icm 2018 sattelite conference in nonlinear...

Contents

Schedule 5

Abstracts - Speakers 7Claudianor ALVES, Existence and concentration of positive solutions for a Schrödinger

logarithmic equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7David ARCOYA, Elliptic nonlocal Kirchhoff problems . . . . . . . . . . . . . . . . 7Julián Fernández BONDER, The concentration-compactness principle for fractional or-

der Sobolev spaces in unbounded domains and applications to the generalized frac-tional Brezis-Nirenberg problem . . . . . . . . . . . . . . . . . . . . . . . . 7

Guy BOUCHITTÉ, Convex relaxation for some free boundary problems . . . . . . . . 7Jean-Baptiste CASTERAS, Stability of ground state and renormalized solutions to a

fourth order Schrödinger equation . . . . . . . . . . . . . . . . . . . . . . . 8Andrea CIANCHI, Nonlinear elliptic equations under minimal data and domain regularity 8Simão CORREIA, Semitrivial vs. fully nontrivial ground-states for systems of M coupled

Schrödinger equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9João Marcos DO Ó, Elliptic equations and systems with critical Trudinger-Moser nonlin-

earities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9Giovany FIGUEIREDO, Existence of positive solution for a planar Schrödinger-Poisson

system with exponential growth . . . . . . . . . . . . . . . . . . . . . . . . 9Juraj FÖLDES, Variational problems arising from fluid dynamics . . . . . . . . . . . 9Marcelo Fernandes FURTADO, Existence and multiplicity of self-similar solutions for

heat equations with nonlinear boundary conditions . . . . . . . . . . . . . . . . 10Filippo GAZZOLA, Periodic solutions and torsional instability in a nonlinear nonlocal

plate equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10Massimo GROSSI, A Morse Lemma for degenerate critical points of solutions of nonlin-

ear equations in R2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10Michal KOWALCZYK, Maximal solutions of the Liouville equation in doubly connected

domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Liliane A. MAIA, Positive multipeak solutions to a zero mass problem in exterior domains 11Eugenio MASSA, Weighted Trudinger-Moser inequalities and associated Liouville type

equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Hayk MIKAYELYAN, Cylindrical optimal rearrangement problem leading to a free bound-

ary problem with non-local obstacle . . . . . . . . . . . . . . . . . . . . . . 12Monica MUSSO, Singularity formation in critical parabolic equations . . . . . . . . . 12Benedetta NORIS, Radial positive solutions for a class of Neumann problems without

growth conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12Filomena PACELLA, Overdetermined problems and constant mean curvature surfaces in

cones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12Edgard A. PIMENTEL, Geometric regularity theory for PDEs . . . . . . . . . . . . 13Angela PISTOIA, The Lane-Emden equation on a planar domain . . . . . . . . . . . 13Alexander QUAAS, Periodic solutions for the one-dimensional fractional Laplacian . . 14Diego RIBEIRO MOREIRA, Regularity of semi-convex supersolutions of fully nonlinear

PDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14David RUIZ, Prescribing Gaussian curvature on a compact surface and geodesic curva-

ture on its boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14Alberto SALDAÑA, On fractional higher-order powers of the Laplacian . . . . . . . . 15

Boyan SIRAKOV, Uniform boundedness of positive solutions of the Lane-Emden equationin dimension two . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Nicola SOAVE, The nodal set of solutions of sublinear equations . . . . . . . . . . . 15Eduardo TEIXEIRA, Free boundaries in rough media . . . . . . . . . . . . . . . . 16Susanna TERRACINI, Regularity of the optimal sets for spectral functionals and the free

boundary for the vectorial Bernoulli problem . . . . . . . . . . . . . . . . . . 16Carlos TOMEI, Positive ground states and global geometry of differential operators . . . 16José Miguel URBANO, Geometric tangential analysis and sharp regularity for degener-

ate pdes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17Gianmaria VERZINI, Asymptotic spherical shapes in some spectral optimization problems 17Alessandro ZILIO, Predator-prey model with competition, the emergence of territoriality 17

Poster Session 18Francisco ALBUQUERQUE, A weighted Trudinger-Moser type inequality and its appli-

cations to quasilinear elliptic problems with critical growth in the whole Euclideanspace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

Marcelo AMARAL, Sharp regularity estimates for quasilinear evolution equations . . . 18Augusto César dos Reis COSTA, A class of critical Kirchhoff problem on the hyperbolic

space Hn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18João Vítor DA SILVA, Sharp regularity estimates for quasi-linear elliptic dead core prob-

lems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19Jefferson A. DOS SANTOS, Existence of positive solutions for a class of semipositone

quasilinear problems through Orlicz-Sobolev space . . . . . . . . . . . . . . . . 19Daniele GARRISI, Ground state stability and uniqueness of normalized standing-waves . 19César LEDESMA, Existence of heteroclinic solution for a class of problems involving the

fractional laplacian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20Gustavo MADEIRA, Concave-convex structure for a class of degenerated or singular

Kirchhoff type problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20Vando NARCISO, Dynamics for a model of flight structures with nonlocal energy damping 20Denilson PEREIRA, Multiple Solutions for Nonvariational Quasilinear Elliptic Systems . 21Adilson PRESOTO, Some progress in studying elliptic problems with exponential nonlin-

earity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21David STOLNICKI, Regularity theory for nonlocal operators . . . . . . . . . . . . . 21

Committees and Sponsors 23

Practical information 25

Schedule

All lectures take place at the conference room Buriti of Mareiro Hotel.

Mon 23 Tue 24 Wed 25 Thu 26 Fri 27

9:00-9:30Registration /

OpeningChair Tomei Teixeira do Ó Gazzola

9:30-10:10 Terracini Gazzola Bouchitté RibeiroMoreira

10:15-10:55 Cianchi Pacella Zilio Kowalczyk

11:00-11:40 Urbano Maia Arcoya do Ó

11:45-14:00 LunchDiscussions

LunchDiscussions

LunchDiscussions

LunchDiscussions

LunchDiscussions

Chair Verzini Cianchi Grossi

14:00-14:40 Tomei Sirakov Bonder

14:45-15:25 Noris Teixeira Correia

15:30-16:10 Furtado Földes Soave

16:15-16:55 Coffee Break

Coffee BreakPosters

AlbuquerqueAmaral

Costada Silva

Coffee BreakPosters

dos SantosGarrisiLedesmaMadeira

Coffee BreakPostersNarcisoPereiraPresotoStolnicki

Chair Pistoia Musso Bonder Bouchitté

17:00-17:40 Musso Pistoia Grossi Verzini

17:45-18:25 Ruiz Casteras Quaas G. Figueiredo

18:30-19:10 Mikayelyan Pimentel Massa Saldaña

20:00 Social Dinner

Tabl

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ICM 2018 Sattelite Conference on Nonlinear PDEs 5

Abstracts - Talks

Existence and concentration of positive solutions for a Schrödinger logarithmicequation

Claudianor ALVES

Universidade Federal de Campina Grande (Brazil)

This talk concerns with the existence and concentration of positive solutions for thefollowing logarithmic elliptic equation

−ε2∆u+V (x)u = u logu2, in RN ,u ∈ H1(RN),

where ε > 0,N > 3 and V is a continuous function with a global minimum. Using varia-tional method developed by A. Szulkin for functionals which are sum of a C1 functionalwith a convex lower semicontinuous functional, we prove, for small enough ε > 0, theexistence of positive solutions and concentration around of a minimum point of V , whenε goes to zero.

Joint work with Daniel C. de Morais Filho (UFCG)

Elliptic nonlocal Kirchhoff problemsDavid ARCOYA

Universidad de Granada (Spain)

We discuss some recent results on nonlinear elliptic Kirchhoff equations, based onjoint works with Antonio Ambrosetti (S.I.S.S.A. Trieste) and with Francisco O. DePaiva and José M. Mendoza (Universidade Federal de São Carlos). Specifically, fora bounded open Ω in RN and under suitable hypotheses on the functions f : Ω×R−→Rand M : [0,∞)−→ R, we study the existence and multiplicity of solutions for boundaryvalue problems like

−M(∫

Ω

|∇u|2dx)

∇u = f (x,u), x ∈Ω,

u(x) = 0, x ∈ ∂Ω.

The concentration-compactness principle for fractional order Sobolev spaces inunbounded domains and applications to the generalized fractional

Brezis-Nirenberg problemJulián Fernández BONDER

Universidad de Buenos Aires (Argentina)

In this talk I will discuss recent joint work with A. Silva and N. Saintier about the well-known concentration–compactness principle for the Fractional Laplacian operator inunbounded domains. As an application we show sufficient conditions for the existenceof solutions to some critical equations involving the fractional p-laplacian in the wholeRn.

ICM 2018 Sattelite Conference on Nonlinear PDEs Abstracts - 7

Convex relaxation for some free boundary problemsGuy BOUCHITTÉ

IMATH Université de Toulon (France)

This lecture is based on a duality scheme for non convex variational problems whichhas been developed in collaboration with I. Fragalà (2016 Arxiv, to appear ARMA 2018)and Minh Phan (Toulon). Typically this theory concerns minimum problems of the kind

inf∫

Ω

(|∇u|p +g(u))dx : u ∈W 1,p(Ω) , u = u0 on ∂Ω

where g : R→ R∪+∞ is a non convex l.s.c function with possibly many jumps. Inthis talk, I will report on two recent related results specific to the case p = 1:- the first one concerns an exclusion principle which states that minimizers take valuesoutside the set g∗∗ < g. This principle allows convex relaxation and then we focus ona multiphase problem that we treat numerically by means of a primal-dual algorithm.- the second one concerns a variant of the Cheeger problem in a convex subset D ⊆R2 for which we construct explicit calibrating fields. This is done by using a locallyLipschiz potential whose trace on ∂D coincides with the normal distance to the cut-locus.

Stability of ground state and renormalized solutions to a fourth orderSchrödinger equation

Jean-Baptiste CASTERAS

Université Libre de Bruxelles (Belgium)

In this talk, we will be interested in standing wave solutions to a fourth order nonlin-ear Schrödinger equation having second and fourth order dispersion terms. This kind ofequation naturally appears in nonlinear optics. In a first time, we will establish the ex-istence of ground-state and renormalized solutions. We will then be interested in theirqualitative properties, in particular their stability. Joint works with Denis Bonheure,Ederson Moreira Dos Santos, Tianxiang Gou, Louis Jeanjean and Robson Nascimento.

Nonlinear elliptic equations under minimal data and domain regularityAndrea CIANCHI

Università di Firenze (Italy)

I will discuss a few aspects of the regularity of solutions to boundary value problemsfor nonlinear elliptic equations and systems of p-Laplacian type. In particular, second-order regularity properties of solutions, and the boundedness of their gradient will befocused. The results to be presented are optimal, in a sense, as far as the regularity ofthe right-hand sides of the equations and the boundary of the underlying domains areconcerned. The talk is based on joint researches with V.Maz’ya.

8 - Abstracts ICM 2018 Sattelite Conference on Nonlinear PDEs

Semitrivial vs. fully nontrivial ground-states for systems of M coupledSchrödinger equations

Simão CORREIA

Institut de Recherche Mathématique Avancée (France)

We consider the system

∆ui−ωiui +M

∑j=1

ki j|u j|p+1|ui|p−1ui = 0 i = 1, ...,M.

over a bounded domain or RN , where p > 0 is Sobolev subcritical and M ∈R. The exis-tence of ground-states (minimal action bound-states) is now completely solved and themain problem is the characterization of ground-states. In the coherent case ωi ≡ 1, wepresent the complete characterization and use it to obtain several results (some known,others new). On the other hand, in the incoherent case, since an explicit characteriza-tion is impossible, we develop some techniques to understand whether the ground-stateshave all components different from zero or not, especially in the case p = 1. If time al-lows, some dynamical aspects of the corresponding nonlinear Schrödinger system willbe considered. A major part of this work has been done in collaboration with H. Tavaresand F. Oliveira.

Elliptic equations and systems with critical Trudinger-Moser nonlinearitiesJoão Marcos DO Ó

Universidade de Brasília (Brazil)

Existence of solutions is established for a class of systems of coupled equations viavariational methods. Typical features of this class of problems are the lack of com-pactness because the unboundedness of the domain and critical growth. Moreover, theassociated functional is strongly indefinite. The main tool used is the Trudinger-Moserinequality combined with a linking theorem.References:[1] D. de Figueiredo, J.M. do Ó, B. Ruf, Elliptic equations and systems with criticalTrudinger-Moser nonlinearities, Discrete Contin. Dyn. Syst. 30 (2011), 455–476.[2] M. de Souza, J.M. do Ó, Hamiltonian elliptic systems in R2 with subcritical andcritical exponential growth, Ann. Mat. Pura Appl. 195 (2016), 935–956.[3] J.M. do Ó, F. Sani, J. Zhang, Stationary nonlinear Schrödinger equations in R2 withpotentials vanishing at infinity, Ann. Mat. Pura Appl. 196 (2017), 363–393.

Existence of positive solution for a planar Schrödinger-Poisson system withexponential growth

Giovany FIGUEIREDO


In this paper we prove the existence of a positive solution with minimal energy fora planar Schrödinger-Poisson system, where the nonlinearity is a continuous functionwith exponential critical growth. We find this solution using minimization technique onthe Nehari manifold.


Variational problems arising from fluid dynamicsJuraj FÖLDES

University of Virginia (USA)

The problems of the fluid dynamics are typically modeled by system of stronglycoupled parabolic equations that are not expected to possess a variational structure.Also, the induced dynamical system is extremely sensitive to change of parameters andinitial conditions, and as such in applications, it is not practical to follow a particularsolution. One way to overcome these difficulties is to consider only typical solutions,which of course must be properly defined. We will discuss two approaches, one basedon the ergodic hypothesis and the other one using the addition of a stochastic forcing.The first approach yields variational problem with infinitely many constraints, whereasthe second one induces a problem with random potential. During the talk, we willdiscuss results, techniques, and tools that help to analyze properties of minimizer suchas symmetry and stability.

The results are based on collaboration with V. Šverák (University of Minnesota), N.Glatt-Holtz (Tulane University), G. Richards (Utah State University), and J. Whitehead(Brigham Young University).

Existence and multiplicity of self-similar solutions for heat equations withnonlinear boundary conditions

Marcelo Fernandes FURTADO


We are going to talk about self-similar solutions in the half-space for linear and semi-linear heat equations with nonlinear boundary conditions. Existence, multiplicity andpositivity of these solutions are analyzed. Self-similar profiles are obtained as solutionsof a nonlinear elliptic PDE with drift term and a nonlinear Neumman boundary condi-tion. We consider subcritical and critical case nonlinearities by employing a variationalapproach and deriving some compact weighted embeddings for the trace operator.

Joint works with Lucas C. Ferreira (UNICAMP), Everaldo S. Medeiros (UFPB) andJoão Pablo P. Silva (UFPA).

Periodic solutions and torsional instability in a nonlinear nonlocal plate equationFilippo GAZZOLA

Politecnico di Milano (Italy)

We consider a thin and narrow rectangular plate having the two short edges hingedand the two long edges free. This plate aims to model the deck of a bridge. We introducea nonlinear nonlocal evolution equation describing the deformation of the plate: weprove well-posedness and existence of periodic solutions. The natural phase space is aparticular second order Sobolev space that can be orthogonally split in two subspacescontaining, respectively, the longitudinal and the torsional movements of the plate. Wegive sufficient conditions for the stability of periodic solutions and of solutions havingonly a longitudinal component. A stability analysis of the so-called prevailing mode isalso performed. Some numerical experiments show that instabilities may occur. This isa joint work with D. Bonheure (Bruxelles) and E. Moreira dos Santos (São Paulo).


A Morse Lemma for degenerate critical points of solutions of nonlinear equationsin R2

Massimo GROSSI

Università di Roma Sapienza (Italy)

We prove a Morse Lemma for degenerate critical points of a function u which satis-fies

∆u = f (u) in B1,

where B1 is the unit ball of R2. Moreover if u admits only one critical point we provesome results about the nondegeneracy of the maximum of u and the shape of the levelsets of u.

Maximal solutions of the Liouville equation in doubly connected domainsMichal KOWALCZYK

Universidad de Chile (Chile)

In this talk we consider the Liouville equation ∆u+λ 2eu = 0 with Dirichlet bound-ary conditions in a two dimensional, doubly connected domain Ω. We show that thereexists a simple, closed curve γ ∈ Ω such that for a sequence λn → 0 there exist a se-quence of solutions uλn such that λ 2

nlog 1

λn

∫Ω

euλn dx→ c0|γ|.

Positive multipeak solutions to a zero mass problem in exterior domainsLiliane A. MAIA


We are going to present some recent results on the existence of positive multipeaksolutions to the nonlinear scalar field equation with zero mass

−∆u = f (u), u ∈ D1,20 (ΩR),

where ΩR := x∈RN : |u|>Rwith R> 0, N > 4, and the nonlinearity f is subcritical atinfinity and supercritical near the origin. We show that the number of positive multipeaksolutions becomes arbitrarily large as R→ ∞.

This is joint work with Monica Clapp, UNAM, México, and Benedetta Pellacci,Università della Campania “Luigi Vanvitelli”, Italy.

Weighted Trudinger-Moser inequalities and associated Liouville type equationsEugenio MASSA

ICMC - USP - São Carlos (Brazil)

We discuss some Trudinger–Moser inequalities with weighted Sobolev norms. Suit-able logarithmic weights in these norms allow an improvement in the maximal growthfor integrability, when one restricts to radial functions.

The main results concern the application of these inequalities to the existence ofsolutions for certain mean-field equations of Liouville-type. Sharp critical thresholdsare found such that for parameters below these thresholds the corresponding functionalsare coercive and hence solutions are obtained as global minima of these functionals. Inthe critical cases the functionals are no longer coercive and solutions may not exist.


We also discuss a limiting case, in which the allowed growth is of double exponentialtype. Surprisingly, we are able to show that in this case a local minimum persists toexist for critical and also for slightly supercritical parameters. This allows to obtainthe existence of a second (mountain-pass) solution, for almost all slightly supercriticalparameters, using the Struwe monotonicity trick. This result is in contrast to the non-weighted case, where positive solutions do not exist (in star-shaped domains) in thecritical and supercritical case.

Cylindrical optimal rearrangement problem leading to a free boundary problemwith non-local obstacle

Hayk MIKAYELYAN

University of Nottingham Ningbo (China)An optimal rearrangement problem in a cylindrical domain Ω=D×(0,1) is consid-

ered, under the constraint that the force function does not depend on the xn variable ofthe cylindrical axis. This leads to a new type of obstacle-type problem, with non-localobstacle,

∆u(x′,xn) = χv>0(x′)+χv=0(x

′)[∂νu(x′,0)+∂νu(x′,1)]

arising from minimization of the functional∫Ω

12|∇u(x)|2 +χv>0(x

′)u(x)dx,

where v(x′)=∫ 1

0 u(x′, t)dt, and ∂νu is the exterior normal derivative of u at the boundary.Several existence and regularity results are proven and it is shown that the comparisonprinciple does not hold for minimizers.

Singularity formation in critical parabolic equationsMonica MUSSO

University of Bath (England) and Universidad Católica de Chile (Chile)In this talk I will discuss some recent constructions of blow-up solutions for a Fujita

type problem for power related to the critical Sobolev exponent. Both finite type blow-up (of type II) and infinite time blow-up are considered. This research program is incollaboration with C. Cortazar, M. del Pino and J. Wei.

Radial positive solutions for a class of Neumann problems without growthconditions

Benedetta NORIS

Université de Picardie Jules Verne (France)For 1 < p < ∞, we consider the following problem

−∆pu = f (u), u > 0 in Ω, ∂νu = 0 on ∂Ω,

where ∆pu = div(|∇u|p−2∇u) and Ω ⊆ RN is either a ball or an annulus. The non-linearity f is possibly supercritical in the sense of Sobolev embeddings; in particular,our assumptions allow to include the prototype nonlinearity f (s) = −sp−1 + sq−1 forevery q > p. We prove existence and multiplicity of non-constant radial solutions andexamine their oscillatory behavior. In case f is subcritical in the sense of Sobolev em-beddings, we also prove a priori bounds of solutions. The talk is based on two papers incollaboration with A. Boscaggin and F. Colasuonno (University of Turin).


Overdetermined problems and constant mean curvature surfaces in conesFilomena PACELLA


In this talk we present some recent results about:

i) characterization of domains in cones which admit a solution of a partial overde-termined problem of Serrin type.

ii) characterization of some constant mean curvature surfaces with boundary.

As in the classical case of closed surfaces, the above questions are strictly relatedand we show that under some hypotheses on the cone or on the surface, the domainsand surfaces characterized by i) and ii) are spherical sectors and spherical caps (respec-tively). Finally, connections with a relative isoperimetric inequality in cones proved in[1] will be described.

The results have been obtained in collaboration with Giulio Tralli (University ofRoma "Sapienza", Italia).References:[1] P.L. Lions and F. Pacella, Isoperimetric inequalities for convex cone. Proc. A.M.S.109 (1990), 477–485.

Geometric regularity theory for PDEsEdgard A. PIMENTEL

PUC-Rio (Brazil)

In this talk we examine the regularity theory of the solutions to a few examples of(nonlinear) PDEs. Arguing through a genuinely geometrical method, we produce regu-larity results in Sobolev and Hölder spaces, including some borderline cases. Our tech-niques relate a problem of interest to another one - for which a richer theory is available -by means of a geometric structure, e.g., a path. Ideally, information is transported alongsuch a path, giving access to finer properties of the original equation. Our examplesinclude elliptic and parabolic fully nonlinear problems, the Isaacs equation, degenerateexamples and the porous medium equation. We close the talk with a discussion on openproblems and further directions of work.

The Lane-Emden equation on a planar domainAngela PISTOIA


I will review some old and new results concerning existence, multiplicity and asymptoticbehaviour of solutions to the classical Lane-Emden equation on a planar domain whenthe exponent of the non-linearity is large.


Periodic solutions for the one-dimensional fractional LaplacianAlexander QUAAS

Universidad Técnica Federico Santa María (Chile)

In this paper we are concerned with the construction of periodic solutions of the non-local problem (−∆)su = f (u) in R, where (−∆)s stands for the s-Laplacian, s ∈ (0,1).We introduce a suitable framework which allows to reduce the search for such solutionsto the resolution of a boundary value problem in a suitable Hilbert space, thereby mak-ing it possible to reach for the usual tools of nonlinear analysis, like bifurcation theoryor variational methods. We obtain some existence theorems which are then enlightenedwith the analysis of some examples. Joint work J. García-Melián and B. Barrios.

Regularity of semi-convex supersolutions of fully nonlinear PDEsDiego RIBEIRO MOREIRA

Universidade Federal do Ceará (Brazil)

In this talk, we discuss some recent developments on the sharp regularity of semi-convex supersolutions of fully nonlinear PDEs. The plan is to present different ap-proaches that led to these results that improve the apriori estimates obtained by Caf-farelli, Kohn, Nirenberg and Spruck in the 80’s and more recently (2006) by C. Imbert.If time allows, we plan to briefly discuss sharp and new estimates on the regularity ofthe convex envelope of supersolutions to fully nonlinear PDEs that extends the resultsof L. Caffarelli in late 80s’. This is a joint work of Alessio Figalli and J. Ederson M.Braga.

Prescribing Gaussian curvature on a compact surface and geodesic curvature onits boundaryDavid RUIZ

Universidad de Granada (Spain)

The problem of prescribing the Gaussian curvature on compact surfaces is a classicone, and dates back to the works of Berger, Moser, Kazdan & Warner, etc. Our aim is toconsider surfaces with boundary where we also prescribe the geodesic curvature of it.This gives rise to a Liouville equation under nonlinear Neumann boundary conditions.

In this talk we address the case of negative Gaussian curvature. We study the ge-ometric properties of the corresponding energy functional, and deduce the existenceof minimum or mountain pass critical points. For that, a compactness result is in or-der. Here the cancellation between the area and length terms make it possible to haveblowing-up solutions with infinite mass. This phenomenon seems to be entirely new inthe related literature.

This is joint work with Andrea Malchiodi (SNS Pisa) and Rafael López Soriano (U.Granada).


On fractional higher-order powers of the LaplacianAlberto SALDAÑA

Karlsruhe Institute for Technology (Germany)

Solutions of linear and nonlinear problems involving the Laplacian and the bilapla-cian (the Laplacian squared) exhibit very different qualitative behaviors in some situa-tions. A well-known example is the lack of general maximum principles for the bilapla-cian. As a consequence, the set of solutions in the higher-order case is much more richand complex, but our current understanding on these problems is still underdeveloped.

In this talk we focus on the following question: What happens between the Laplacianand the bilaplacian? To answer this, we study higher-order fractional powers of theLaplacian. This talk aims to be an introduction to these operators and we survey somerecently obtained results.

Uniform boundedness of positive solutions of the Lane-Emden equation indimension two

Boyan SIRAKOV

PUC-Rio (Brazil)

We prove that positive solutions of the Lane-Emden equation in a two-dimensionalsmooth bounded domain are uniformly bounded for all large exponents. A consequenceis an integral bound which implies sharp asymptotic for such solutions. Joint work withNikola Kamburov.

The nodal set of solutions of sublinear equationsNicola SOAVE


We are concerned with the nodal set of solutions to sublinear equations of the form

−∆u = λ+

(u+)q−1−λ−

(u−)q−1 in B1

where λ+,λ− > 0, q ∈ (0,2), B1 = B1(0) is the unit ball in RN , N > 2, and u+ :=maxu,0, u− := max−u,0 are the positive and the negative part of u, respectively.This class includes equations with sublinear (1 < q < 2), discontinuous (q = 1), and sin-gular (q ∈ (0,1)) nonlinearity. In both cases the right hand side is not locally Lipschitzcontinuous as function of u.

In this talk we present results regarding: (a) the validity of the unique continuationprinciple; (b) the finiteness of the vanishing order at every point and the complete char-acterization of the order spectrum; (c) a weak non- degeneracy property; (d) the partialregularity of the nodal set of any solution: the nodal set is a locally finite collection ofregular codimension one manifolds up to a residual singular set having Hausdorff di-mension at most N− 2 (locally finite when N = 2) and a partial stratification theorem.Ultimately, the main features of the nodal set are strictly related with those of the solu-tions to linear (or superlinear) equations, with two remarkable differences. First of all,the admissible vanishing orders can not exceed the critical value 2/(2− q). At thresh-old, we find a multiplicity of homogeneous solutions, yielding the non-validity of anyestimate of the (N− 1)-dimensional measure of the nodal set of a solution in terms ofthe vanishing order.


The proofs are based on Weiss-type monotonicity formulæ, blow-up arguments andthe classification of homogenous solutions.

The talk is based on joint works with Tobias Weth and Susanna Terracini.

Free boundaries in rough mediaEduardo TEIXEIRA

University of Central Florida (USA)

I will discuss recent endeavors to understand free boundary problems modeled incomplex materials.

Regularity of the optimal sets for spectral functionals and the free boundary forthe vectorial Bernoulli problem

Susanna TERRACINI

Universtità di Torino (Italy)

In this talk we deal with the regularity of optimal sets for a shape optimizationproblem involving a combination of eigenvalues, under a fixed volume constraints. Asa model problem, consider

min

λ1(Ω)+ · · ·+λk(Ω) : Ω⊆ Rd, open , |Ω|= 1,

where λi(·) denotes the eigenvalues of the Dirichlet Laplacian and | · | the d-dimensionalLebesgue measure. We prove that any minimizer Ωopt has a regular part of the topo-logical boundary which is relatively open and C1,α regular and that the singular parthas Hausdorff dimension smaller than d− d∗, where d∗ > 3 is the minimal dimensionallowing the existence of minimal conic solutions to the blow-up problem.

We examine the link between this and the problem of regularity of the free boundaryfor a vector-valued Bernoulli problem, with no sign assumptions on the boundary data.More precisely, given an open, smooth set of finite measure D ⊆ Rd , Λ > 0 and ϕi ∈H1/2(∂D), we deal with

min

k

∑i=1

∫D|∇vi|2 +Λ

∣∣∣ k⋃i=1

vi 6= 0∣∣∣ : vi = ϕi on ∂D

.

We prove that, for any optimal vector U = (u1, . . . ,uk), the free boundary ∂ (∪ki=1ui 6=

0)∩D is made of a regular part, which is relatively open and locally the graph of aC∞ function, a (one-phase) singular part, of Hausdorff dimension at most d− d∗, for ad∗ ∈ 5,6,7, and by a set of branching (two-phase) points, which is relatively closedand of finite H d−1 measure having a stratified structure itself.

These are joint works with Dario Mazzoleni and Bozhidar Velichkov.References:[1] D. Mazzoleni, S. Terracini and B. Velichkov, Regularity of the optimal sets for spec-tral functionals, Geom. Funct. Anal. 27 (2017), no. 2, 373–426[2] D. Mazzoleni, S. Terracini and B. Velichkov, Regularity of the free boundary for thevectorial Bernoulli problem, preprint 2018


Positive ground states and global geometry of differential operatorsCarlos TOMEI

PUC-Rio (Brazil)

The usual Ambrosetti-Prodi theorem relies on the positivity of the ground state ofthe free Laplacian for part of its argument. It turns out that more is obtained: for rathergeneral differential operators F , if F(u) = F(v) then necessarily u dominates v or vice-versa (said differently, a trichotomy holds: either u < v, u = v or u > v pointwise).The hypotheses hold also for the global cusps F of Ruf, and Dancer-Church-Timourian.The approach gives rise in particular to global folds associated to Markov chains, andother contexts where eigenvector positivity comes up. Joint work with M. Calanchi (U.Milano) and A. Zaccur.

Geometric tangential analysis and sharp regularity for degenerate pdesJosé Miguel URBANO

Universidade de Coimbra (Portugal)

We give an overview of a series of recent results on the sharp regularity of solutionsfor some degenerate elliptic and parabolic equations, obtained using tangential analysismethods.

Asymptotic spherical shapes in some spectral optimization problemsGianmaria VERZINI


We study the positive principal eigenvalue of a weighted problem associated withthe Neumann Laplacian. This analysis is related to the investigation of the survivalthreshold in population dynamics. When trying to minimize such eigenvalue with re-spect to the weight, one is lead to consider a shape optimization problem, which isknown to admit spherical optimal shapes only in trivial cases. We investigate whetherspherical shapes can be recovered in general situations, in some singular perturbationlimit. We also consider a related problem, where the diffusion is triggered by a frac-tional s-Laplacian, and the optimization is performed with respect to the fractional orders ∈ (0,1]. These are joint works with Dario Mazzoleni and Benedetta Pellacci.

Predator-prey model with competition, the emergence of territorialityAlessandro ZILIO

Université Paris Diderot (France)

I will present a series of works in collaboration with Henri Berestycki, dealing withsystems of predators interacting with a single prey. The system is linked to the Lotka-Volterra model of interaction with diffusion, but unlike more classic works, we are in-terested in studying the case where competition between predators is very strong: inthis context, the original domain is partitioned in different sub-territories occupied bydifferent predators. The question that we ask is under which conditions the predatorssegregate in packs and whether there is a benefit to the hostility between the packs.More specifically, we study the stationary states of the system, the stability of the solu-tions and the bifurcation diagram, and the asymptotic properties of the system when theintensity of the competition becomes infinite.


Abstracts - Poster Session

A weighted Trudinger-Moser type inequality and its applications to quasilinearelliptic problems with critical growth in the whole Euclidean space

Francisco ALBUQUERQUE

Universidade Estadual da Paraíba (Brazil)

We establish a version of the Trudinger-Moser inequality involving unbounded ordecaying radial weights in weighted Sobolev spaces. In the light of this inequality andusing a minimax procedure we also study existence of solutions for a class of quasilinearelliptic problems involving exponential critical growth.

This is a joint work with Sami Aouaoui - Institut Supérieur des Mathématiques Ap-pliquées et de l’Informatique de Kairouan (Tunisia) - to appear in Topological Methodsin Nonlinear Analysis.

Sharp regularity estimates for quasilinear evolution equationsMarcelo AMARAL

Universidade da Integração Internacional da Lusofonia Afro-brasileira (Brazil)

We establish sharp geometric C1+α regularity estimates for bounded weak solutionsof evolution equations of p-Laplacian type. Our approach is based on geometric tan-gential methods, and makes use of a systematic oscillation mechanism combined withan adjusted intrinsic scaling argument.

A class of critical Kirchhoff problem on the hyperbolic space Hn

Augusto César dos Reis COSTA

Universidade Federal do Pará (Brazil)

We investigate questions on the existence of positive solution for a class of the crit-ical Kirchhoff-type problems in Hyperbolic space. By the use of the stereographicprojection the problem becomes a singular problem on the boundary of the open ballB1(0) ⊆ Rn. The Hardy inequality, in a version due to the Brezis-Marcus, combinedwith the mountain pass theorem due to Ambrosetti-Rabinowitz are used to obtain a pos-itive solution. One of the difficulties is to find a range where the Palais Smale converges,because our equation involves a nonlocal term coming from the Kirchhoff term.


Sharp regularity estimates for quasi-linear elliptic dead core problemsJoão Vítor DA SILVA

Universidade de Buenos Aires (Argentina)

In this work we study geometric regularity estimates for quasi-linear elliptic equa-tions of p-Laplace type (1 < p < ∞) with strong absorption condition:

−div(Φ(x,u,∇u))+λ0(x)uq+(x) = 0 in Ω⊆ RN ,

where Φ : Ω×R+×RN → RN is a vector field with an appropriate p-structure, λ0 is anon-negative and bounded function and 0 6 q < p−1. Such a model permits existenceof solutions with dead core zones, i.e, a priori unknown regions where non-negativesolutions vanish identically. We establish sharp and improved Cγ regularity estimatesalong free boundary points, namely F0(u,Ω) = ∂u > 0 ∩Ω, where the regularityexponent is given explicitly by γ = p

p−1−q 1. Some weak geometric and measuretheoretical properties as non-degeneracy, uniform positive density and porosity of freeboundary are proved. The approach employed is novel even to dead core problemsgoverned by the p-Laplace operator −∆pu+λ0uqχu>0 = 0 for any λ0 > 0.

This is a joint work with A.M. Salort (CONICET/Universidad de Buenos Aires). Toappear in Calc. Var. Partial Differential Equations (2018) 57: 83.https://doi.org/10.1007/s00526-018-1344-8.

Existence of positive solutions for a class of semipositone quasilinear problemsthrough Orlicz-Sobolev space

Jefferson A. DOS SANTOS


In this paper we show the existence of weak solution for a class of semipositoneproblem of the type

−∆Φu = f (u)−a in Ω,u(x) > 0 in Ω,

u = 0 on ∂Ω,(P)

where Ω ⊆ RN , N > 3, is a smooth bounded domain, f : [0,+∞)→ R is a continuousfunction with subcritical growth, a > 0, ∆Φu stands for the Φ-Laplacian operator. Byusing variational methods, we prove the existence of solution for a small enough.

This is a joint work with Claudianor O. Alves (UAMat/UFCG) and Angelo R. F. deHolanda (UAMat/UFCG).

Ground state stability and uniqueness of normalized standing-wavesDaniele GARRISI

University of Leeds (England)

We illustrate different results of stability for the non-linear Schrödinger equationand the non-linear Klein-Gordon equation with relation to two main problems, (1) thestability of the ground-state, which is the collection of all the standing-waves minimiz-ing the energy functional E over a mass-type constraint S, (2) the stability of each of thesingle standing-waves; (1) is a consequence of the Concentration-Compactness Lemma.We describe a result of stability of ground-state for the coupled non-linear Klein-Gordonequation (D. G., Adv. Nonlinear Stud., 2012) with the aid of a symmetric rearrangement


technique (J. Byeon, J. Differential Equation, 2000); (2) seems related to the problem ofuniqueness and non-degeneracy standing-waves. For two pure-powers non-linearitiesfor the non-linear Schrödinger equation (D. G. and V. Georgiev, Discrete Contin. Dyn.Syst., 2017), such uniqueness holds, just as pure-powers non-linearities (T. Cazenaveand P. L. Lions, Comm. Math. Phys., 1982).

Existence of heteroclinic solution for a class of problems involving the fractionallaplacian

César LEDESMA

Universidad Nacional de Trujillo (Peru) and Universidade Federal de Campina deGrande (Brazil)

In this paper we study the existence of heteroclinic solution for a class of nonlocalproblem of the type

(−∆)αu+a(εx)V ′(u) = 0, x ∈ R,lim

x→−∞u(x) =−1 and lim

x→+∞u(x) = 1,

where V,a : R→ R are continuous function verifying some technical conditions. Forexample a can be asymptotically periodic and potential V can be the Ginzburg-Landaupotential, that is, V (t) = (t2−1)2.

Concave-convex structure for a class of degenerated or singular Kirchhoff typeproblems

Gustavo MADEIRA

Universidade Federal de São Carlos (Brazil)

It is established in this work the convex-concave structure introduced by Ambrosetti,Brezis and Cerami the for a class of Kirchhoff elliptic problems where nonlocal termmay be degenerated, discontinuous or singular at origin.

Dynamics for a model of flight structures with nonlocal energy dampingVando NARCISO

Universidade Estadual de Mato Grosso do Sul (Brazil)

This paper is concerned with well-posedness and long-time dynamics for a class ex-tensible beams with nonlocal Balakrishnan-Taylor energy damping. The related modeldescribes vibrations in nonlinear extensible beams arising in connection with models ofoscillation in pipes and supersonic panel flutter. Our main results feature the study ofthe nonlinear dynamical system generated by the system.


Multiple Solutions for Nonvariational Quasilinear Elliptic SystemsDenilson PEREIRA


We establish the existence of at least three nontrivial solutions for a nonvariationalquasilinear elliptic system with homogeneous Dirichlet boundary condition. Two ofthese solutions are of opposite constant sign and the third one is nodal in an appropriatesense provided that a suitable location occurs. The approach combines the methods ofsub-supersolution and Leray-Schauder topological degree.

Joint work with Dumitru Motreanu and Abdelkrim Moussaoui.

Some progress in studying elliptic problems with exponential nonlinearityAdilson PRESOTO

Universidade Federal de São Carlos (Brazil)

Since the Brezis and Benilan’s seminal work [1] on Thomas-Fermi problem, the ellipticproblems involving L1 or measure data has developed in several ways. We are interestedin understanding the phenomenon of breaking of existence solutions of

−∆u = eu +1 = µ, in Ω,

u = 0, on ∂Ω.

when the measure evaluated in singletons surpass 4πH n−2. We approximate the datumby a suitable convergent sequence in the measure sense in order to obtain a convergentsequence of solutions. The limit will be determined in terms only of datum. We alsodiscuss the cases when the data has signal and the counterpart of system.[1] PH. Bénilan; H. Brézis, Nonlinear problems related to the Thomas-Fermi equation.J. Evol. Equ. 3 (2003), no. 4, 673–770.[2] H. Brézis; M. Marcus; A. C. Ponce, Nonlinear elliptic equations with measure re-visited. In J. Bourgain, C. E. Kenig, and S. Klainerman (eds.), Mathematical aspects ofnonlinear dispersive equations. Ann. of Math Studies 163, Princeton University Press,Princeton, NJ, 2007, 55–109.[3] A. C. Ponce; A. E. Presoto, Limit solutions of the Chern-Simons equation. NonlinearAnal. 84 (2013), 91–102.

Regularity theory for nonlocal operatorsDavid STOLNICKI

PUC-Rio (Brazil)

In this poster, we put forward a brief introduction to the nonlocal diffusion operators.Our toy-model is the fractional Laplacian. We start with an excursion to the realm ofstochastic analysis, motivating the fractional Laplacian by the study of the infinitesimalgenerator of a Lévy process. Then, we present a number of properties and proceed withthe establishment of an ABP estimate and the Harnack inequality. As an application,we examine the regularity of the solutions in Hölder spaces.

This is based on the author’s M.Sc. thesis, directed by Prof. Boyan Sirakov andProf. Edgard Pimentel.


Committees and Sponsors

Scientific CommitteeDjairo Guedes de Figueiredo (Universidade Estadual de Campinas/Brazil)Maria J. Esteban (Université Paris-Dauphine/France)Paul H. Rabinowitz (University of Wisconsin/USA)Susanna Terracini (Università di Torino/Italy)

Organizing CommitteeDenis Bonheure (Université libre de Bruxelles/Belgium)Ederson Moreira dos Santos (Universidade de São Paulo/Brazil)Sergio H. Monari Soares (Universidade de São Paulo/Brazil)Hugo Tavares (Faculdade de Ciências da Universidade de Lisboa/Portugal)

SponsorsThis conference was sponsored by the Brazilian agencies CNPq and CAPES, by theInternational Mathematical Union, by ERC Advanced Grant 2013 n. 339958 “Com-plex Patterns for Strongly Interacting Dynamical Systems - COMPAT”, by the Belgiumagency FNRS, and by the Portuguese research center CMAFcIO and FCT/Portugalgrant UID/MAT/04561/2013.


Practical information

From airport to Mareiro Hotel∣∣ To go from the airport to Mareiro Hotel, Avenida

Beira Mar 2380, we suggest to take a taxi (which costs around 60 reais) or use the UberApp to call a driver (which costs around 30 reais and works very well). We inform thatthere is no bus line connecting directly the airport to the hotel’s district (Beira Mar).

Restaurants∣∣ The one in the hotel; Self-service per kilo at Hotel Brasil Tropical

(300 meters from Mareiro Hotel); Rasco Steakhouse (300 meters from Mareiro Hotel)Padaria Ideal Abolição (bread, fruits, lunch self-service per kilo, water, 350 meters fromMareiro Hotel); Didi Rei dos Mares (Sea food, 600 meters from Mareiro Hotel); CocoBambu Meireles (1,5 km from Mareiro Hotel); Coco Bambu Beira Mar (1,5 km fromMareiro Hotel); Sorveteria 50 Sabores (Ice Cream House, 600 meters from MareiroHotel).

Social Dinner∣∣ It will take place on Tuesday 24, at 20h, at Rasco Steakhouse,

Avenida Beira Mar 2500, Loja 9 (300 meters from Mareiro Hotel).

Tourism∣∣ Beaches: Praia de Iracema (1,5 km from Mareiro Hotel); Praia do Beach

Park (25 km from Mareiro Hotel). Feirinha Beira Mar (handcrafts, 200 meters fromMareiro Hotel, every evening). Mercado Central de Fortaleza (handcrafts, 3 km fromMareiro Hotel, take a taxi or a Uber driver); Emcetur (handcrafts, 500 meters from Mer-cado Central de Fortaleza); Museu da Cachaça (42 km from Mareiro Hotel, http://site.ipark.tur.br/museu-da-cachaca/); Chico Carangueijo at Praia doFuturo (Comedy show and Forró music at Thursday night; dinner and lunch; take ataxi or a Uber driver).

Maps∣∣ Map of the neighborhood of Mareiro Hotel. From left to right: 1-Padaria

Ideal Abolição (Avenida Abolição 1920), 2-Mareiro Hotel (Avenida Beira Mar 2380),3-Pizza Vignoli (Rua Silvia Jatahy 529), 4-Hotel Brasil Tropical (Avenida da Abolição2323), 5-Rasco Steakhouse (Avenida Beira Mar 2500), 6-Didi Rei dos Mares (Rua Vis-conde de Mauá 200), 7-Feirinha Beira Mar, 8-Sorveteria 50 Sabores (Avenida BeiraMar 2982), 9-Coco Bambu Meireles (Rua Canuto de Aguiar 1317), Coco Bambu BeiraMar (Avenida Beira Mar 3698). See the map on the next page.

ICM 2018 Sattelite Conference on Nonlinear PDEs Practical information - 25

http://site.ipark.tur.br/museu-da-cachaca/

http://site.ipark.tur.br/museu-da-cachaca/

26 - Practical information ICM 2018 Sattelite Conference on Nonlinear PDEs

icm 2018 sattelite conference in nonlinear...

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