variational and topological methods: theory, applications...
TRANSCRIPT
Variational and Topological Methods: Theory,Applications, Numerical Simulations, and Open
Problems
Northern Arizona UniversityFlagsta!, Arizona, USA
June 6–9, 2012
Contents
Plenary speakers 1Monica Clapp: Elliptic boundary value problems with critical
and supercritical nonlinearities. Part I. . . . . . . . . . . 2Monica Clapp: Elliptic boundary value problems with critical
and supercritical nonlinearities. Part 2. . . . . . . . . . . 3Pavel Drabek:Fredholm Alternative for the p-Laplacian: Bifur-
cation Approach (1st plenary lecture) . . . . . . . . . . . 4Pavel Drabek: Fredholm Alternative for the p-Laplacian: Varia-
tional Approach (2nd plenary lecture) . . . . . . . . . . . 4Jacques Giacomoni: (Some) Basic methods for solving quasilinear
parabolic equations (an introductory lecture for graduatestudents and postdocs) . . . . . . . . . . . . . . . . . . . . 5
Jacques Giacomoni: An Overview of (recent) results about quasi-linear singular elliptic and parabolic problems (advancedlecture) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
Jean-Michel Rakotoson: Linear elliptic equation with singulardata: the role of New Hardy inequalities, rearrangement,and the behaviour of the solution. (Part I, II, III). . . . . 7
Peter Takac: I. (Simple) Basic Variational and Topological Meth-ods (an introductory lecture for graduate students andpostdocs) . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Peter Takac: II. Regular and Singular Systems with the p- andq-Laplacians (an advanced lecture, suitable also for grad-uate students and postdocs) . . . . . . . . . . . . . . . . . 9
Sessions 10T.V. Anoop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Vieri Benci . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Jirı Benedikt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Alfonso Castro . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12Houssam Chrayteh . . . . . . . . . . . . . . . . . . . . . . . . . . 12Jorge Cossio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12David G. Costa . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13Ionut Danaila . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
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Ann Derlet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14Pavel Drabek . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14Carsten Erdmann . . . . . . . . . . . . . . . . . . . . . . . . . . . 14Djairo Guedes de Figueiredo . . . . . . . . . . . . . . . . . . . . . 15Jacques Giacomoni . . . . . . . . . . . . . . . . . . . . . . . . . . 15Francois Genoud . . . . . . . . . . . . . . . . . . . . . . . . . . . 16Francois Genoud . . . . . . . . . . . . . . . . . . . . . . . . . . . 16Jean-Pierre Gossez . . . . . . . . . . . . . . . . . . . . . . . . . . 17Jerome Goddard II . . . . . . . . . . . . . . . . . . . . . . . . . . 17Christopher Grumiau . . . . . . . . . . . . . . . . . . . . . . . . . 18David Hartenstine . . . . . . . . . . . . . . . . . . . . . . . . . . 18Jesus Hernandez . . . . . . . . . . . . . . . . . . . . . . . . . . . 18Arturo Hidalgo . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19Jay Hineman . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19Gabriela Holubova . . . . . . . . . . . . . . . . . . . . . . . . . . 20Haydi Israel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20Huiqiang Jiang . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20Parimah Kazemi . . . . . . . . . . . . . . . . . . . . . . . . . . . 21Ian Knowles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21Sheldon Lee . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22Olimpio Hiroshi Miyagaki . . . . . . . . . . . . . . . . . . . . . . 22Nsoki Mavinga . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22Jochen Merker . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23Anna Maria Micheletti . . . . . . . . . . . . . . . . . . . . . . . . 23J. W. (Monty) Montgomery . . . . . . . . . . . . . . . . . . . . . 24Petr Necesal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24John W. Neuberger . . . . . . . . . . . . . . . . . . . . . . . . . . 24Rosa Pardo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25Vıctor Padron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25Bartosz Protas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26Dhanya Rajendran . . . . . . . . . . . . . . . . . . . . . . . . . . 27Robert Renka . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27Walter Richardson . . . . . . . . . . . . . . . . . . . . . . . . . . 28Stephen Robinson . . . . . . . . . . . . . . . . . . . . . . . . . . 28Bryan Rynne . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29Claudio Saccon . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29Dora Salazar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30Paul Sauvy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30Ra!aella Servadei . . . . . . . . . . . . . . . . . . . . . . . . . . . 31Ian Schindler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32Ratnasingham Shivaji . . . . . . . . . . . . . . . . . . . . . . . . 32Daryl Je!rey Springer . . . . . . . . . . . . . . . . . . . . . . . . 33Jim Swift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33Peter Takac: III. Special Solution Methods for Singular Sys-
tems(an advanced lecture) . . . . . . . . . . . . . . . . . . 34Lourdes Tello . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
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Sweta Tiwarii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35Christophe Troestler . . . . . . . . . . . . . . . . . . . . . . . . . 35
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Plenary speakers
Elliptic boundary value problems with critical and supercritical non-linearities. Part I.Monica ClappUniversidad Nacional Autonoma de Mexico, [email protected]
We consider problem
!"u = |u|p!2 u in #, u = 0 on !#,
where # is a bounded smooth domain in RN , N " 3, and p " 2" where 2" :=2N
N!2 is the critical Sobolev exponent.In the first lecture we present a detailed discussion of the critical case p = 2" :
its relevance, its lack of compactness and its geometric properties, and we givean overview of the classical existence and nonexistence results.
We also present some recent multiplicity results for domains which are ob-tained as small perturbations of a given domain, and discuss the techniqueswhich have been used so far to obtain multiplicity, particularly the symmetrymethod, which will also be applied in the second lecture.
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Elliptic boundary value problems with critical and supercritical non-linearities. Part II.Monica ClappUniversidad Nacional Autonoma de Mexico, [email protected]
We consider problem
!"u = |u|p!2 u in #, u = 0 on !#,
where # is a bounded smooth domain in RN , N " 3, and p " 2" where 2" :=2N
N!2 is the critical Sobolev exponent.In the second lecture we present a recent multiplicity result for the critical
exponent p = 2" in domains with nontrivial topology that have only finitesymmetries. This result provides, for each dimension N , many new examplesof domains in which the problem has a prescribed number of solutions: one ofthem positive and the rest sign changing. It is joint work with J. Faya.
We also present some recent nonexistence results in the supercritical casep > 2". Bahri and Coron showed that if # has nontrivial homology the problemhas a positive solution for p = 2". But this is not enough to guarantee existencein the supercritical case: for p " 2(N!1)
N!3 Passaseo exhibited domains carryingone nontrivial homology class in which no nontrivial solution exists. Here wegive examples of domains whose homology becomes richer as p increases in whichno nontrivial solution exists.
Finally, we present some multiplicity results for p > 2" obtained via varia-tional methods and the Hopf fibrations.
The nonexistence and multiplicity results for p > 2" were obtained in colab-oration with J. Faya and A. Pistoia.
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Fredholm alternative for the p-Laplacian: Bifurcation approach. PartI.Pavel DrabekUniversity of West Bohemia in [email protected]: Petr Girg, Peter Takac, Michael Ulm
In this lecture we study the homogeneous p-Laplacian equation in higherdimension. These results generalize well-known Fredholm alternative for thelinear Laplace equation subject to homogeneous Dirichlet boundary conditionsnear the principal eigenvalue. We explain the bifurcation approach and illustratesome existence and multiplicity results. We point out the link between threedi!erent cases: 1 < p < 2, p = 2 and p > 2.
Fredholm alternative for the p-Laplacian: Variational approach. PartII.Pavel DrabekUniversity of West Bohemia in [email protected]: Gabriela Holubova, Peter Takac
In this lecture we discuss the Fredholm alternative for the p-Laplacian nearthe principal eigenvalue from the variational point of view. We study the geome-try of energy functional associated with the boundary value problem in questionand point out some questions connected with the Palais-Smale (P.-S. for short)condition. We sketch the proofs of the existence of critical points without P.-S.condition in hands.
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(Some) Basic methods for solving quasilinear parabolic equations (anintroductory lecture for graduate students and postdocs)Jacques GiacomoniLab. of Math and Appl. UMR-CNRS 5142, BP 1155 64013 Pau Cedex [email protected]
In this lecture, we introduce di!erent methods for proving existence andeventually uniqueness (by weak comparison principle) to quasilinear parabolicequations involving p-Laplace operator (that is !"pu = !# · (|#u|p!2#u),with 1 < p < $). We adress a particular attention to methods using themonotonicity of the operator as the Theory of m-accretive operators in BanachSpaces and the Rothe’s Method (semi-discretization in time). We also presentthe ”Ladisenskaya’ type approximations method based on a regularization of thesecond-order quasilinear elliptic operator. We apply these di!erent methods onthe following problems:
1. Prototype equations as
(Pt)!
ut !"pu = f(x, u) in (0, T ) % #,u = 0 on (0, T )% !#, u(0, x) = u0(x) in #,
with suitable assumptions on f .
2. Barenblatt-p(x)-equations of the type:
(P) :
"#
$
f(., !tu) !"pu = g in Q = (0, T )% #,u = 0 on $ =]0, T [%%,u(0, .) = u0 in #,
where "pudef=
%di=1 !xi
&|!xiu|
pi(x)!2 !xiu'
is the anisotropic p(x)-Laplaceoperator.
3. Quasilinear parabolic equations with singular absorption as:
(S)!
ut !"pu = !f(t)"{u>0}u!! in Q = #% (0, T ],
u = 0 on $T = !#% [0, T ], u(x, 0) = u0(x) in #,
with # & (0, 1), u0 & L#(#)'W 1,p0 (#) and f a continuous non increasing
positive function.
In the above equations, # is an open bounded domain with smooth boundary (C2,") in RN.
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An Overview of (recent) results about quasilinear singular elliptic andparabolic problems (advanced lecture)Jacques GiacomoniLab. of Math and Appl. UMR-CNRS 5142, BP 1155 64013 Pau Cedex [email protected]
We will be mainly concerned by the following quasilinear and singular parabolicproblem
(Pt)
"#
$ut !"pu = 1
u! + f(x, u) in QTdef= (0, T ) % #
u = 0 on $T = [0, T ]% !#, u > 0 in QT
u(0, x) = u0(x) in #
and the corresponding stationary problem with 1 < p < $, # a bounded domainwith smooth boundary and suitable assumptions on $, f and u0. We will discussthe following issues:
1. Existence and uniqueness / multiplicity of weak and very weak solutions;
2. Regularity of weak solutions in respect to $; existence of mild and strongsolutions;
3. Global behaviour of global weak solutions and stabilization.
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Linear elliptic equation with singular data: the role of New Hardyinequalities, rearrangement, and the behavior of the solution. (PartI, II, III).Jean-Michel RakotosonUniversite de Poitiers, [email protected]
The main purpose of my talks is the study of(!"u = f in #,
u = 0 on !#,(1)
I will show how the tools developed in functional analysis as Hardy inequalities,rearrangement, will help for obtaining properties on u.
So Part I, (dedicated to a large public) is devoted to define the three notionsof solution widely used for (1):
1. weak solution,
2. strong solution,
3. very weak solution.
We will then prove how to obtain weak and strong solutions. Namely in 1D, weshall attempt to give explicit computations to illustrate some general results.Then, to derive some properties on the weak solutions, we shall introduce thenotion of monotone and relative rearrangement, and basic properties relatedLorentz spaces.
The second part (Part II) is devoted essentially to the existence, uniquenessand regularity of the very weak solution for (1). Namely, the regularity resultwill be proved in Lorentz spaces and partly based on a classical Hardy inequality.Some examples of application should be given
Part III advanced talk. In this talk, we shall be interested in the blow-upphenomena related to the very weak solution associated to (1). For doing this,we shall introduce new Hardy inequalities namely in Hardy space or Zygmumdspaces. We shall use them to prove some blow-up results.
All of those discussions depends on membership of f , we shall end the talkwith a blow-up result proved in a non constructive way.
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(Simple) Basic variational and topological methods (an introductorylecture for graduate students and postdocs). Part I.Peter TakacUniversitaet Rostock, Rostock, [email protected]
We begin with the formulation of the Dirichlet boundary value problem thatinvolves a (p ! 1)-homogeneous quasilinear elliptic operator:
! div(a(x, u,#u)) + b(x, u(x),#u(x)) = 0 in #; u = 0 on !# . (2)
Here, the vector field ! () a(x, u, !) : RN ) RN behaves like |!|p!2! for |!| ) 0and |!| ) $. This problem is considered in a bounded domain # * RN witha C2-boundary !#. The canonical example of a(x, u, !) = |!|p!2! yields thep-Laplacian which will be given special attention and it will be used quite oftenin order to keep the notation and many results and explanations more tractable.The corresponding energy functional (to the Dirichlet p-Laplacian)
J#(u) =1p
)
!|#u|p dx ! %
p
)
!|u|p dx !
)
!f(x)u dx (3)
on the Sobolev space W 1,p0 (#) will play a decisive role in our variational approach
to the Dirichlet problem
! div(|#u|p!2#u) = % |u|p!2u + f(x) in #; u = 0 on !# . (4)
Here, f & L#(#) is a given function and % & R is the spectral parameter.In our topological methods for problem (4) we will discuss and employ the
strong maximum and comparison principles for the p-Laplacian. The strongcomparison principle, in a desired generality, is still an open problem. Never-theless, we will be able to formulate a few “su&ciently” general versions suitablefor our applications.
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Regular and singular systems with the p- and q-Laplacians (an ad-vanced lecture, suitable also for graduate students and postdocs).Part IIPeter TakacUniversitaet Rostock, Rostock, [email protected]
We treat systems of two power-coupled equations with the p- and q-Laplacians:(!"pu = u±a1v±b1 in #; u|$!= 0, u > 0 in #,
!"qv = v±a2u±b2 in #; v|$! = 0, v > 0 in #,(5)
with the sign “+” (“!”, respectively) corresponding to the regular (singular)system. We begin by comparing such systems to a single equation through theirhomogeneity properties. We reformulate the problem of a (principal) eigenvaluefor a single equation into a problem of a (principal) eigencurve for a system oftwo coupled equations. The first (principal) eigencurve will be obtained from anonlinear version of the Kreın-Rutman theorem, for the homogeneous regularcase
(p ! 1 ! a1)(q ! 1 ! a2) ! b1b2 = 0 . (6)
For the subhomogeneous regular case
(p ! 1 ! a1)(q ! 1 ! a2) ! b1b2 > 0 , (7)
we present a solution method by Schauder’s fixed point theorem using an or-derded pair of sub- and supersolutions. Also the singular case will be treatedby similar methods.
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Sessions
A Note on generalized Hardy-Sobolev inequalitiesT.V. AnoopIndian Institute of Science, Bangalore, [email protected]
In this talk we are primarily concerned with finding a class of weight func-tions g so that the following generalized Hardy-Sobolev inequality holds:
)
!gu2 + C
)
!|#u|2 u & H1
0 (#),
for some C > 0; where is a bounded domain in R2: By making use of Mucken-houpt condition for the one dimensional weighted Hardy inequalities we identifya rearrangement invariant Banach function space so that the above integral in-equality holds for all weight functions in it. Some related results and possiblegeneralizations will also be discussed.
Ultrafunctions and generalized solutionsVieri BenciUniversity of Pise, [email protected]
The theory of distribution provides generalized solutions for problems whichdo not have a classical solutions. However, there are problems which do nothave solutions, not even in the space of distributions. As model problem youmay think of
!"u = up!1, u > 0, p " ((2N)/(N ! 2))
with Dirichlet boundary conditions in a bounded open set. Using this problemas model, we construct a function space where to look for generalized solutions.The functions in this space are called ”ultrafunctions”.
Proof by interval arithmeticJirı BenediktZapadoceska univerzita v Plzni, Czech [email protected]: P. Drabek, P. Girg, P. Takac
The purpose of the talk is to show that special self-validated numericalcomputations (interval arithmetic) can be regarded as a rigorous mathemati-cal proof. We focus on proofs of joint results with P. Drabek, P. Girg and P.Takac.
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A saddle point principle and applicationsAlfonso CastroHarvey Mudd College, California, [email protected]
For for functionals that are bounded above by a number & in a closed sub-space X and that are bounded from below by a number # > & on the boundaryof set that is linked with X it is proven that they have either a critical pointor almost critical points. The projection of the gradient of the functionals re-stricted to the orthogonal complement of X is assumed to be of Fredholm type.Applications are given to the study of the range of a wave operator with non-monotone nonlinearity.
Qualitative properties of eigenvectors related to multivoque operatorsHoussam ChraytehPoitiers University, [email protected]
We introduce the notion of '-multivoque Leray-Lions operators that arestrongly monotonic on a Banach-Sobolev function space V and we study thegeneralized eigenvalue problem Au = %!j(u). Here !j denotes the subdi!eren-tial in the sense of convex analysis or more generally in the sense of H. Clarke.Connected with this problem, we also study a minimization problem with con-straint then we give some qualitative properties of solutions by using relativerearragement theory.
Existence of multiple solutions for a nonlinear Dirichlet problem viabifurcationJorge CossioUniversidad Nacional de Colombia, [email protected]: Sigifredo Herron and Carlos Velez, Universidad Nacional de Colom-bia.
In this talk we study the existence of multiple solutions for the nonlinearelliptic boundary value problem
!"u + f(u) = 0 in #,
u = 0 on !#,(8)
where # * RN , N " 2, is a bounded and smooth domain, and f : R ) R is aLipschitz function. First we study (8) when the nonlinearity is asymptoticallylinear and then when it has an arbitrary behavior for large values of the argu-ment. Our proofs use extensively the global bifurcation theorem and bifurcation
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from infinity. Additionally, when we can apply the Lyapunov-Schmidt reductionmethod, we show the existence of multiple solutions, we give an exact numberof solutions, and we provide qualitative properties of these solutions.
Homoclinic solutions for a class of singular problemsDavid G. CostaUniversity of Nevada Las Vegas, [email protected]: Hossein Tehrani
We show existence of infinitely many homoclinic orbits at the origin for aclass of singular second-order Hamiltonian systems. We use variational methodsunder the assumption that the potential satisfies the so-called ”Strong-Force”condition.
E!ective gradient methods for the computation of vortex states inBose-Einstein condensatesIonut DanailaUniversity of Rouen, [email protected]: Parimah Kazemi
We compute vortex states of a rotating Bose-Einstein condensate by directminimization of the Gross-Pitaevskii energy functional. We extensively comparedi!erent versions of the Newton method with improved steepest descent meth-ods based on Sobolev gradients. Advantages and drawbacks of each methodare summarized. We also show that a high spatial accuracy scheme is compul-sory to accurately capture configurations with quantized vortices. We presentnumerical setups using 6th order finite di!erence schemes and finite elementswith mesh adaptivity that were successfully used to compute a rich variety ofdi&cult cases with quantized vortices.
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Ground state solutions for a nonlinear problem involving mean cur-vature operators in Minkowski spaceAnn DerletCeReMath, Universite Toulouse 1, Toulouse, [email protected]: Denis Bonheure
We investigate the existence of solutions of a prescribed mean curvatureequation in Minkowski space. We obtain a positive radial solution by minimizingthe energy functional associated to our problem over some suitable subset. Ascaling argument enables us to find an estimate on the gradient of minimizers,and to derive the Euler-Lagrange equation.
The second eigenfunction of the p-Laplacian on the disc is not radialPavel DrabekUniversity of West Bohemia in Pilsen, Czech [email protected]: Jirı Benedikt, Petr Girg
In this talk we report on the joint result of the speaker, J. Benedikt and P.Girg, where we prove that the second eigenfunction of the p-Laplacian, p > 1,on the disc is not radial. Our computer-assisted proof is a combination ofasymptotic analysis for p approaching both infinity and 1 and the applicationof the interval arithmetic.
A nonlinear Black-Scholes Equation - Existence, uniqueness and im-plementationCarsten ErdmannUniversity of [email protected]
We consider a Black-Scholes equation, which arises in nonlinear option pric-ing models in combination with the jump approach, in particular, we consider
Vt !12
(2
(1 ! %(t, S)SVSS)2S2VSS ! (r ! q)SVS + rV !
!)
R/0)(dy) [V (t, Sey) ! V (t, S) ! S(ey ! 1)VS(t, S)] = 0
(t, S) & (0, T ]% (0,$)
where ) is a Levy measure. We show existence and uniqueness of a classi-cal solution by using the fixpoint theorem of Banach and results about linear
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parabolic integro di!erential equations. Furthermore, we construct a finite dif-ference scheme for this kind of problem and prove its convergence.
Sharp pointwise estimates for functions in the Sobolev spaces of radialfunctions defined in a ballDjairo Guedes de FigueiredoUniversidade Estadual de Campinas, [email protected]
We prove sharp pointwise estimates for functions in the Sobolev spaces ofradial functions defined in a ball. As a consequence, we obtain some imbed-dings of such Sobolev spaces in weighted Lq-spaces. We also prove similarimbeddings for Sobolev spaces of functions with partial symmetry We applythese imbeddings to obtain radial solutions and partially symmetric solutionsfor a biharmonic equation of the Hnon type under both Dirichlet and Navierboundary conditions.
Existence of singular solutions to semilinear critical problems in R2
Jacques GiacomoniLab. of Math and Appl. UMR-CNRS 5142, BP 1155 64013 Pau Cedex [email protected]
We study the existence of singular solutions for the following semilinearproblem involving nonlinearities with super exponential growth:
(P#)!
!"u = %(h(u)eu") in #,
u > 0 in #, u|$! = 0.
where # * R2 is a bounded domain with smooth boundary, % > 0, 1 + & + 2and h is a smooth ”perturbation” of et"
as t ) +$. Next, we focus on the crit-ical case, i.e. & = 2 and discuss the properties of singular solutions (regularity,Morse index). We exploit these results to prove the existence of many turningpoints in the continua of solutions to the related bifurcation problem.
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Global bifurcation and stability for the asymptotically linear NLSFrancois GenoudHeriot-Watt University, Edinburgh, [email protected]
For the stationary asymptotically linear NLS, I will show how global asymp-totic bifurcation of positive solutions can be obtained by topological arguments,using recent degree theory. In the one-dimensional case, the set of positive so-lutions is actually a smooth curve, bifurcating from the line of trivial solutionsat one end, and from infinity at the other end. I will show that the standingwaves of the time-dependent NLS are orbitally stable along this global curve.
Bifurcation along curves for the p-Laplacian in the unit ballFrancois GenoudHeriot-Watt University, Edinburgh, [email protected]
I will give a complete description of the set of solutions of the problem!!(rN!1*p(u$))$ = %rN!1f(r, u), 0 < r < 1,
u$(0) = u(1) = 0,
where *p(+) := |+|p!2+, + & R, % > 0, and f & C1([0, 1]%R). Under appropriateassumptions, all non-trivial solutions lie on two smooth curves of respectivelypositive and negative solutions, bifurcating from the first eigenvalue of the ho-mogeneous problem
!!(rN!1*p(u$))$ = %rN!1f0(r)*p(u), 0 < r < 1,
u$(0) = u(1) = 0,
where f0(r) := lim%%0 f(r, +)/*p(+) > 0. My approach involves a local bifurca-tion result a la Crandall-Rabinowitz, and global continuation arguments relyingon monotonicity properties of f . The global properties of the solution set arethus obtained by a purely analytical method, whereas previous results on thisproblem used topological arguments.
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Maximum and antimaximum principles: beyond the first eigenvalueJean-Pierre GossezUniversite Libre de Bruxelles, [email protected]: J.Fleckinger and F. de Thelin (Toulouse)
Consider the Dirichlet problem
!"u = µu + fin#, u = 0on!#,
with # a smooth bounded domain in RN .The well-known maximum and antimaximum principles give informations on
the sign of the solution u when the parameter µ varies near the first eigenvalue%1 of the corresponding homogenous problem. Our purpose in this talk is tointroduce an analogue of these two principles when µ varies near a higher eigen-value %k. Nodal domains play a central role in our study, as well as, in somecases, the Payne conjecture relative to the nodal line of a second eigenfunctionin the plane.
Population models with di!usion, strong Allee e!ect, and nonlinearboundary conditionsJerome Goddard IIAuburn University [email protected]: E. Lee & R. Shivaji
We discuss the steady state solutions of a di!usive population model withstrong Allee e!ect, namely,
!"u = a(x)u + b(x)u2 ! m(x)u3 ! ch(x); #
&(u)!u
!,+ [1 ! &(u)] u = 0; !#
where # is a subset of Rn with n " 1, a(x), b(x), and m(x) are Hlder continuousfunctions such that b(x), m(x) are strictly positive on the closure of # witha(x) < 0 for some x in #, c " 0, &(u) : R !) [0, 1] is a non-decreasing smoothfunction, and $u
$& is the outward normal derivative. Our study is focused ona population that satisfies a certain nonlinear boundary condition and on itspersistence when constant yield harvesting is introduced. We establish ourexistence results by the method of sub-super solutions.
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Lane-Emden problems: asymptotic behavior of low energy nodal so-lutions.Christopher GrumiauUniversity of Mons, [email protected]: Massimo Grossi Filomena Pacella
We study the nodal solutions of the Lane Emden Dirichlet problem !"u =|u|p!1u defined on a smooth bounded domain A in R2 and p > 1. We considersolutions up satisfying p
*A |#up|2 ) 16-e as p ) +$ (*). We are interested
in the shape and the asymptotic behavior as p ) +$.First we prove that (*) holds for least energy nodal solutions. Then we obtain
some estimates and the asymptotic profile of this kind of solutions. Finally, insome cases, we prove that pup can be characterized as the di!erence of twoGreen’s functions and the nodal line intersects the boundary of A, for large p.
On functions satisfying a mean/median value propertyDavid HartenstineWestern Washington University, [email protected]
Motivated by the mean value property characterizing harmonic functions,we consider functions satisfying a functional equation involving the mean andmedian operators. For the Dirichlet problem, continuous solutions are shownto be unique and existence is guaranteed when a sub/supersolution pair can befound. Connections with p-harmonic functions will be discussed. This is jointwork with Matthew Rudd.
Positive and compact support solutions for some singular nonlinearelliptic equations with absorptionJesus HernandezUniversidad Autonoma de Madrid, [email protected]
We provide an overview on some recent work and open problems concerningexistence and multiplicity of both positive and compact support solutions for aclass of nonlinear singular elliptic problems involving a singular absoption term.We also study the case of the p- Laplacian, in particular in one dimension. Veryweak solutions in the sense od Dıaz-Rakotoson are considered as well.
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Numerical approximation of nonlinear systems based on advection-di!usion-reaction equations.Arturo HidalgoUniversidad Politecnica de Madrid, [email protected]: M. Dumbser
In this work we extend the high order ADER finite volume schemes, that wasintroduced for sti! hyperbolic balance laws in M. Dumbser, C. Enaux, E.F. Toro(2008) to nonlinear systems of advection-di!usion-reaction equations with sti!algebraic source terms. In this new scheme we develop an e&cient formulationof the local space-time discontinuous Galerkin predictor, using a nodal approachwhose interpolation points are tensor-products of Gauss-Legendre quadraturepoints, as proposed in: A. Hidalgo and M.Dumbser (2011). We apply ournumerical method to some systems of advection–di!usion–reaction equationsfor linear model systems and the compressible Navier–Stokes equations withchemical reactions.
Well-posedness of nematic liquid crystal flow in L3U(R3)
Jay HinamanUniversity of Kentucky, Lexington, [email protected]: Changyou Wang
I will present recent well-posedness results for a version of the Ericksen-Leslie hydrodynamic flow model for nematic liquid crystals in three dimensionswith initial data that is uniformly locally L3(R3) integrable. In this setting,the hydrodynamic flow of nematic liquid crystals is a coupled system in whichthe incompressible Navier-Stokes equations specify the velocity of a fluid whichpossesses microstructure that evovles based on the transported heat flow ofharmonic maps. Specifically, we show that weak solutions with small L3
U-normare smooth, and as a consequence, the local well-posedness for initial data withsmall L3
U-norm is established. I will also discuss open problems related to thehydrodynamic flow of nematic liquid crystals.
19
The Fucık spectrum: Asymptotic behaviorGabriela HolubovaUniversity of West Bohemia, Pilsen, Czech [email protected]: Petr Necesal
We study the structure of the Fucık spectrum of a linear operator in a generalsetting. We focus mainly on the asymptotic behavior of the Fucık branchesand on the relation of their asymptots and so called K-eigenvalues and Paretoeigenvalues.
Well-posedness and long time behavior of a perturbed Cahn-HilliardsystemHaydi IsraelPoitiers [email protected]
Our aim in this talk is to discuss the well-posedness (existence and unique-ness of solutions) and the asymptotic behavior, in terms of finite-dimensionalattractors (global and exponential attractors)for a perturbed Cahn-Hilliard sys-tem.
On a two phase free boundary problem arising from material scienceHuiqiang JiangUniversity of [email protected]
Let # be a bounded open domain in Rn, n " 2 and $ be a q dimensionalsmooth submanifold of Rm with 0 + q < m. We use M!," to denote thecollection of all pairs of (A, u) such that A * # is a set of finite perimeter andu & H1 (#, Rm) satisfies
u (x) & $ a.e. x & A.
We consider the energy functional
E! (A, u) =)
!|#u|2 + P! (A) , (9)
defined on M!,", where P! (A) denotes the perimeter of A inside # in the senseof De Giorgi, i.e.,
P! (A) = Hn!1 (!"A ' #) ,
where !"A is the reduced boundary of A and Hn!1 is the (n ! 1)-dimensionalHausdor! measure. This free boundary problem can be viewed as a coupling
20
of harmonic mapping and minimal surface. The existence and regularity of theenergy minimizers will be discussed.
Convergence of the Levenberg-Marquardt methodParimah Kazemi3519 Roma Lane, Middleton, [email protected]: Robert Renka
Here we will discuss a generalized Levenberg-Marquardt method based on aSobolev gradient. We discuss results related to global existence and uniquenessof the gradient flow as well as asymptotic convergence.
A variational approach for the inverse Black-Scholes problemIan KnowlesUniversity of Alabama at [email protected]: Ajay Mahato
The generalized Black-Scholes equation provides a realistic and reasonablysimple model for pricing equity options. The corresponding inverse problem,which consists of using published prices of options to recover market parameterssuch as the local volatility of the underlying equity, and interest rates, may beseen as providing a market view of the future behaviour of these quantities.We discuss a variational approach to this inverse problem that involves stableminimization using convex functionals.
21
Random Walks and continuum couplingSheldon LeeViterbo [email protected]: Don Estep, Simon Tavener
A random walk simulation can be thought of as discrete version of the di!u-sion equation. In this talk, we begin with a basic overview of random walks, andtheir connection to the di!usion equation. Next, we discuss basic deterministiccoupling, along with stability and convergence of such a coupling. Finally, wediscuss how a random walk region may be coupled to a continuous region, andthe relevant stability and convergence issues that arise.
Steklov-Neumann eigenvalue problem for a class of elliptic systemand applicationsOlimpio Hiroshi MiyagakUniversidade Federal de Juiz de Fora, [email protected]: J.D. B. De Godoi, R.S. Rodrigues
We will study two class of the eigenvalue problems of the Steklov and Neu-mann.
The results obtained will be applied to get multiplicity results for a classof the elliptic system with nonlinear boundary condition. These results extendthat obtained by Mavinga and Nkashama, in the scalar case.
Strong bounded solutions for nonlinear parabolic equationsNsoki MavingaSwarthmore College, [email protected]: Marius Nkashama
We are concerned with the existence of bounded solutions existing for alltimes for nonlinear parabolic equations with nonlinear boundary conditions ona domain that is bounded in space and unbounded in time (the entire real line).We establish (one- sided) a priori estimates for solutions to linear boundary valueproblems, and derive a weak maximum principle which is valid on the entirereal line in time. We then use comparison techniques, a priori estimates, andnonlinear approximation methods to prove the existence and, in some instances,positivity and uniqueness of bounded solutions existing for all times.
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On a variational principle for doubly nonlinear reaction-di!usion equa-tions and blow-up/extinctionJochen MerkerUniversity of Rostock - Institute of Mathematics, [email protected]
In the first part of the talk it is shown that doubly nonlinear reaction-di!usion equations
!
!tb(u) ! div(a(#u)) = f(u)
on a domain # * Rn originate from variation of the energy balance of a thermo-dynamic system in equilibrium. In the second part, blow-up of the Lm-norm of uand extinction of u is discussed in the case b(u) = |u|m!2u, a(#u) = |#u|p!2#u,f(u) = |u|q!2u. Particularly, we show that blow-up and extinction occur simul-taneously for 1 < p < m < q < p& := p(1 + (m/n)), i.e. in the case offast di!usion (p < m) and superlinear (max(m, p) < q) but subcritical growth(q < p&) of the nonlinearity (even if p > 2 or q < 2).
Some generic properties of nondegeneracy of critical pointsAnna Maria MichelettiUniversita di Pisa, [email protected]
We consider a compact smooth Riemannian manifold M of finite dimensionn. Given a metric g0 on M we show that, if g lies in a neighbourhood of g0,then, generically with respect to the metric g, all the positive solutions u inH1
g0(M) with u di!erent from the constant function 1 of the equation:
!.2"gu + u = |u|p!2u
are nondegenerate. Here 2 < p < (2n)/(n!2). We apply these results to obtainsome estimate on the number of solutions of the previous equation using Morsetheory. We also show a genericity result for nondegenerate critical points of theRobin function with respect to deformations of the domain.
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Necessary and su"cient conditions for mixed PDEsJ. W. (Monty) MontgomeryUniversity of North Texas, [email protected]
A set of supplementary conditions (”ladder” conditions) are defined withrespect to rectangular mixed domains so that when paired with Tricomi-typePDEs the problem may be reduced to an elliptic PDE under Dirichlet condi-tions and a resulting hyperbolic PDE under initial-boundary value (IBV) con-ditions. A conjecture is posed on the conditions being necessary and su&cientfor Tricomi’s equation. Numerical evidence will be presented in support of saidconjecture, as well as evidence which suggests the Tricomi-Dirichlet problem tobe overdetermined.
The Fucık spectrum for matricesPetr NecesalDepartment of Mathematics, Faculty of Applied Sciences, University of WestBohemia, Univerzitnı 22, CZ-306 14, Pilsen, Czech [email protected]: Gabriela Holubova
We investigate the structure and qualitative properties of the Fucık spectrumfor matrices. We consider the discrete problem
"2u(t ! 1) + &u+(t) ! #u!(t) = 0, t & T,
with Neumann and Dirichlet boundary conditions. We provide the analyticreconstruction of the Fucık spectrum of the corresponding discrete operators.Finally, we propose the suitable numerical algorithms for finding approximationsof the particular Fucık spectra.
A linear condition determining local or global existenceJohn W. NeubergerDepartment of Mathematics, University of North Texas, Denton, [email protected]
Suppose that for a given system of autonomous ODEs or PDEs on a spaceX , one has at least local existence and uniqueness. Let m be the stoppingtime function, that is, if x is an initial value, then m(x) is the time for whichexistence ceases. Suppose m is continuous. Such a system generates at least alocal semigroup. In this talk, a linear condition is given on the Lie generatorA of the system. This condition characterizes global existence. Results of anumerical investigation are presented. Some of this work is in collaborationwith John M. Neuberger and James Swift.
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Localization phenomena in degenerate logistic equationRosa PardoDepartamento de Matematica Aplicada, Universidad Complutense de Madrid,28040 Madrid, [email protected]: J. M. Arrieta, A. Rodriguez-Bernal
Let us consider the following problem"#
$
ut !"u = %u ! n(x)u' in #, t > 0u = 0 on !#, t > 0
u(0) = u0 " 0(10)
in a bounded domain # * RN , N " 1, where n(x) " 0 in #, is a Holder function,and / > 1, % & R.
Betwen 1798 and 1826 Malthus publish “An Essay on the Principle of Pop-ulation”, where the incresing rate of a population is proportional to the exis-tent population. Verhulst criticize this model and proposed the logistic model.On the region where n(x) = 0, the growth rate is Malthusian, an exponen-tial growth, and could be expected to be unbounded. In the region wheren(x) > n0 > 0 it is a logistic growth, and could be expected to be limited. Thequestion is what kind of behavior can be expected. The answer is known whenthe interior of the set where n(x) = 0 is an open set of regular boundary.
When n(x) is a Holder function, then the region n(x) = 0 is a compact andnothing prevents from having empty interior. We are interested in analyzingpossible interactions that vanish on compact sets, with not necessarily smoothboundary, and whose interior can be empty, and to study possible relationsbetween n(x) and the dimension of the region over which n(x) = 0.
This type of nonlinearity has applications in the nonlinear Schrodinger equa-tion and the study of Bose - Einstein condensates. In this case, this phenomenonexplains the fact that the ground state presents a strong localization in the spa-tial region in which n(x) = 0, see [V. M. Perez-Garcıa and R. Pardo. Local-ization phenomena in nonlinear Schrodinger equations with spatially inhomo-geneous nonlinearities: theory and applications to Bose-Einstein condensates.Phys. D, 238(15):1352–1361, 2009. ].
On thermochemical equilibrium of reacting plasmasVıctor PadronNormandale Community College, [email protected]: Alfonso Castro, Harvey Mudd College, and Miguel Ibanez, Univer-sidad de Los Andes, Merida, Venezuela
25
In a recent paper with A. Castro we classify and study the stability of theradial steady state solutions of the equation
!u
!t= #(uk#u) + %(um ! un) in B(0, 1) % R+, (11)
under the initial-boundary conditions
u(·, 0) = u0 in B(0, 1), u(·, t) = 1 in !B(0, 1) % R+, (12)
where B(0, 1) is the unit ball in RN , and k, m, n are given constants.This problem arises in the investigation of thermal structures in plasma
physics. The study of thermal (or more general thermochemical) equilibriumand stability of plasmas plays an important role in understanding the originof astrophysical structures. Our study includes the construction of bifurcationdiagrams that describe the stability properties of both homogeneous and het-erogeneous steady states, specifying this way the type of inhomogeneities thatdevelops at the onset of thermal instability.
In this talk we discuss extensions of this work to the study of gasses in whichchemical reactions are in progress. This accounts for an addition of one morevariable to the equations of gas dynamics denoting the degree of the extent ofthe chemical reaction.
Probing fundamental bounds in hydrodynamics using variational op-timization methodsBartosz ProtasDepartment of Mathematics and Statistics, McMaster [email protected]: Diego Ayala (Department of Mathematics and Statistics, McMasterUniversity)
This work demonstrates how the modern methods of PDE-constrained op-timization can be used to assess sharpness of a class of fundamental functionalestimates in fluid mechanics. These estimates concern bounds on the instanta-neous rate of growth and finite-time growth of quadratic quantities such as theenstrophy and palinstrophy in viscous incompressible flows. Sharpness of suchestimates is inherently related to the problem of singularity formation in the3D Navier-Stokes system. In our presentation we will first review earlier resultsof Lu & Doering (2008) and Ayala & Protas (2011) concerning the maximumgrowth of enstrophy the 1D Burgers equation. We will then present severalnew results regarding the maximum growth of palinstrophy in 2D flows and willdiscuss some questions concerning sharpness of the corresponding analytical es-timates. While it is well known that solutions of 1D Burgers equations and2D Navier-Stokes equation evolving from smooth initial data remain smooth forall times, the question whether the best available estimates for the maximum
26
growth of enstrophy and palinstrophy are sharp is both interesting and relevant.One reason is that such estimates are derived using similar mathematical tech-niques as in the 3D case where blow-up cannot be ruled out. We will show hownew insights regarding these problems can be obtained by formulating them asvariational PDE optimization problems which can be solved computationallyusing suitable adjoint–based gradient descent (or ascent) methods. In particu-lar, we will discuss certain topological features of the families of vorticity fieldsmaximizing the instantaneous rate of growth of palinstrophy in 2D. In o!eringa systematic approach to finding flow solutions closest to saturating a givenanalytical bound, the proposed approach provides a bridge between theory andcomputation.
Global bifurcation results for exponential type problems in 2 dimen-sionDhanya RajendranTIFR Centre for Applicable Mathematics, [email protected]: J. Giacomoni (Universite de Pau), S. Prashanth (TIFR CAM), K.Saoudi
We study the positive solutions to the following singular elliptic problem ofexponential type growth posed in a bounded smooth domain # * R2 with theDirichlet boundary conditions.
!"u = %(u!( + h(u)eu"
) in #
Here, 1 + & + 2, 0 < $ < 3 , % " 0 and h(t) is assumed to be a smooth“perturbation” of et"
as t ) $
A Sobolev trust-region method for numerical solution of the Ginzburg-Landau equationsRobert RenkaUniversity of North [email protected]: Parimah Kazemi
We describe a generalized Levenberg-Marquardt method for computing criti-cal points of the Ginzburg-Landau energy functional which models superconduc-tivity. The algorithm is a blend of a Newton iteration with a Sobolev gradientdescent method, and is equivalent to a trust-region method in which the trust-region radius is defined by a Sobolev metric. Numerical test results demonstratethe method to be remarkably e!ective, achieving a quadratic rate of convergencewhile allowing an arbitrary initial estimate.
27
On the variational characterization of the Fucik spectrum: Extendingbeyond Castro’s ”strip”Stephen RobinsonWake Forest [email protected]: Pavel Drabek
In a recent paper Castro and Chang used a reduction method combinedwith a minimization argument to characterize Fucik Spectrum pairs ,(&,#),assuming that %k < & < %k+1, where %k,%k+1 represent consecutive eigenvaluesof a linear di!erential operator such as the Laplacian or the D’Alembertian. Weshow that under certain circumstances this characterization can be extendedto a larger strip %k < & < %k+2, where the characterization now relies on aminimax argument applied to the reduced functional.
Landesman-Lazer conditions at half-eigenvalues of the p-LaplacianBryan RynneDepartment of Mathematics, Heriot-Watt University, Edinburgh, EH14 4AS,[email protected]
We study the existence of solutions of a resonant, p-Laplacian Dirichletproblem in 1-dimension, with a ‘jumping’ nonlinearity. Using a shooting argu-ment, we derive ‘Landesman-Lazer’ type conditions for the existence of solu-tions. The results are related to those involving the Fredholm alternative forthe p-Laplacian near points on the Fucik spectrum, which has been investigatedvariationally. The results here extend the variational results.
Multiplicity results for nonvariational elliptic problems with singularnonlinearity. A monotonicity approachClaudio SacconUniversity of Pisa, [email protected]
We consider problems of the type:+
ij
(aij(x)DiDju + bi(x)Diu) + g(x, u) = f(x, u) in #, u = 0 on !#
where the linear part is uniformly elliptic (not necessarily symmetric) and whereg(x, u) is increasing and singular in u. We regard the problem as Ag(u) = Nf (u)where Ag = A+!G , G$
s(x, s) = g(x, s) ( so Ag is a maximal monotone operator),and Nf , NG are Nemetskii operators. This allows to look for (weak) solutions asfixed points of id!Rg ,Nf , Rg being the resolvent of Ag (provided the elliptic
28
operator is coercive): this is a nice setting for the Leray Schauder degree. Thissetting is also quite e!ective if sub or super solutions of the problem are known.In this case we can use them as “natural constraints” including them, in somesense, in the operator Ag, and look for solutions of the original problem lyingabove a subsolution or below a supersolution. The latter is also useful forproviding regularity of the solutions, which in the applications we deal with areproved to be C2(#) ' C0(#).
We use this approach to prove that the problem:+
ij
(aij(x)DiDju + bi(x)Diu) ! uq = %up in #, u = 0 on !#
where !3 < q < 0, 1 < p < 2" ! 1, and some regularity is assumed on aij , bi,has two solutions in W1,2
0 (#)'C2(#)'C0(#). for % small. To get this result weneed to prove a uniform bound of the solutions of the problem, when % staysaway from zero.
With the same ideas we can prove that+
ij
(aij(x)DiDju + bi(x)Diu) = %f(u) in #, u = 0 on !#
where f(u) - !uq near zero with !3 < q < !1, f(u) - !up at infinitywith !1 < p < 1, but f(u0) > 0 at some point u0, has three solutions inW1,2
0 (#) ' C2(#) ' C0(#) for % large.
Multiple sign changing solutions of nonlinear elliptic problems in ex-terior domainsDora SalazarUniversidad Nacional Autonoma de [email protected]: Monica Clapp (Universidad Nacional Autonoma de Mexico)
We consider the problem
!"u + (V# + V (x))u = |u|p!2 u, u & H10 (#),
where # is an exterior domain in RN , V# > 0, V & C0(RN ), infRN V > !V#and V (x) ) 0 as |x| ) $. Under symmetry conditions on # and V, and someassumptions on the decay of V at infinity, we show that there is an e!ect of thetopology of the orbit spaces of certain subsets of the domain on the number oflow energy sign changing solutions to this problem.
29
Existence of compact support solutions to a quasilinear and singularproblemPaul SauvyUniversite de Pau et des Pays de lAdour, [email protected]: Jacques Giacomoni, Habib Maagli
Let # be a bounded domain of RN , N " 2 with a C2 boundary !#. Weconsider the following quasilinear elliptic problem:
(P#)
"#
$
!"pu = K(x)(%uq ! ur), in #;
u|$! = 0, u " 0 in #,
where p > 1 and "pudef= div
,|#u|p!2#u
-denotes the p-Laplacian operator. In
this study, % > 0 is a real parameter, the exponents q and r satisfy !1 < r <q < p ! 1 and K : # !) R is a positive function having a singular behaviournear the boundary !#. Precisely,
K(x) = d(x)!kL(d(x)) in #,
with 0 < k < p, L a positive perturbation function and d(x) the distance ofx & # to !#.By using a sub- and super-solution technique, we discuss the existence of positivesolutions or compact support solutions of (P#) in respect to the blow-up ratek of K. Precisely, we prove that if k < 1 + r, (P#) has at least one positivesolution for % > 0 large enough, whereas it has only compact support solutionsif k " 1 + r.
Nonlocal equations of elliptic type: A variational approachRa!aella ServadeiUniversity of Calabria, [email protected]: Enrico Valdinoci
Aim of this talk will be to present some results which extend the validityof some existence theorems known in the classical case of the Laplacian to thenon-local framework, i.e. to fractional equations modelled by
!(!")su ! %u = |u|q!2u in #u = 0 in Rn \ # ,
where 0 < s < 1 is fixed and (!")s is the fractional Laplace operator defined,up to normalization factors, as
!(!")su(x) =12
)
Rn
u(x + y) + u(x ! y) ! 2u(x)|y|n+2s
dy , x & Rn ,
30
while # is an open, bounded subset of Rn, n > 2s, with Lipschitz boundary,% is a positive parameter, 2" = 2n/(n ! 2s) is the fractional critical Sobolevexponent and 2 < q + 2" .
In particular, in the critical setting (i.e. q = 2") our theorems may be seenas the extension of the classical Brezis-Nirenberg result to the case of non-localfractional operators.
All these results were obtained in collaboration with Enrico Valdinoci in [1,2, 3, 4, 5].
[1] R. Servadei, The Yamabe equation in a non-local setting, preprint,submitted for publication.
[2] R. Servadei and E. Valdinoci, Mountain Pass solutions for non-localelliptic operators, J. Math. Anal. Appl., 389, 887–898 (2012).
[3] R. Servadei and E. Valdinoci, Variational methods for non-localoperators of elliptic type, preprint, submitted for publication, available athttp://www.math.utexas.edu/mp arc-bin/mpa?yn=11-131.
[4] R. Servadei and E. Valdinoci, The Brezis-Nirenberg result for thefractional Laplacian, preprint, submitted for publication, available athttp://www.math.utexas.edu/mp arc-bin/mpa?yn=11-196.
[5] R. Servadei and E. Valdinoci, Fractional Laplacian equations withcritical Sobolev exponent, in preparation.
Holder-regularity and singular elliptic equationsIan SchindlerUniversite des Sciences Sociales, Toulous, [email protected]: J. Giacomoni, I. Schindler and P. Takac
We prove Holder regularity for weak solutions to singular quasilinear ellipticequations of the type
"#
$!"pu =
K(x)u(
+ g(x) in #;
u|$! = 0, u > 0 in #,(P)
where # is an open bounded domain with smooth boundary, 1 < p < $, $ > 0,K & L1
loc(#) satisfies 0 + K(x) a.e. x & #, and g & L#(#).
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A uniqueness result for a singular nonlinear eigenvalue problemRatnasingham ShivajiUniversity of North Carolina at [email protected]: Alfonso Castro, Eunkyung Ko
We consider positive solutions to classes of nonlinear elliptic boundary valueproblems when the reaction term has a singularity at the origin. We establishthe uniqueness of the positive solution when a parameter is large.
Semilinear elliptic PDE on manifoldsDaryl Je!rey SpringerNorthern Arizona University, Flagsta!, [email protected]: John M. Neuberger
In this work we aim to solve the PDE "Su + f(u) = 0, where "S is theLaplace-Beltrami operator on a given manifold S. Our approach here is numer-ical. We find solutions using a combination of numerics and variational theory.We begin by introducing the Gradient Newton Galerkin Algorithm (GNGA).The Gradient Newton Galerkin Algorithm is a method for finding approxima-tions of critical points of a functional. This method was introduced by Neubergerand Swift to solve PDE of the form "Su + f(u) = 0 on regions # , with zeroDirichlet or Neumann boundary conditions. We use the GNGA to solve PDEon manifolds by first finding eigenfunctions of the Laplace-Beltrami operatoron our given manifold. We also introduce the Closest Point Method (CPM) forfinding eigenfunctions on manifolds when the required eigenfunctions are notknown in closed form.
Newton’s method and symmetry for semilinear elliptic PDE on thecubeJim SwiftNorthern Arizona University, Flagsta!, [email protected]: John M. Neuberger, Nandor Sieben
We seek discrete approximations to solutions u : # ) R of semilinear ellipticpartial di!erential equations of the form "u + fs(u) = 0, where fs is a one-parameter family of nonlinear functions and # is a domain in Rd. The mainachievement of this paper is the approximation of solutions to the PDE on thecube # = (0,-)3 . R3. There are 323 possible symmetries of functions on thecube, which fall into 99 conjugacy classes. Our automated symmetry analysis
32
helps us find solutions with 50 of these 99 symmetry types. As a preliminary,we compute solutions to the PDE on the square.
This article extends the authors’ work in Automated Bifurcation Analysis forNonlinear Elliptic Partial Di!erence Equations on Graphs (Int. J. of Bifurcationand Chaos, 2009), wherein they combined symmetry analysis with modifiedimplementations of the gradient Newton-Galerkin algorithm (GNGA, Neubergerand Swift) to automatically generate bifurcation diagrams and solution graphicsfor small, discrete problems with large symmetry groups. The code described inthe current paper is e&ciently implemented in parallel, allowing us to investigatea relatively fine-mesh discretization of the cube. We use the methodology andcorresponding library presented in our paper An MPI Implementation of a Self-Submitting Parallel Job Queue (preprint, 2010).
Special solution methods for singular systems. Part IIIPeter TakacUniversitaet Rostock, Rostock, [email protected]
We treat systems of two singular power-coupled equations with the p- andq-Laplacians:
".#
.$
!"pu =1
ua1vb1in #; u|$!= 0, u > 0 in #,
!"qv =1
va2ub2in #; v|$! = 0, v > 0 in #,
(13)
for both, the homogeneous
(p ! 1 + a1)(q ! 1 + a2) ! b1b2 = 0 , (14)
and the subhomogeneous case
(p ! 1 + a1)(q ! 1 + a2) ! b1b2 > 0 . (15)
Such problems arise, for instance, in models of pseudo-plastic flows, chemicalreactions, morphogenesis (Gierer-Meinhardt system), population dynamics,and many other applications. We begin by reformulating this system of twopower-coupled equations as an abstract fixed point problem for a homogeneousor subhomogeneous mapping in the positive cone of a strongly orderded Banachspace of C1(#)-functions satisfying the Hopf maximum principle. A suitableorderded pair of sub- and supersolutions provides an invariant conical shell forSchauder’s fixed point theorem in the subhomogeneous case (15).
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On a parabolic problem with a dynamic and di!usive boundary con-dition for the coupling surface – deep ocean temperatures involvinglatent heat.Lourdes TelloUniversidad Politecnica de Madrid, [email protected]: J.I. Diaz, A. Hidalgo
We study a global climate model for the coupling of the mean surface tem-perature with the deep ocean temperature. The nonlinear model presents somenonstandard facts: the boundary condition, representing the mean surface tem-perature, is not only of dynamic type (involving the time derivative of the traceof the solution) but also a surface di!usive term. The model includes alsosome delicate nonlinear terms such as the coalbedo e!ect and the latent heat,which here are formulated in terms of suitable (multivalued) maximal monotonegraphs of R2. We prove the existence of bounded weak solutions and show somenumerical experiences. Other qualitative properties of the solutions will be alsopresented.
On W 1,p(x) versus C1(#) local minimizers of functionals related to p(x)-Laplacian and it’s applicationSweta TiwariiIndian Institute of Technology Delhi, [email protected]: K. Sreenadh (Indian Institute of Technology Delhi, India)
In this talk we discuss the embedding theorems in variable exponent spacesand other preliminary results required to show the existence and multiplicityof solutions for elliptic problem with variable exponent. In particular we studythe result on W 1,p(x) versus C1 local minimizers of functionals related to p(x)-Laplacian.Let # be a bounded domain in RN , N " 2 with smooth boundary. We considerthe functional J : W 1,p(x)(#) ) R defined as
J(u) =)
!
|#u|p(x)
p(x)+
)
!
|u|p(x)
p(x)!
)
!F (x, u) !
)
$!G(x, u)d(,
where p(x) be a Holder continuous function satisfying 1 < p(x) + N in # and
F (x, t) =) t
0f(x, s)ds, G(x, t) =
) t
0g(x, s)ds. Under appropriate assumptions
on f(x, t) and g(x, t) which include some critical cases, we show that a C1,)(#)local minimizer u0 /0 0 of J is also a W 1,p(x) local minimizer.
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Multiplicity and symmetry of positive solutions to semi-linear ellipticproblems with Neumann boundary conditionsChristophe TroestlerUniversite de Mons, [email protected]: D. Bonheure; C. Grumiau
In this talk we will examine how the number of positive solutions to !"u+u = |u|p!2u on #, with Neumann boundary conditions, changes as the exponentp increases. In particular, when # is a ball, we will show that this problempossesses a large number of positive solutions as well as degenerate solutions.
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