second case). I will first describesome aspects concerning the former. For these we have a complete theory in the caseof first order operators which is in complete analogy with that of regular boundaryvalue problems for elliptic operators. Concerning complexes, I will state a theorem
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and illustrate some of the issues by contrasting the with the analogous problems in thecase of an elliptic cone operator. If time permits, I will also give some applications.
Klein–Gordon equation and growth of Lie algebrasof vector fields
D. V. MillionshchikovLomonosov Moscow State University and Steklov Mathematical Institute of RAS,
Moscow, Russia
The notion of characteristic Lie algebra of a hyperbolic system of PDE was intro-duced by Leznov, Shabat and Smirnov in 1982 and in the recent years characteris-tic Lie algebras of various hyperbolic systems were actively studied by Zhiber’s andHabibullin’s schools. We will study the growth of the characteristic Lie algebra forthe Klein–Gordon equation uxy = f(u).
We show that the characteristic Lie algebras of the Sinh–Gordon and Tzitzeica
equations are isomorphic to the positive parts of two Kac–Moody affine algebras A(1)1
and A(2)2 , respectively [2, 3]. Both these Lie algebras are in the classification list [1]
of narrow naturally graded Lie algebras g = ⊕+∞i=1 gi of the “width 3/2” (dim gi +
dim gi+1 ≤ 3) [1]. A Lie algebra g = ⊕+∞i=1 gi is called naturally graded if [g1, gi] =
gi+1, i ≥ 1. Shalev and Zelmanov proposed to call “narrow” those positively gradedLie algebras that have homogeneous subspaces gi at most two-dimensional. NarrowLie algebras have slow linear growth (slow growth of the dimension of the subspacespanned by all commutators of length n in generators of g).
The research was made under the support of the RSF grant No 14-11-00414.
References[1] Millionshchikov, D. V., “Narrow positively graded Lie algebras”, Doklady Mathe-
matics (to appear in 2018).[2] Millionshchikov, D. V., “Characteristic Lie algebras of Sine-Gordon and Tzitizeica
equations”, Russian Mathematical Surveys, 72:6, 2017, 1174–1176.[3] Millionshchikov, D., “Lie Algebras of Slow Growth and Klein–Gordon PDE”, Al-
gebr. Represent. Theor., 2018, 1–33, doi:10.1007/s10468-018-9794-4.
Semi-classical asymptotics for the two-dimensionalradially symmetric Dirac equation
D. S. MinenkovIshlinsky Institute for problems in mechanics, Moscow, Russia
The current study is related to the question of how much the tip of a scanningtunneling microscope affects the measured values. Microscope measures the tunnelcurrent between the tip and the crystal surface, and this current depends on the localdensity of states in the crystal. To perform the study the stationary Schrodinger andDirac equations on the plane with a radially symmetric potential U(r) are considered.
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Using semiclassical asymptotic forms for the generalized eigenfunctions of these equa-tions, the local density of states corresponding to these asymptotics is determined andanalyzed. It is shown that in the case of the Schrodinger equation the tip significantlydistorts the measured density only for small energies while in the case of the Diracequation finite distortions appears for all energies.
Derivations of group algebrasand Hochschild cohomology
A. S. MishchenkoLomonosov Moscow State University, Moscow, Russian Federation
A description of the algebra of outer derivations of a group algebra of a finitelypresented discrete group is given in terms of the Cayley complex of the groupoid ofthe adjoint action of the group. This task is a smooth version of Johnson’s prob-lem concerning the derivations of a group algebra. It is shown that the algebra ofouter derivations is isomorphic to the group of the one-dimensional cohomology withcompact supports of the Cayley complex over the field of complex numbers.
On the other hand the group of outer derivation is isomorphic to the one di-mensional Hochschild cohomology of the group algebra. Thus the whole Hochschildcohomology group can be described in terms of the cohomology of the classifying spaceof the groupoid of the adjoint action of the group under the suitable assumption ofthe finiteness of the supports of cohomology groups.
The report presents the results partly obtained jointly with A. Arutyunov, andalso with the help of A.I. Shtern.
References[1] Arutyunov, A. A., Mishchenko, A. S., “Smooth Version of Johnson’s Problem Con-
cerning Derivations of Group Algebras”, arXiv:1801.03480 [math.AT]. (Sub-mitted to Mathematical sbornik).
Completeness of perturbed systemsof trigonometric functions
in the space of Lebesgue integrable functions
E. I. MoiseevMSU by M.V.Lomonosov, Moscow, Russia
D. A. GulyaevMSU by M.V.Lomonosov, Moscow, Russia
The paper addresses the completeness of perturbed systems of sines and cosineswith non integer indices and, moreover, complex, in the space of Lebesgue integrablefunctions. The criteria for the variation from the integer value have been found suchthat sine and cosine systems are complete.
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This work was supported by a grant from the Russian Science Foundation (ProjectNo. 16-11-10194).
References[1] Zygmund, A., Trigonometric Series. Vol. 1, Cambridge University Press, 2002, .[2] Moiseev, E. I., “On the basis property of systems of sines and cosines”, Soviet
Mathematics - Doklady, 29:2, 1984, 296–300.[3] Devdariani, G. G., “The basis property of a trigonometric system of functions”,
Differentsial’nye uravneniya, 22:1, 1986, 168–170.[4] Moiseev, E. I., Gulyaev, D. A., “The completeness of the eigenfunctions of the Tri-
comi problem for the Lavrent’ev–Bitsadze equation with the Frankl gluing condi-tion”, Integral Transforms and Special Functions, 27:11, 2016, 893–898.
[5] Kolmogorov, A. N., Fomin, S. V., Elements of the theory of functions and func-tional analysis, Graylock Press, 1965.
[6] Bateman, H., Erdelyi, A., Higher transcendental functions. Vol. 2., New York,McGraw-Hill, 1953.
Reduction of computation of the signatureof a G-manifold in the simplest case
Morales Melendez, QuitzehCONACYT - Universidad Pedagogica Nacional, Oaxaca, Mexico
In [3] it was defined a C∗-algebra signature for an algebraic Poincare complex.This definition can be applied to the l2-complex of an invariantly triangulated smoothco-compact G-manifold, for a discrete group G. In [2] it is shown that this is an (al-gebraic) homotopy invariant. In [1] it was shown that this is an algebraic bordisminvariant. It will be discussed the task of reduction of the computation of this signa-ture using the Conner–Floyd fixed points construction in the simplest case of a groupG with a single normal cyclic subgroup of prime order.
References[1] Barcenas, N., Morales Melendez, Q., “Analytical signatures and proper actions”.
Preprint: arXiv:1610.02118 (2016).[2] Higson, N., Roe, J., “Mapping surgery to analysis I”, K-theory, 33:4, 2005, 277–
299.[3] Mishchenko, A. S., “The theory of almost algebraic Poincare complexes and local
combinatorial Hirzebruch formula.”, Acta Appl. Math., 68:1-3, 2001, 5–37.
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On absence of global positive solutionsof correlated-noise elliptic inequalities
(noncoercive case)
A. B. MuravnikJSC “Concern “Sozvezdie”, Voronezh, Russia, and Peoples Friendship University of
Russia, Moscow, Russia
For the inequality
∆u−n∑j=1
αj(x, u)
(∂u
∂xj
)2
≥ ω(x, u) (1)
considered in Rn, the following result is true:
Theorem 1. Suppose that αj(x, s) ≥1
s, j = 1, n. Suppose that there exist a non-
negative function k(x), a positive constant R0, and a positive function θ defined on
[R0,+∞) such that limt→∞
θ(t) = ∞,θ(t)
t2is a nonincreasing function, and the in-
equality k(x) ≥ θ(|x|)|x|2
holds outside the ball |x| < R0. Suppose that there exists a
constant p from (1,+∞) such that ω(x, s) ≥ k(x)sp. Then the inequality (1) has noglobal positive solutions.
This work was financially supported by the Ministry of Education and Science ofthe Russian Federation on the program to improve the competitiveness of Peoples’Friendship University (RUDN University) among the world’s leading research andeducation centers in the 2016–2020, by the Russian Foundation for Basic Research(grant No. 17-01-00401), and by the President Grant for the Government Support ofthe Leading Scientific Schools of the Russian Federation, No. 4479.2014.1.
On fractional Neumann Laplacians in the half-space
A. I. NazarovPDMI RAS and St. Petersburg State University, St. Petersburg, Russia
We consider different fractional Neumann Laplacians of order s ∈ (0, 1) on do-mains Ω ⊂ Rn, namely, the Restricted Neumann Laplacian (−∆NΩ)sR, the Semire-stricted Neumann Laplacian (−∆NΩ)sSr and the Spectral Neumann Laplacian (−∆NΩ)sSp.In particular, we are interested in the attainability of Sobolev constants for theseoperators when Ω is a half-space.
The talk is based on the joint work with Roberta Musina [1]. This work waspartially supported by RFBR, grant 17-01-00678.
References[1] Musina, R., Nazarov, A. I., “Sobolev inequalities for fractional Neumann Lapla-
cians on half spaces” (to appear in Advances in Calculus of Variations). Preprint:arXiv:1708.01567.
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Propagation and blowing-up of non-plane frontsin Burgers-type equations with modular advection
N. N. NefedovDepartment of Mathematics, Faculty of Physics, Lomonosov Moscow State University,
Moscow, Russia
We present recent results for initial boundary value problem for some classes ofBurgers-type equations, for which we investigate moving fronts by using the devel-oped comparison technique. These results are based on a further development of theasymptotic comparison principle (see [1] and references therein). For the consideredinitial boundary value problems the existence of moving fronts and their asymptoticapproximation were investigated. The results were illustrated by the problem
ε∂2u
∂x2−A(u, x, t)
∂u
∂x− ∂u
∂t= f(u, x, t, ε), x ∈ (0, 1), 0 < t ≤ T, (1)
u(0, t, ε) = u0(t), u(1, t, ε) = u1(t), t ∈ [0, T ],
u(x, 0, ε) = uinit(x, ε), x ∈ [0, 1].
An asymptotic approximation of solutions with a moving front for specific forms ofequation (1) in the case of modular advection term and nonlinear amplification isconstructed. The considered problem extends results of the paper [2] for non-planefronts case. The influence exerted by nonlinear amplification on front propagationand blowing-up is determined. The front localization and the blowing-up time areestimated.
This work was supported by the Russian Science Foundation (project No. 18–11–00042).
References[1] Nefedov, N. N., “Comparison principle for reaction-diffusion-advection problems
with boundary and internal layers”, Lecture Notes in Comput. Sci. (Springer-Verlag, Heidelberg), 8236, 2013, 62–72.
[2] Nefedov, N. N., Rudenko, O. V., “On front motion in a Burgers-type equationwith quadratic and modular nonlinearity and nonlinear amplification”, DokladyMathematics, 97, 2018, 99–103.
Lie algebroids and Lie algebra bundles
Nguyen Le AnhHanoi, Vietnam
A Lie algebroid is a vector bundle with a Lie structure on all vector sections asso-ciated with a bundle map called anchor to have the Leibniz identity. A transitive Liealgebroid is one in which the anchor is fiberwise surjective. Every Lie algebra bundleis a trivial Lie algebroid with a null anchor, so that the existence of a transitive Liealgebroid is important. For a Lie algebra, the problem of existence of a transitive Liealgebroid is equivalent to the Mackenzie obstruction to be trivial for the Mishchenko
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Lie algebra bundle, i.e., the Lie algebra bundle with structural group consisting of allautomorphisms for the Lie algebra with the topology that is discrete on its quotientgroup by the ad-automorphism subgroup. The Mackenzie obstruction is trivial forthe case in which the Lie algebra is the direct sum of the quotient algebra and thecenter.
Whitney-Sullivan constructions for transitive Liealgebroids: the polynomial case
A. S. MishchenkoMoscow State University, Moscow, Russia
J. R. OliveiraUniversity of Minho, Braga, Portugal
H. Whitney [3] and D. Sullivan [2] considered several variants of cochain complexesdefined over cell-like spaces by taking different notions of differential forms. Theyshowed that the cohomology of those cochains complexes is isomorphic to the classicalcohomology of the polytope underlying the cell-like space. Based in Rham–Sullivantheorem for cell spaces, it is proved that, for each transitive Lie algebroid A on atriangulated compact manifold M , the inclusion mapping from the cochain algebraof all polynomial forms on A to the cochain algebra of all piecewise smooth forms onA induces an isomorphism in cohomology.
This work was supported by MICINN, Grant MTM2014-56950-P.
References[1] Mishchenko, A. S., Oliveira, J. R., “Whitney-Sullivan constructions for transitive
Lie algebroids — smooth case” (to appear).[2] Sullivan, D., “Infinitesimal computations in topology”, Publ. I.H.E.S., 47, 1977,
269–331.[3] Whitney, H., Geometric Integration Theory, Princeton University Press, 1957.
Dense quasi-free subalgebras of the Toeplitz algebra
A. Yu. PirkovskiiHigher School of Economics, Moscow, Russia
The Toeplitz algebra (i.e., the universal C∗-algebra T generated by an isometry)has several interesting dense locally convex subalgebras, for example, the algebraicToeplitz algebra Talg and the smooth Toeplitz algebra Tsmth. Such subalgebras, aswell as the Toeplitz algebra itself, play an important role in bivariant K-theory and incyclic homology. The motivation for our talk comes from the observation (probablydue to Ralf Meyer) that Talg is quasi-free in the sense of Cuntz and Quillen. On theother hand, T is not quasi-free by a general result of Oleg Aristov. Now a naturalquestion is whether or not Tsmth is quasi-free. To answer this question, we introducea family TP,Q of dense locally convex subalgebras of T indexed by Kothe power sets
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P,Q satisfying some natural conditions. Our main result gives a sufficient condition(in terms of P and Q) for TP,Q to be quasi-free. As a corollary, we show that thesmooth Toeplitz algebra Tsmth and the holomorphic Toeplitz algebra Thol are quasi-free.
This is a part of a joint project with O. Aristov.
On the sum of narrow and compact operators
M. A. PlievSouthern Mathematical Institute of The Russian Academy of Science, Vladikavkaz,
Russia
Linear narrow operators in function spaces and vector lattices were studied bymany authors (see [5]). The notion of the narrow operator was extended to thesetting of lattice-normed spaces in [6]. It was proved in [3] that the sum of twonarrow operators in generally is not a narrow operator. Nevertheless, some particularcases were investigated in [2, 3]. The aim of this short notes is to present a new resultin this direction. For the standard information we refer to the monograph [1].
An element y of a vector lattice E is said to be a fragment of an element x ∈ E,if y ⊥ (x − y). The notation y v x means that y is a fragment of x. The set of allfragments of the element x ∈ E is denoted by Fx. Two fragments y, z of x are saidto be mutually complemented if y ⊥ z and x = y + z.
Let E be a vector lattice, and let F be a real linear space. An operator T : E → Fis said to be orthogonally additive if T (x + y) = Tx + Ty for any disjoint elementsx, y ∈ E.
Let E be an atomless vector lattice and F be a normed space. An orthogonallyadditive operator T : E → F is said to be narrow if for any ε > 0 and x ∈ E thereexist mutually complemented fragments x1, x2 of x such that ‖T (x1)− T (x2)‖ < ε.
Let E be a vector lattice. A net (vα)α∈Λ ⊂ E is said to be laterally convergent tov ∈ E if v = (o) − limα vα and |vβ − vγ | ⊥ |vγ | for all β, γ ∈ Λ, β ≥ γ. Let E be avector lattice and F be a normed space. An operator T : E → F is said to be:
• laterally-to-norm continuous provided T sends laterally convergent nets in E tonorm convergent nets in F ;
• C-compact if a set T (Fx) is relatively compact in F for every x ∈ E.
The following theorem is the main result of this notes.
Theorem 1. Let E be an atomless vector lattice, F be a Banach space, S : E → Fbe an orthogonally additive narrow operator and T : E → F be a laterally-to-normcontinuous C-compact orthogonally additive operator. Then R = S + T is a narrowoperator as well.
This work was supported by RFBR (the grant number 17-51-12064).
References[1] Aliprantis, C. D., Burkinshaw, O., Positive Operators, Springer, Dordrecht, 2006.[2] Humenchuk, H. I., “On the sum of narrow and finite-rank orthogonally additive
operator”, Ukrainian Math. J., 67:12, 2016, 1831–1837.
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[3] Mykhaylyuk, V., “On the sum of a narrow and a compact operators”, J. Funct.Anal., 266, 2014, 5912–5920.
[4] Mykhaylyuk, V., Popov, M., “On sums of narrow operators on Kothe functionspaces”, J. Math. Anal. Appl., 404, 2013, 554–561.
[5] Popov, M., Randrianantoanina, B., Narrow Operators on Function Spaces andVector Lattices, De Gruyter Studies in Mathematics 45, De Gruyter, 2013.
[6] Pliev, M., “Narrow operators on lattice-normed spaces”, Open Mathematics, 9:6,2011, 1276–1287.
Cyclic homology of crossed-product algebras
Raphael PongeSeoul National University, Seoul, South Korea
There is a great amount of work on the cyclic homology of crossed-product al-gebras, by Baum–Connes, Brylinski–Nistor, Connes, Crainic, Elliott–Natsume–Nest,Feigin–Tsygan, Getzler–Jones, Nest, Nistor, among others. However, at the excep-tion of the characteristic map of Connes from the early 80s, we don’t have explicitchain maps that produce isomorphisms at the level of homology and provide us withgeometric constructions of cyclic cycles in the case of group actions on manifolds orvarieties.
The aim of this talk is to present the construction of explicit quasi-isomorphismsfor crossed products associated with actions of discrete groups. Along the way werecover and clarify various earlier results (in the sense that we obtain explicit chainmaps that yield quasi-isomorphisms). In particular, we recover the spectral sequencesof Feigin–Tsygan and Getzler–Jones, and derive an additional spectral sequence.
In the case of group actions on manifolds we have an explicit description of cyclichomology and periodic cyclic homology. In the finite order case, the results areexpressed in terms of what we call “mixed equivariant homology”, which interpolatesgroup homology and de Rham cohomology. This is actually the natural receptaclefor a cap product of group homology with equivariant cohomology. As a result takingcap products of group cycles with equivariant characteristic classes naturally gives toa geometric construction of cyclic cycles. For the periodic cyclic homology we recoverearlier results of Connes and Brylinski–Nistor via a Poincare duality argument. Forthe non-periodic cyclic homology the results seem to be new. In the infinite ordercase, we fix and simplify the misidentification of cyclic homology by Crainic.
In the case of group actions on smooth varieties we obtain the exact analogues ofthe results for group actions on manifolds. In particular, in the special case of finitegroup actions on smooth varieties we recover recent results of Brodzki–Dave–Nistorvia the construction of an explicit quasi-isomorphism.
This work was supported by Basic Research Grant 2016R1D1A1B01015971 ofNational Research Foundation of Korea.
References[1] Ponge, R., “The cyclic homology of crossed-product algebras, I”, C. R. Acad. Sci.
Paris, ser. I, 355, 2017, 618–622.[2] Ponge, R., “The cyclic homology of crossed-product algebras, II”, C. R. Acad.
Sci. Paris, ser. I, 355, 2017, 623–627.
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[3] Ponge, R., “Cyclic homology and group actions”, J. Geom. Phys., 123, 2018,30–52.
[4] Ponge, R, “Para-S-modules and perturbation lemmas”. Preprint, arXiv, Septem-ber 2018.
[5] Ponge, R., “Eilenberg–Zilber theorem and cup product for biparacylic modules”.In preparation.
Combinatorial Ricci flow for degenerate metrics
Th. Yu. PopelenskyLomonosov Moscow State University, Moscow, Russia
In 1998 R.Hamilton published a paper in which he proved that the Ricci flow forany closed orientable surface not diffeomorphic to the sphere for any metric convergesto a metric of constant curvature. He proved the same statement for a sphere providedan initial metric has positive Gauss curvature. Few years later B.Chow proved theconvergence of the Ricci flow to a metric of constant curvature on a sphere for anyinitial metric.
In 2003 B.Chow and F.Luo introduced the combinatorial Ricci flow for triangu-lated surfaces. They gave a complete description of the asymptotic behaviour of thesolution of the combinatorial Ricci flow under certain assumption. Both the Euclideanand hyperbolic background geometry were considered. Their theory uses Thurstonscircle packing metrics on triangulated surfaces with weights on the edges of the trian-gulation. Under assumption that weights belong to [0, 1] the Chow-Luos conditionsof convergence of the Ricci flow coincide with the Thurstons condition of existence ofa constant curvature metric.
In our papers with R.Pepa we have weaken condition of non-negativity of theweights in such a way that the theorem about convergence of the combinatorial Ricciflow for any initial condition still holds. On the other hand, we found several examplesof triangulated surfaces with some negative weights such that under the Ricci flowsome initial metrics degenerate. At the same time numerical simulation shows someregularity in behaviour as time tends to infinity.
In the present talk we give a definition of degenerate circle packing metrics andprove under conditions similar to the Chow-Luo conditions the convergence of thecombinatorial Ricci flow to a constant curvature metric for any initial condition.
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Semiclassical theoryof electronic Veselago lensing in graphene
Koen ReijndersRadboud University, Institute for Molecules and Materials, Nijmegen, The Netherlands
Mikhail KatsnelsonRadboud University, Institute for Molecules and Materials, Nijmegen, The Netherlands
The motion of low-energy charge carriers in graphene is governed by the Diracequation. Electron-hole interfaces in this material are able to focus electrons andtherefore behave as electronic analogs of Veselago lenses [1]. The intensity near thefocus can be expressed in terms of the Green’s function. For quasi-one-dimensionalinterfaces, this Green’s function can be determined analytically. It is given by anintegral expression, which we study using the stationary phase approximation. Weobserve that, even for large values of the dimensionless semiclassical parameter, theleading-order approximation in terms of the Pearcey function correctly predicts theposition of the intensity maximum. We use this observation to study two sourcesof aberrations in graphene Veselago lenses. The first of these is the second-orderterm in graphene’s Hamiltonian, the so-called trigonal warping term. This termsignificantly modifies the classical trajectories [2] and destroys the ideal focus thatis present for the Dirac equation [3]. Using the stationary phase approximation, wecan however still obtain a quasi-analytical formula for the position of the intensitymaximum, which is in very good agreement with numerical results. The secondsource of aberrations is the initial sublattice polarization. This polarization leads toa sideward shift of the focus [4], which we study by considering corrections to theleading-order approximation.
References[1] Cheianov, V. V., Fal’ko, V., Altshuler, B. L., “The focusing of electron flow and a
Veselago lens in graphene p-n junctions”, Science, 315, 2007, 1252.[2] Garcia-Pomar, J. L., Cortijo, A., Nieto-Vesperinas, M., “Fully valley-polarized
electron beams in graphene”, Phys. Rev. Lett., 100, 2008, 236801.[3] Reijnders, K. J. A., Katsnelson, M. I., “Diffraction catastrophes and semiclassical
quantum mechanics for Veselago lensing in graphene”, Phys. Rev. B., 96, 2017,045305.
[4] Reijnders, K. J. A., Katsnelson, M. I., “Symmetry breaking and (pseudo)spin po-larization in Veselago lenses for massless Dirac fermions”, Phys. Rev. B., 95, 2017,115310.
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Elliptic functional differential equationwith contracted and shifted argument
Leonid RossovskiiRUDN University, Moscow, Russia
Abubakar TovsultanovChechen State University, Grozny, Russia
Let p > 1, K ⊂ Rn be a compact set, and Ω ⊂ Rn a bounded domain such thatp−1Ω−K ⊂ Ω. Cosider also any regular complex-valued Borel measure ν concentratedon K, ν ∈ (C(K))∗. By ν denote its characteristic function, ν(ξ) =
∫K
e−ihξ dν(h).
Define an operator with contracted and shifted argument as follows:
Tu(x) =
∫K
u(p−1x− h) dν(h).
Lemma 1. The operator T is a bounded linear operator in the Sobolev spaces Hs(Ω)s ∈ R with
ρ(T ) = ρ(T ;L2(Rn)) = pn/2 limm→∞
supξ∈Rn
|ν(ξ)ν(pξ) · · · ν(pm−1ξ)|1/m.
If α ∈ C with |α| < 1/ρ(T ), then the operator I + αT : Hs(Ω) → Hs(Ω) has abounded inverse for all s > 0. If, in addition, α satisfies the stronger inequality|α| < 1/(psρ(T )) for some positive s, then the operator I + αT : H−s(Ω) → H−s(Ω)is boundedly invertible as well.
We consider the boundary value problem
−∆(u(x) + αTu(x)) = f(x) (x ∈ Ω), (1)
u|∂Ω = 0, (2)
where f ∈ L2(Ω) and the (generalized) solution is understood as a function u ∈ H1(Ω)satisfying the identity∑n
j=1((u+ αTu)xj , vxj )L2(Ω) = (f, v)L2(Ω) (v ∈ H1(Ω)).
Theorem 1. If |α| < p/ρ(T ), then problem (1) and (2) has a unique solution forany function f ∈ L2(Ω). If, in addition, f ∈ Hk(Ω) and ∂Ω ∈ Ck+2 (k nonnegativeinteger), then u ∈ Hk+2(Ω).
Some specific examples are considered. It is also shown that for large α problem (1)and (2) may have infinitely many generalized solutions for any function f ∈ L2(Ω)(in the sense that these solutions form an infinite-dimensional linear manifold).
In [1], the theory of boundary value problems for elliptic differential- differenceequations is constructed. Elliptic functional differential equations with contractedand expanded argument are studied in [2]. The principal limitation of the latter workis the assumption that all the contractions have the same center, which essentiallynarrows the range of problems considered. In the present talk, we study the equationwith contractions whose centers are distributed over the domain.
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References[1] Skubachevskii, A. L., “Elliptic Functional-Differential Equations and Applica-
tions” in Vol. 91 of Oper. Theory Adv. Appl. Birkhauser Verlag, Basel, 1997.[2] Rossovskii, L. E., “Elliptic functional differential equations with contractions and
extensions of independent variables of the unknown function” in Vol. 223 ofJ. Math. Sci. Springer US, New York, 2017.
Quasi-classical quantum mapsof semi-hyperbolic type
Hanen LouatiUniversite de Tunis El-Manar, Departement de Mathematiques, 1091 Tunis, Tunisia
Michel RouleuxAix Marseille Univ, Universite de Toulon, CNRS, CPT, Marseille, France
Let M = Rn or possibly a Riemannian, non compact manifold. We considersemi-excited resonances for a h-pseudo-differential operator H(x, hDx;h) on L2(M)induced by a nondegenerate periodic orbit γE of semi-hyperbolic type, contained insome energy surface ΣE . We may think of H(x, hDx;h) as Schrodinger operator withStark effect, with M = Rn, or H(x, hDx;h) as the geodesic flow on an axially sym-metric manifold M , extending Poincare example of Lagrangian systems with 2 degreeof freedom. By semi-hyperbolic, we mean that the linearized Poincare map PE asso-ciated with γE has at least one eigenvalue of modulus greater (or less) than 1, and bynon-degenerate that 1 is not an eigenvalue. We look for semi-excited resonances nearE, i.e. with imaginary part of magnitude hs, with 0 < s < 1. It is known that theseresonances are given by the zeroes of a determinant ζ(z, h) = det(I −M(z, h)) whereM(z, h) is the monodromy operator. We make here this result more precise, in provid-ing a first order asymptotics for Bohr-Sommerfeld quantization rule associated withγE in terms of the (real) longitudinal and (complex) transverse quantum numbers, in-cluding the action integral, the sub-principal 1-form and Gelfand-Lidskii (or Cohnley–Zehnder) index. This index enters only when PE has some eigenvalue of modulus 1.In the latter case, a theorem of Lewis and Birkhoff shows that an infinite number ofperiodic orbits cluster near γE , with periods approximately multiples of the primitiveperiod of γE . Extending some result by Sjostrand and Zworski, we propose also a gen-eralized Poisson formula expressing Tr f(H/h) microlocally near γE , as a truncatedseries (depending on h) of integrals of the form Tr
∫f(z/h)M(z, h)k d
dzM(z, h)dz.
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How to hear the corners of a drum
Zhiqin LuUniversity of California, Irvine, USA
Medet NursultanovUniversity of Gothenburg, Gothenburg, Sweden
Julie RowlettChalmers University of Technology, Gothenburg, Sweden
David SherDePaul University, Chicago, IL, USA
Analytically computing the spectrum of the Laplacian is impossible for all buta handful of classical examples. Consequently, it can be tricky business to deter-mine which geometric features are spectrally determined; such features are known asgeometric spectral invariants. Weyl demonstrated in 1912 that the area of a planardomain is a geometric spectral invariant. In the 1950s, Pleijel proved that the perime-ter is also a spectral invariant. Kac, and McKean & Singer independently proved inthe 1960s that the Euler characteristic is a geometric spectral invariant for smoothlybounded domains. At the same time, Kac popularized the isospectral problem for pla-nar domains in his article, “Can one hear the shape of a drum?” Colloquially, one saysthat one can “hear” spectral invariants. Hence the title of this talk in which we willshow that the presence, or lack, of corners is spectrally determined. In the process, wewill see how a certain “corner contribution” to the heat trace is obtained by explicitlycalculating the Green’s functions for infinite sectors with Dirichlet, Neumann, Robin,and mixed boundary conditions. Moreover, using microlocal techniques, we will seethat this corner contribution is universal. Finally, we will show how the results gen-eralize to surfaces. This talk is based on current joint work with M. Nursultanov andD. Sher, and previous joint work with Z. Lu.
References[1] Lu, Z., Rowlett, J., “One can hear the corners of a drum”, Bull. London Math.
Soc., 48:1, 2016, 85–93.[2] Nursultanov, M., Rowlett, J., Sher, D., “How to hear the corners of a drum”,
Matrix Annals (published online May, 2018).
Dirichlet problem for mixed type equationwith characteristic degeneration
Yu. K. SabitovaSterlitamak Branch of Bashkir State University, Sterlitamak, Russia
Consider the elliptic-hyperbolic equation
Lu = uxx + (sgn y) |y|nuyy + a|y|n−1uy − b2u = 0 (1)
in the rectangular domain D = (x, y)| 0 < x < l,−α < y < β, where l > 0, α > 0,β > 0, 0 < a < 1, n < 1 are given positive real numbers.
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The Dirichlet problem. Find a function u(x, y) satisfying the following condi-tions:
u(x, y) ∈ C2(D+ ∪D−) ∩ C(D); (2)
limy→0+0
yauy(x, y) = limy→0−0
(−y)auy(x, y); (3)
Lu(x, y) ≡ 0, (x, y) ∈ D+ ∪D−; (4)
u(x, β) = f(x), u(x,−α) = g(x), 0 ≤ x ≤ l, (5)
u(0, y) = 0, −α ≤ y ≤ β; (6)
u(0, y) = u(l, y), −α ≤ y ≤ β, (7)
where D− = D ∩ y < 0, D+ = D ∩ y > 0, f(x) and g(x) are given sufficientlysmooth functions, f(0) = f(l) = 0, and g(0) = g(l) = 0.
In the present paper based on works [1], [2], necessary and sufficient conditions ofuniqueness of a solution to problem (2)–(7) for Eq. (1) in a rectangular domain aremore clearly identified. A solution is constructed in the form of a series with respect tothe system of eigenfunctions of the corresponding one-dimensional spectral problem.When substantiating the convergence of the series, a problem of small denominatorswith respect to 2αq/l, q = (2−n)/2, arises. In this connection, we establish estimatesabout separation of a small denominator from zero with the corresponding asymp-totics for rational and irrational values of the number 2αq/l, which permit one toestablish the convergence of the constructed series in the class of regular solutions (2)and (3).
This work was supported by RFBR grant No. 18-31-00111.
References[1] Sabitov, K. B., Syleimanova, A. H., “Dirichlet Problem for mixed type equation
with characteristic degeneration”, Izvestiya Vysshikh Uchebnykh Zavedenii. Math,No 4, 2007, 45–53.
[2] Sabitova, Yu. K., “Boundary Value problem with nonlocal integral condition formixed type equations with degeneracy on the transition line”, Math. Notes, 98:3,2015, 454–465.
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Steady-state motion of charged dust particlesunder gravitational forces and forces of inertia
T. SalnikovaLomonosov Moscow State University and RUDN University, Moscow, Russia
S. StepanovInstitution of Russian Academy of Sciences Dorodnicyn Computing Centre of RAS and
Lomonosov Moscow State University, Moscow, Russia
V. VedenyapinKeldysh Institute of Applied Mathematics (Russian Academy of Sciences) and RUDN
University, Moscow, Russia
We study the time evolution of an ensemble of charged dust particles. Theyevolve according to the Vlasov–Poisson equations. The problem we have to solvehere consists in adding to the electric field produced by the charged particles thegravitational forces and the forces of inertia. This setting emerges from the problemof elusive Kordylewski clouds - dust particles in vicinity of the Lagrange librationpoints of the Earth-Moon system [1]. The mathematical approach is based on thereduction of the stationary Vlasov equation by means of energy substitution, andfollowing analysis of the system of non-linear elliptic equations [2].
References[1] Salnikova, T., Stepanov, S., “Effect of electromagnetic field on Kordylewski clouds
formation”, AIP Conference Proceedings, 1959, 2018, 020004-1-020004-6.[2] Vedenyapin, V., Negmatov, M., Fimin, N., “Vlasov-type and Liouville-type equa-
tions, their microscopic, energetic and hydrodynamical consequences”, Izvestiya:Mathematics, 81:3, 2017, 505–541.
Analytic and algebraic indices of elliptic operatorsassociated with quantized canonical transformations
Anton SavinRUDN University, Moscow, Russia
Elmar SchroheLeibniz University Hannover, Hannover, Germany
Boris SterninRUDN University, Moscow, Russia
We consider elliptic operators associated with discrete groups of quantized canon-ical transformations as defined in [1] In order to be able to apply results from alge-braic index theory, we define the localized algebraic index of the complete symbol ofan elliptic operator. With the help of a calculus of semiclassical quantized canonicaltransformations, a version of Egorov’s theorem and a theorem on trace asymptoticsfor semiclassical Fourier integral operators we show that the localized analytic index
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and the localized algebraic index coincide. As a corollary, we express the Fredholmindex in terms of the algebraic index for a wide class of groups, in particular, for finiteextensions of Abelian groups.
References[1] Savin, A., Schrohe, E., Sternin, B., “Elliptic Operators Associated with Groups
of Quantized Canonical Transformations”, arXiv:1612.02981. To apear in BullSci. Math.
Some applications of asymptotic solutionsof Boussinesq type equations for the approximationof 2011 tsunami mareograms from DART stations
S. Ya. Sekerzh–Zen’kovichIshlinsky Institute for Problems in Mechanics RAS, Moscow, Russia
A. A. TolchennikovIshlinsky Institute for Problems in Mechanics RAS, Moscow, Russia
In article [1] an asymptotic solution of the linearized Boussinesq equation (from thetheory of gravitational water waves) with a localized initial condition was obtained.In article [2], the accuracy of this asymptotic solution was estimated. We apply theseformulas to calculate the waves of the Japanese tsunami in 2011 and compare themwith the real records of the deep–water stations DART. This work was supported bythe grant of the RNF 16-11-10282.
References[1] Dobrokhotov, S. Yu., Sergeev, S. A., Tirozzi, B., “Asymptotic solutions of the
Cauchy problem with localized initial conditions for linearized two-dimensionalBoussinesq-type equations with variable coefficients”, Russian Journal of Mathe-matical Physics, 20:2, 2013, 155–171.
[2] Sekerzh-Zen’kovich, S. Ya., “Estimation of accuracy of an asymptotic solution ofthe generalized Cauchy problem for the Boussinesq equation as applied to thepotential model of tsunami with a “simple” source”, Russian Journal of Mathe-matical Physics, 24:4, 2017, 535–543.
Quantum Hall effect and noncommutative geometry
Armen SergeevSteklov Mathematical Institute, Moscow, Russia
We present an interpretation of quantum Hall effect in terms of noncommutativegeometry. The essence of this effect is that for very low temperatures in the graphof dependence of Hall conductivity on filling factor there appear horizontal plateau,corresponding to integer (in appropriate units) values of the conductivity. In other
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words, the Hall conductivity for low temperatures is “quantized.” Using noncommu-tative geometry ideas, it is possible to construct a cyclic cocycle which integralityimplies the quantization of Hall conductivity.
Asymptoticsof the solution explicit difference scheme
for the wave equation with localized initial data
S. A. SergeevIshlinsky Institute for problems in mechanics and Moscow Institute of Physics and
Technology, Moscow, Russia
We consider the Cauchy problem for the wave equation with localized data
utt = c2(x)uxx, u|t=0 = V (x/µ), ut|t=0 = 0, x ∈ R,
where V (y) is a smooth fast-decaying function with derivations and the parameterµ 1 is the parameter of the localization, function c(x) is smooth and bounded. Theasymptotic solution of this problem can be constructed with the help of the Maslov’scanonical operator.
On the other hand the solution of this problem can be obtained via the numericalmethods due to solving the difference scheme. The difference scheme can be writtenas a pseudo-differential operator due to shifting operator Tu = eh
∂∂xu, where h 1
is a step of discretization in the scheme.Thus the difference scheme can be also studied with the help of the Maslov’s theory
and the asymptotic solution of the difference scheme can be studied. We investigatethe asymptotic solution of the explicit difference scheme. We compare the numericalsolution and asymptotic solution of the difference scheme. Even the explicit schemeis unstable one can provide interesting results of such comparison depending the ratiobetween the step of the difference scheme h and parameter of localization µ.
This work was supported by the Russian Science Foundation (project 16-11-10282).
References[1] Dobrokhotov, S. Yu., Nazakinskii, V. E., “Punctured Lagrangian manifolds and
asymptotic solutions of the linear water wave equations with localized initial con-ditions”, Mathematical Notes, 101:5-6, 2017, 1053–1060.
[2] Danilov, V. G., Maslov, V. P., “Pontryagin’s duality principle for calculation of aneffect of Cherenkov’s type in crystals and difference schemes. I”, Proceedings ofthe Steklov Institute of Mathematics, 166, 1986, 143–177.
[3] Danilov, V. G., Maslov, V. P., “Pontryagin’s duality principle for calculation of aneffect of Cherenkov’s type in crystals and difference schemes. II”, Proceedings ofthe Steklov Institute of Mathematics, 167, 1986, 103–116.
[4] Danilov, V. G., Zhevandrov, P. N., “On Maslov’s method for constructing com-bined asymptotics for h-pseudodifferential equations”, Mathematics of the USSR-Izvestiya, 34:2, 1990, 425–439.
[5] Maslov, V. P., Operator Methods (Operatornye metody) in Russian. Nauka,Moscow, 1973.
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Laplacians and wave equationson two-dimensional polyhedra
A. I. ShafarevichLomonosov Moscow State University, Moscow, Russia
We study Laplacians and wave equations on 2D polyhedra. Laplacians are definedby setting self-adjoint boundary condition in vertices. We study kernels of theseoperators and prove that they can be expressed in terms of the Mittag-Leffler problemon the corresponding Riemann surface. We also obtain trace formulas for the heatkernel. For the wave equation, we describe certain exact formulas for the solutionof the Cauchy problem in terms of special integral transform as well as asymptoticsupport of localized asymptotic solutions. We also discuss statistics of localized waves.
Sobolev multipliers and their applicationsto differential and integral operators
Tatyana ShaposhnikovaRoyal Institute of Technology, Stockholm, Sweden
The talk is a survey of the theory of pointwise multipliers in spaces of differen-tiable functions which was developed together with Vladimir Mazya and summarisedin our book “Theory of Sobolev multipliers and their applications to differential andintegral operators”, Springer, 2009. In particular, I shall discuss explicit characteri-zations of multipliers in pairs of Sobolev spaces, Bessel potential spaces, and Besovspaces. Sharp results on essential norms and compactness properties of the multi-pliers, as well as trace, extension and composition theorems for the multipliers willbe presented. I shall also give various applications of our theory to differential andintegral operators, addressing such topics as regularity of the boundary for ellipticboundary value problems and continuity of general differential operators in pairs ofSobolev spaces.
Bruhat order and the symmetric Toda flowon real Lie groups
Georgy SharyginMoscow State University, Moscow, Russia
My talk will describe a next step of our joint investigation with Yu. Chernyakovand A. Sorin, see [1, 2, 3]. It is dedicated to the generalizations of the theory of Todaflow, relating it with the theory of Lie groups and algebras. Namely, I will describethe way this system can be defined for arbitrary real form of a semisimple Lie algebra,so that the role of orthogonal group will fall on the maximal compact subgroup K ofthe corresponding real group G.
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The dynamical system that I will talk about “lives” on the real flag space G/K ofthe real group; it has been studied in papers [4, 5, 6] and many others. In particular,it was shown there that it is equal to the gradient flow of a Morse function (in genericcase). Its constant points correspond to the elements of Weil group W of G andwe shall show that (for normal real forms) trajectories of the system are “governed”by the Bruhat order on W ; an interesting consequence of this fact is that the realBruhat cells intersect (transversally) the dual cells if and only if the correspondingelements in W are comparable in Bruhat order. So far this fact has been establishedonly in complex case; and in the case G = SLn(R) we used this property (whichcan be proved by the virtue of matrix representation) to describe the trajectories ofToda flow. There are many evidences that a similar property should hold for non-split real forms as well, although the size of the intersections should also depend onthe dimensions of eigenspaces. Besides this, if time permits, I will also describe theinvariants of this system, which will depend on the structure of the representationsof G.
References[1] Chernyakov, Yu. B., Sharygin, G. I., Sorin, A. S., “Bruhat Order in Full Symmetric
Toda System”, Commun. Math. Phys, 330, 2014, 367399, arXiv:1212.4803.[2] Chernyakov, Yu. B., Sharygin, G. I., Sorin, A. S., “Phase portraits of the general-
ized full symmetric Toda systems on rank 2 groups”, Theor. Math. Phys., 193:2,2017, 15741592, arXiv:1212.4803.
[3] Chernyakov, Yu. B., Sharygin, G. I., Sorin, A. S., “Bruhat order in the Toda systemon so(2, 4): an example of non-split real form”, arXiv:1712.0913.
[4] De Mari, F., Pedroni, M., “Toda flows and real Hessenberg manifolds”, J. Geom.Anal., 9:4, 1999, 607–625.
[5] Bloch, A. M., Gekhtman, M., “Hamiltonian and gradient structures in the Todaflows”, J. Geom. Phys., 27, 1998, 230–248.
[6] Faybusovich, L., “Toda flows and isospectral manifolds”, Proceedings of the Amer-ican Mathematicals Society, 115:3, July 1992.
On perturbations of self-adjoint and normaloperators: analytical aspects
Andrey A. ShkalikovDepartment of Mechanics and Mathematics, Lomonosov Moscow State University,
Moscow, Russia
We shall talk about spectral properties of operators of the form A = T +B, whereB is a non-symmetric operator subordinated to a self-adjoint or normal operator T .An operator B is said to be T − p-subordinated (0 6 p < 1) if the domain of Bcontains the domain of T and
‖Bx‖ 6 b‖Tx‖p ‖x‖1−p ∀ x ∈ D(T ) ⊂ D(B), b = const.
There is a great number of results (in particular, by J. Birkhoff, T. Carleman,M. V. Keldysh, F .Brauder, S. Agmon, V. B. Lidskii, I. Ts. Gohberg and M. G. Krein,
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F. S. Markus, V. I. Matsaev, B.S.Mityagin) which say about completeness and ba-sis property of the eigenfunctions and preserving of the eigenvalue asymptotics ofself-adjoint operators under p-subordinated perturbations.
We introduce new concepts of local subordination and local subordination in thesense of quadratic forms and prove new theorems which involve new technique andcan be applied for concrete problems in more general situation.
In the second part we will discuss problems on perturbations of a self-adjointoperator T with continuous spectrum whose spectrum consists of infinitely manysegments [σk, σk+1]∞k=1 separated by gaps: dist(σk, σk+1]) > const.
The work is supported by the Russian Science Foundation, grant No 17-11-01215.
Study of some general classes of estimatorsfor estimating population mean
in compromised imputation
B. K. SinghDepartment of Mathematics, North Eastern Regional Institute of Science and
Technology, India
U. GogoiDepartment of Mathematics, North Eastern Regional Institute of Science and
Technology, India
In this paper, three generalized classes of estimators in compromised imputationhave been suggested and their corresponding point estimators of the population meanare obtained. Their biases, mean square error (MSE) expressions and percentage rel-ative efficiency (PRE) are obtained and their optimum estimators are compared withthe sample mean, ratio and compromised method of imputation and necessary con-ditions are derived. Further, theoretical results have been verified through empiricalstudies with the help of some natural population data sets to compare their efficien-cies.
Stationary and nonstationary solutions of mixedproblems for Vlasov–Poisson equations
A. L. SkubachevskiiRUDN University, Moscow, Russia
A mathematical model of high temperature rarefied plasma in thermonuclear fu-sion reactor is described by the Vlasov-Poisson equation. If considerable number ofcharged particles reach the wall of the vacuum chamber of the reactor, then eitherthe reactor wall will be destroyed, or the high-temperature plasma will be cooled.We consider a two-component rarefied plasma with external magnetic field in infinitecylinder. To obtain plasma confinement an external magnetic field is used, i.e. we
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are looking for solutions with supports, which are situated at some distance fromthe cylindrical boundary. It were constructed stationary solutions with this property.It is also proved that for sufficiently strong magnetic field and sufficiently small ini-tial density distribution functions the Vlasov-Poisson system has a unique classicalsolution with support being at some distance from the boundary [1, 2].
The publication was prepared with the support of the “RUDN University Program5-100” and the Russian Foundation for Basic Research, grant number 17-01-00401.
References[1] Skubachevskii, A. L., “Vlasov–Poisson equations for a two-component plasma in
a homogeneous magnetic field”, Russian Math. Surveys., 69:2, 2014, 291–330.[2] Skubachevskii, A. L., Tsuzuki, Y., “Classical Solutions of the Vlasov-Poisson
Equations with External Magnetic Field in a Half-Space”, Comp.Math. andMath.Phys., 57:3, 2017, 541–557.
Locally bounded finally precontinuousfinite-dimensional quasirepresentationsof connected locally compact groups
A. I. ShternLomonosov Moscow State University, Moscow, Russia
We describe the structure of finite-dimensional locally bounded finally precontin-uous quasirepresentations of arbitrary connected locally compact groups.
References[1] Shtern, A. I., “Locally bounded finally precontinuous finite-dimensional quasirep-
resentations of connected locally compact groups”, Sb.: Math, 208:10, 2017, 1557–1576.
Lower bounds of Lipschitz constants on foliations
Guangxiang SuNankai University, Tianjin, P. R. China
In this talk, we will discuss Llarull’s theorem in the foliation case and a lowerbound of the Lipschitz constant of the map M → Sn in the foliation case under thespin condition.
References[1] Llarull, M., “Sharp estimates and the Dirac operator”, Math. Ann., 310, 1998,
55–71.[2] Su, G., “Lower bounds of Lipschitz constants on foliations”. Preprint:
arXiv:1801.06967.
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Approximate solution of fractional order partialdifferential equations and its coupled systems
using operational matrices approach
I. TalibVirtual University of Pakistan, Lahore, Pakistan
In this article, an accurate numerical method based on operational matrices offractional order derivatives and integrals in the Caputo and Riemann-Liouville sensesof two-parametric orthogonal shifted Jacobi polynomials is proposed for studying theapproximate solutions of FOPDEs. The technique is extended herein to generalizedclasses of fractional order coupled systems having mixed partial derivatives terms.One salient aspect of this article is the development of a new operational matrixfor mixed partial derivatives in the sense of Caputo. Furthermore, as a result of thecomparative study, some results presented in the literature are extended and improvedin the investigation herein.
References[1] Oldham, K. M., Spanier, J., The Fractional Calculus, Academic Press, New York,
1974.[2] Doha, E. H., Bhrawy, A. H., Ezz-Eldien, S. S., “A new Jacobi Operational matrix:
an application for solving fractional differential equations”, Applied MathematicalModeling, 36:10, 2012, 4931–4943.
[3] Khalil, H., Khan, R. A., “A new method based on Legendre polynomials for solu-tion of system of fractional order partial differential equations”, Int. J. Comput.Math., 91:12, 2014, 2554–2567.
[4] Talib, I., Belgacem, F. B. M., Asif, N. A., Khalil, A., “On mixed deriva-tives type high dimensional multi-term fractional partial differential equationsapproximate solutions”, American Institute of Physics, 1798, 020024, 2017,doi:10.1063/1.4972616.
[5] Saadatmandi, A., “Bernstein operational matrix of fractional derivatives and itsapplications”, Appl. Math. Model., 38, 2014, 1365–1372.
[6] Bhrawy, A. H., Alofi, A. S., “The operational matrix of fractional integration forshifted Chebyshev polynomials”, Appl. Math. Lett., 26, 2013, 25–31.
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Solution of the equations describing the plasmain a gas discharge
Brunello TirozziFSN Department, Create Consortium, Physics Department, Rome, Italy
Paolo BurattiFSN Department, CR-Enea, Frascati, Italy
Franco AlladioFSN Department, CR-Enea, Frascati, Italy
Paolo MicozziFSN Department, CR-Enea, Frascati, Italy
The Protosphera project [1] is aimed at producing hot toroidal plasma around acenterpost plasma discharge. Toroidal magnetic field is produced by current drivenby a DC voltage between electrodes placed at the edges of the centerpost discharge.This work presents a preliminary analysis of momentum balance in the centerpostplasma. Plasma density and magnetic field data indicate that a multifluid descrip-tion of the plasma can be applied, but Larmor gyration, electron-ion collisions andplasma-neutrals collisions have to be kept into account as a whole. A set of fluid equa-tions sufficiently general to provide a realistic description is presented. Solutions insimplified geometry are given, which provide insight into the dominant mechanisms.We found reasonable hypothesis in order to deal with the complicated non-linearmechanisms acting in the system like electron-ion collisions and plasma-neutrals col-lisions.
References[1] Alladio, F., Micozzi, P., Tirozzi, B., Buratti, P., et al., “PROTO-SPHERA, with-
out toroidal magnets, produces and confines plasma tori inside magnetostaticfields”, submitted to Nature Communications, NCOMMS-18-27536.
Asymptotics of the solution of the Klein–Gordonequation with localized initial conditions
A. A. TolchennikovIshlinsky Institute for Problems in Mechanics RAS, Moscow, Russia
The talk will be devoted to the Klein–Gordon equation with localized initial con-ditions. We will give the asymptotics of the solution and simplify the asymptoticformulas for various relations between the mass parameter and the width of the ini-tial condition. It will be shown that for some simple initial condition the asymptoticsof the solution is expressed in terms of the Hankel function of the first kind. Thiswork was supported by the grant of the RNF 16-11-10282.
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Pseudo-differential operators, equations, and ellipticboundary value problems
V. B. VasilyevBelgorod National Research University, Belgorod, Russia
We study Fredholm properties for special types of pseudo-differential operatorswhich are constructed by their local representatives, so-called operators of local type [1,4].
Let M be a smooth m-dimensional manifold with boundary ∂M which has somesingularities Mk, k = 0, 1, . . . ,m− 2. There are some smooth compact sub-manifoldsMk of dimension 0 ≤ k ≤ m − 2 on the boundary ∂M of the manifold M whichare singularities of the boundary. These singularities are described by local repre-sentatives of the operator A in a point x ∈ M in the chart U 3 x in the followingway
u(x) 7−→∫D
∫Rm
eiξ·(x−y)A(ψ−1(x), ξ)u(y)dξdy, x ∈ D,
where A(x, ξ) is an elliptic symbol defined on a special bundle over the manifoldM , ψ : U → D is a diffeomorphism, and the canonical domain D has a special formdepending on a location of the point x on manifold M .
Such an operator A will be considered in Sobolev–Slobodetskii spaces Hs(M), andlocal variants of such spaces will be spaces Hs(D) where D is one of the followingcanonical domains Rm,Rm+ = x ∈ Rm : x = (x′, xm), xm > 0,W k = Rk × Cm−k,and Cm−k is a convex cone in Rm−k non-including a whole straight line. For eachpoint x ∈ ∂M we define and assume the existence of special factorization for theelliptic symbol A(x, ξ) with index æk(x), k = 0, 1, . . . ,m − 1 [3, 5], and æm−1(x) isan index of factorization in Eskin’s sense [2] for smoothness points x ∈ ∂M .
Theorem. If
|æk(x)− s| < 1/2, ∀x ∈Mk, k = 0, 1, . . . ,m− 1,
then the operator A : Hs(M)→ Hs−α(M) is a Fredholm operator.
Remark. If an ellipticity property does not hold on sub-manifolds Mk then one canconsider a modification of the operator A [3, 5].
This work was supported by the State contract of the Russian Ministry of Educa-tion and Science (contract No 1.7311.2017/8.9).
References[1] Simonenko, I. B., A local method in the theory of translation invariant operators
and their envelopes, CVVR, Rostov on Don, 2007 (Russian).[2] Eskin, G., G. Boundary value problems for elliptic pseudodifferential equations,
AMS, Providence, 1981.[3] Vasilyev, V. B., Wave factorization of elliptic symbols: theory and applications.
Introduction to the theory of boundary value problems in non-smooth domains,Kluwer Academic Publishers, Dordrecht-Boston-London, 2000.
[4] Vasil’ev, V. B., “Regularization of multidimensional singular integral equations innon-smooth domains”, Trans. Moscow Math. Soc., 59, 1998, 65–93.
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[5] Vasilyev, V. B., “Model elliptic boundary-value problems for pseudodifferentialoperators in canonical non-smooth domains”, J. Math. Sci., 234, 2018, 397–406.
The non-trivial Liouville foliationof a three-dimensional torus
V. V. VedyushkinaLomonosov Moscow State University, Moscow, Russia
Consider a part of the plane bounded by arcs of confocal quadrics. Such billiard isintegrable. More precisely, the straight lines containing the segments of the polygonalbilliard trajectory are tangent to a certain quadric (ellipse or hyperbola). This quadricbelongs to the same class of confocal quadrics as the quadrics whose arcs form theboundary of the domain (billiard). Such billiard we call elementary billiard.
Let B0 be an elementary billiard which has an empty intersection with the focalline and bounded by two arcs of hyperbolas (convex and nonconvex) and two arcs ofellipses. We glue a number of such billiards into a two-dimensional torus ∆T (B0).
Distinguish on it a convex parallel and a meridian — a union of convex gluingedges lying on elliptic and hyperbolic segments, respectively. These two parallels andtwo meridians break the ∆T (B0) torus into four leaf regions. We denote them byα, β, γ and δ. At the same time, the pairs of sheets α, γ and β, δ are glued alonghyperbolic borders, denoted by a (convex) and b (nonconvex). Pairs of sheets α, βand γ, δ are glued along elliptic borders, denoted by c (convex) and d (non-convex).
Let n and k be mutually simple positive integers, and k < n. Let σ denotethe permutation (1 2 . . . n). Consider n copies of billiards ∆T (B0). We numberthe billiards with numbers from 1 to n and continue the numbering on their sheetsα, β, γ, δ. On the union of these billiards, we define the following motion. Onthe convex elliptic boundary a, the material point, when moving along the sheet αi(corresponding to βi), after impact hits the sheet γσ(i) (corresponding to δσ(i)), andwhen moving on the sheet γi (corresponding to δi), after the impact, moves to thesheet ασ−1(i) (corresponding to βσ−1(i)). On the convex hyperbolic boundary of c, thematerial point when moving along the sheet αi (respectively γi) after impact goes tothe sheet βσk(i) (respectively δσk(i)), and when moving on the sheet βi (respectively δi)after the impact, it moves to the sheet ασ−k(i) (respectively γσ−k(i)). On nonconvexboundaries b and c, the motion is defined without changing billiard ∆T (B0). Wedenote this billiard (billiard book) by T (∆(B0), n, k).
Theorem. The Fomenko–Zieshang invariant classifying the Liouville foliation of theQ3 isoenergy surface for an integrable topological billiard ∆T (B0) has the form shownin the figure 1a), topological billiard books ∆T ((B0), n, k) has the form shown in thefigure 1b).
This work was supported by the program “Leading Scientific Schools” (grant no.NSh-6399.2018.1).
References[1] Bolsinov, A. V., Fomenko, A. T., Integrable Hamiltonian Systems. Geometry,
Topology, Classification, Chapman & Hall/CRC, Boca Raton, 2004.
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n=-1 n=-1
n=-1n=-1
r=k/n
r=k/n
r=-k/n
r=-k/n
=1
=1
=1
=1
W (f)2 W (g)2
W (f)2W (g)2
n=0 n=0
n=0 n=0
r=0 =1
=1
=1
=1
W (f)2 W (g)2
W (f)2W (g)2
r=0 r=0
r=0
а) б)
Figure 1: Fomenko–Zieschang invariants of a series of topological billiards ∆T (B0)(a) and topological billiard books (b).
[2] Kozlov, V. V., Treshchev, D. V., Billiards: A genetic introduction to the dynamicsof systems with impacts, Translations of Mathematical Monographs, vol. 89, Amer.Math. Soc., 1991.
[3] Dragovic, V., Radnovic, M., Poncelet Porisms and Beyond, Springer Birkhauser,Basel, 2011.
[4] Fokicheva, V. V., “A topological classification of billiards in locally planar domainsbounded by arcs of confocal quadrics”, Sb. Math., 206:10, 2015, 1463–1507.
The resolvent expansion of geometric operatorson stratified spaces
Boris VertmanUniversity of Oldenburg, Germany
In this overview talk I will present a “direct” approach to the resolvent expansionof Laplace type operators on stratified spaces with iterated cone edge singularities.The paradigm is that on the model singularity one has a second order Sturm-Liouvilleoperator with operator valued coefficients. This point of view has the advantage ofbeing very elementary, microlocal analysis is only needed in the smooth interior. Thisis an joint work with Luiz Hartmann and Matthias Lesch.
Asymptotic of tunneling for Schrodinger equationwith hyperbolic frequency resonance
E. V. VybornyiNational Research University “Higher School of Economics”, Moscow, Russia
M. V. KarasevNational Research University “Higher School of Economics”, Moscow, Russia
We consider the spectral problem for a two-dimensional Schrodinger operator Hwith hyperbolic resonance. Let a Wick symbol H = H(z∗|z) be given by a sum ofhomogeneous functions in (z, z)-coordinates
H =1
~(H0 +H1 +H2 + . . . ), degHj = j + 2, N ≥ 2,
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where the harmonic summand H0 is just the hyperbolic oscillator
H0(z|z) = ω+z+z+ − ω−z−z−.
Under assumption that normal frequencies ω± > 0 of the harmonic part H0 are inresonance ω+ : ω− = k+ : k−, we study the spectral problem for the operator Hin the semiclassical approximation ~ → 0. The resonance leads to the appearanceof a non-Lie algebra of symmetries (operators commuting with H0) with polynomialcommutation relations.
Applying the operator averaging to the an harmonic term H1 + H2 + . . ., we ob-tain the effective operator E defined on irreducible representations of the symmetryalgebra. We show that a bounded eigenstate of E can be localized in different regionsof the classical phase space. In this case, we obtain an asymptotic for the correspond-ing tunnel energy splitting. This energy splitting is similar to the well-known tunnelsplitting that appears in case of one-dimensional Schrodinger operator H = p2 +V (x)with a double-well potential V (x). We also express the asymptotic of slitting in termsof the tunneling action and complex instantiations.
The details can be found in [1]. This work was supported by the Program forFundamental Research of Higher School of Economics.
References[1] Karasev, M., Novikova, E., Vybornyi, E., “Instantons via breaking geometric
symmetry in hyperbolic traps”, Mathematical Notes, 102:6, 2017, 776–786.
Fried conjecture for Morse–Smale flow
Jianqing YuUniversity of Science and Technology of China, Hefei, China
The relation between the spectrum of the Laplacian and the dynamical flow ona closed Riemannian manifold is one of the central themes in differential geometry.Fried [3] conjectured a relation between the analytic torsion, which is an alternatingproduct of regularized determinants of the Hodge Laplacians, and the Ruelle dynam-ical zeta function. We will formulate and show this conjecture for Morse-Smale flow.Our proof relies on the Cheeger-Muller [2, 4]/Bismut-Zhang [1] theorem. This is jointwork with Shu Shen.
This work was supported by NSFC (No. 11771411).
References[1] Bismut, J.-M., Zhang, W., An extension of a theorem by Cheeger and Muller,
Asterisque, vol. 205, 1992, pp. 235.[2] Cheeger, J., “Analytic torsion and the heat equation”, Ann. of Math. (2), 109:2,
1979, 259–322.[3] Fried, D., “Lefschetz formulas for flows”, The Lefschetz centennial conference,
Part III (Mexico City, 1984), Contemp. Math., vol. 58, Amer. Math. Soc., Provi-dence, RI, 1987, pp. 19–69.
[4] Muller, W., “Analytic torsion and R-torsion of Riemannian manifolds”, Adv. inMath., 28:3, 1978, 233–305.
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Positive scalar curvature on foliations
Weiping ZhangChern Institute of Mathematics, Nankai University, China
The classical Lichnerowicz theorem states that the Hirzebruch A-roof genus ofa compact spin manifold vanishes if the underlying manifold carries a Riemannianmetric of positive scalar curvature. We will describe various generalizations of thisresult to the case of foliations.
The Maslov index in symplectic Banach spaces
Bernhelm Booss-BavnbekRoskilde University, Roskilde, Denmark
Chaofeng ZhuNankai University, Tianjin, P. R. China
We consider a curve of Fredholm pairs of Lagrangian subspaces in a fixed Banachspace with continuously varying weak symplectic structures. Assuming vanishingindex, we obtain intrinsically a continuously varying splitting of the total Banachspace into pairs of symplectic subspaces. Using such decompositions we define theMaslov index of the curve by symplectic reduction to the classical finite-dimensionalcase. We prove the transitivity of repeated symplectic reductions and obtain theinvariance of the Maslov index under symplectic reduction, while recovering all thestandard properties of the Maslov index. As an application, we consider curves ofelliptic operators which have varying principal symbol, varying maximal domain andare not necessarily of Dirac type. For this class of operator curves, we derive adesuspension spectral flow formula for varying well-posed boundary conditions onmanifolds with boundary and obtain the splitting formula of the spectral flow onpartitioned manifolds.
References[1] Booß-Bavnbek, B., Zhu, C., “The Maslov index in symplectic Banach spaces”,
Mem. Amer. Math. Soc., 252, doi:10.1090/memo/1201.
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Научное издание
МЕЖДУНАРОДНАЯ КОНФЕРЕНЦИЯ ПО ДИФФЕРЕНЦИАЛЬНЫМ УРАВНЕНИЯМ
С ЧАСТНЫМИ ПРОИЗВОДНЫМИ И ПРИЛОЖЕНИЯМ,
ПОСВЯЩЁННАЯ ПАМЯТИ ПРОФЕССОРА Б.Ю. СТЕРНИНА
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