| Tipo do Evento: | SEMINÁRIO |
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| Período do Evento: | 28/07/2020 a 15/12/2022 (Evento encerrado) |
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The purpose of this webinar is to maintain contact with researchers from around the world in the specific area of nonlinear analysis, partial differential equations and geometric analysis. The talks will be held mainly on Tuesdays 4p.m. (GMT -3) via Google Meet. Since 2020, we had a total of 42 talks by different researchers from all over the world. Please check our program for details and recordings of these talks. The 2022 season started on march 08. See the program for the confirmed schedule so far. NEXT TALK Yasuhito Miyamoto (The University of Tokyo - Japan) 29/012/2022 - 4p.m. local time (-3:00 GMT) Title: Structure of positive radial solutions for semilinear elliptic Dirichlet problems with general supercritical growth Abstract: We are concerned with a classification of the bifurcation diagrams of the positive solutions of semilinear elliptic Dirichlet problems with supercritical growth in a ball. In this case the bifurcation diagram is a curve parametrized by the maximum value of a solution. We study a relationship among the growth rate, the bifurcation curve and a positive radial singular solution. |
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Next talks Yasuhito Miyamoto (The University of Tokyo - Japan) 29/012/2022 - 4p.m. local time (-3:00 GMT) Title: Structure of positive radial solutions for semilinear elliptic Dirichlet problems with general supercritical growth Abstract: We are concerned with a classification of the bifurcation diagrams of the positive solutions of semilinear elliptic Dirichlet problems with supercritical growth in a ball. In this case the bifurcation diagram is a curve parametrized by the maximum value of a solution. We study a relationship among the growth rate, the bifurcation curve and a positive radial singular solution. ----------------------------------------------------------------- Past talks 28/07/2020 - Olimpio Miyagaki (UFSCAR - São Carlos - Brazil) Title: L^2-supercritical and H^{s/2}-subcritical and critical fractional NLS equations with partial confinement. Abstract: We study the existence and stability of standing waves for a L^2-supercritical and H^{s/2}-subcritical and critical NLS equations involving fractional laplacian operator with a partial confinement. This result complements or extends for nonlocal case, the existence result and stability of stand waves obtained in the papers by J. Bellazzini, N. Boussaid, L. Jeanjean and N. Visciglia, Comm. Math. Phys. 253 (2017) and in M. Ohta, Comm. Pure Appl. Anal. 17 (2018), where there were considered the local case with subcritical exponent. ----------------------------------------------------------------- 11/08/2020 - Edcarlos Domingos da Silva (UFG - Goiânia - Brazil) Title: Quasilinear elliptic problems in the whole R^N. Abstract: In this talk we shall discuss some recent results for quasilinear elliptic problems in the whole R^N. More specially, we consider quasilinear elliptic problems involving the Schrödinger equation. The nonlinear term can be superlinear or asymptotically linear at infinity. Furthermore, we consider also nonlinearities with subcritical or critical growth. The main difficult is to ensure the linking geometry for the associated energy functional taking into account an eigenvalue problem. Another difficulty is to restore the compactness required in variational methods. In order to overcome these difficulties we employ some tools such as Lion's Concentration Compactness Principle together with a fine analysis on the Palais-Smale sequences. Moreover, we also consider quasilinear elliptic systems where the coupling term can be superlinear or asymptotically linear at infinity in a suitable sense. These works are joint with Jefferson S. Silva (UFG), Maxwell L. Lizette (UFG), Marcelo F. Furtado (UnB), José Carlos de Albuquerque (UFPE). link to the seminar recording: https://youtu.be/h1QYj-iU4J4 ----------------------------------------------------------------- 18/08/2020 - Jefferson Abrantes dos Santos (UFCG - Campina Grande - Brazil) Title: A limiting free boundary problem for a degenerate operator in Orlicz-Sobolev spaces. Abstract: A free boundary optimization problem involving the phi-Laplacian in Orlicz-Sobolev spaces is considered for the case where "phi" does not satisfy the natural conditions introduced by Lieberman. A minimizer u_phi having non-degeneracy at the free boundary is proved to exist and some important consequences are established, namely, the Lipschitz regularity of u_phi along the free boundary, the locally uniform positive density of positivity set of u_phi and that the free boundary is porous with porosity r>0 and has finite (N-r)-Hausdorff measure. The method is based on a truncated minimization problem in terms of the Taylor polynomial of "phi" of order 2k. The proof demands to revisit the Lieberman's proof of a Harnack inequality and verify that the associated constant with this inequality is independent of k provided that k is sufficiently large. ----------------------------------------------------------------- 25/08/2020 - Liliane Maia (UnB - Brasília - Brazil) Title: Nonlinear elliptic systems with local and nonlocal terms. Abstract: We revisit weakly coupled nonlinear Schrödinger systems (NLS) with cubic or higher order power terms of local type and present some recent work on nonexistence, existence and asymptotic behaviour with respect to a parameter of ground state solutions for a class of systems with local and nonlocal nonlinear terms. Particularly, we illustrate with systems of this type which model Bose-Einstein condensates and Hartree-Fock-Slater methods in Physics or population dynamics in Biology. This is work in collaboration with Gaetano Siciliano (USP/BR) and Pietro D'Avenia (Politecnico di Bari/IT). ----------------------------------------------------------------- 01/09/2020 - Pedro Ubilla (USACH - Santiago - Chile) Title: Elliptic systems involving Schrodingeroperators with vanishing potentials. Abstract: Inspired by a well-known work by Brezis-Kamin (1992), we prove the existence of a bounded positive solution for some elliptic system involving Schrodinger operators. Furthermore, by imposing additional hypotheses on the nonlinearities we obtain a second solution using variational methods. In this context we consider two particular systems: one gradient-type and the other, Hamiltonian-type. This is a joint work with Denilson Pereira(UFCG) and Juan Arratia(USACH). link to the seminar recording: https://youtu.be/FL98nvxEDbo ----------------------------------------------------------------- 08/09/2020 - Gaetano Siciliano (USP - São Paulo - Brazil) Title: Existence of solutions for a singular elliptic problem. Abstract: In the talk we show a multiplicity result of positive solutions for a nonlocal elliptic equation which is singular in the nonlinearity. The problem is settled in a bounded domain with Dirichlet boundary conditions. The proof mixes elementary tools in variational methods and topological arguments. ----------------------------------------------------------------- 15/09/2020 - Leonelo Iturriaga (UTFSM - Santiago - Chile) Title: An existence result for quasilinear equations involving local conditions. Abstract: In this talk we study the existence of weak solutions of a quasilinear equation involving local conditions on the operator, as well as on the nonlinearity, only at zero. The existence result relies considering a parameter and estimates on the gradient of the eventual solutions. ----------------------------------------------------------------- 29/09/2020 - Pedro Gaspar (UChicago - Chicago - USA) Title: Soluções instáveis de menor energia para uma classe de problemas semilineares em variedades de curvatura positiva com simetrias. Abstract: A caracterização de soluções de menor energia é um problema clássico e ativo em diversos ramos de Equações Diferenciais Parciais e Análise Geométrica. Nesta palestra, discutiremos caracterizações variacionais e geométricas para soluções instáveis de menor energia para certas EDPs semilineares em variedades de curvatura de Ricci não-negativa com simetrias. Um importante exemplo é a equação de Allen-Cahn, a qual possui profundas conexões com hipersuperfícies mínimas e de curvatura média constante. Para essa equação, veremos como caracterizar completamente tais soluções em uma esfera. Veremos também como utilizar essa informação para estudar os primeiros valores críticos de uma sequência min-max para a energia associada e um problema de bifurcação. Esse é um trabalho conjunto com Rayssa Caju, Marco A.M. Guaraco e Henrik Matthiesen. ----------------------------------------------------------------- 06/10/2020 - João Pablo Pinheiro da Silva (UFPA - Belém - Brazil) Title: Existência e Multiplicidade de soluções para um problema de quarta ordem com crescimento crítico Abstract: Neste trabalho vamos estender os resultados clássicos obtimos por por Brézis e Nirenberg em seu trabalho "Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (1983)" para um operador de quarta ordem, mais especificamente mostramos que se a dimensão N for maior do que 5 e o parâmetro relacionado com a equação está entre zero e o primeiro auto-valor do operador Bi-Laplaciano com condições de Navier, então existe uma solução positiva, se o parâmetro for maior do que ou igual a esse primeiro auto valor, então não existe solução positiva e se o domínio for estrelado e o parâmetro for não positivo então não há solução positiva, mostramos também que quando N=5 e o domínio é uma bola então não há solução positiva quando o parâmetro está perto da origem. No que concerne a multiplicidade usando uma técnica introduzida por Benci e Cerami mostramos que quando o parâmetro é pequeno podemos relacionar o número de soluções positivas da equação com a topologia do domínio. ----------------------------------------------------------------- 13/10/2020 - Marcos Pimenta (UNESP - Presidente Prudente - Brazil) Title: Anisotropic 1-Laplacian problems with unbounded weights Abstract: In this work we prove the existence of nontrivial bounded variation solutions to quasilinear elliptic problems involving a weighted 1-Laplacian operator. A key feature of these problems is that weights are unbounded. One of our main tools is the well known Caffarelli-Kohn-Nirenberg’s inequality, which is established in the framework of weighted spaces of functions of bounded variation (and that provides us the necessary embeddings between weighted spaces). Additional tools are suitable variants of the Mountain Pass Theorem as well as an extension of the pairing theory by Anzellotti to this new setting. ----------------------------------------------------------------- 27/10/2020 - Feliciano Vitório (UFAL - Maceió - Brazil) Title: Prescribing the curvature of Riemannian manifolds with boundary Abstract: Let M be a compact connected surface with boundary. We prove that the signal condition given by the Gauss-Bonnet theorem is necessary and sufficient for a given smooth function f on the boundary of M (resp. in M) to be geodesic curvature of the boundary (resp. the Gauss curvature) of some flat metric in M (resp. metric in M with geodesic boundary). In order to provide analogous results for this problem with n>2 we prove some topological restrictions which imply, among other things, that any function that is negative somewhere on the boundary of M (resp. in M) is a mean curvature of a scalar flat metric in M (resp. scalar curvature of a metric in M and minimal boundary with respect to this metric). As an application of our results, we obtain a classification theorem for manifolds with boundary. This is a joint work with Tiarlos Cruz (UFAL). ----------------------------------------------------------------- 10/11/2020 - Jesse Ratzkin (Universaet der Wuerzburg - Germany) Title: Foliations of an asymptotically flat end by critical capacitors Abstract: I will explain a folitation of an asymptotically flat end of a Riemannian manifold by hypersurfaces which are critical points of a functional arising naturally in potential theory. Each of these hypersurfaces is a perturbation of a large coordinate sphere and admits a solution to an overdetermined boundary value problem. In a key step we must invert the Dirichlet-to-Neumann operator of the Laplacian. ----------------------------------------------------------------- 17/11/2020 - Erwin Topp Paredes (USACH - Chile) Title: Some results on periodic homogenization for fractional Hamilton-Jacobi equations Abstract: In this talk we present some results on periodic homogenization for nonlocal Hamilton-Jacobi-Bellman equations. We start showing homogenization for evolution problems in two cases: the first one concerns equations where the gradient dominates, and the second concerns the ``critical case", in which there is a competition among the diffusion and the transport term. We conclude the talk with a third result showing a rate of convergence in the case the diffusion dominates. link to the seminar recording: https://youtu.be/WDDRcruyZ2c ----------------------------------------------------------------- 08/12/2020 - Claudianor Alves (UFCG - Campina Grande - Brazil) Title: Super-critical Neumann problems on unbounded domains Abstract: In this paper, by making use of a new variational principle, we prove existence of nontrivial solutions for two different types of semilinear problems with Neumann boundary conditions in unbounded domains. Namely, we study elliptic equations and Hamiltonian systems on the unbounded domain $\Omega=\R^{m}\times B_r$ where $B_r$ is a ball centered at the origin with radius $r$ in $\mathbb{R}^{n}$. Our proofs consist of several new and novel ideas that can be used in broader contexts. This is a joint work with Abbas Moameni that was accepted for publication in Nonlinearity ----------------------------------------------------------------- 16/03/2021 - Gabrielle Nornberg (USP - São Carlos - Brazil) Title: Principal spectral curves for Lane-Emden fully nonlinear type systems Abstract: In this talk we discuss the phenomenon of two principal spectral curves for Lane-Emden systems with fully nonlinear structure. Joint work with Ederson Moreira dos Santos (ICMC-USP), Delia Schiera (La Sapienza Università di Roma), and Hugo Tavares (Técnico de Lisboa) ----------------------------------------------------------------- 23/03/2021 - Rayssa Caju (UFPB - João Pessoa - Brazil) Title: Existência e comportamento assintótico de soluções singulares de sistemas do tipo-Yamabe Abstract: Nosso principal objetivo é estudar sistemas de equações de Schrödinger que, do ponto de vista da geometria conforme, são extensões de equações do tipo-Yamabe fortemente acopladas. Mais especificamente, estudaremos o comportamento assintótico de soluções positivas próximo a uma singularidade isolada e, utilizando as informações obtidas nesse estudo, provaremos existência de soluções singulares utilizando métodos de colagem. Esse tipo de problema exemplifica a rica interação entre a geometria e análise assintótica. Trabalho em colaboração com João Marcos do Ó e Almir Silva Santos. ----------------------------------------------------------------- 13/04/2021 - Carlos Alberto Santos (UnB - Brasília - Brazil) Title: Multiplicity for a very-singular superlinear quasilinear problem via bifurcation theory Abstract: A p-Laplacian elliptic problem in the presence of both strong singular and (p-1)-superlinear nonlinearities is considered. We employ bifurcation theory, approximation techniques and sub-supersolution method to establish the existence of an unbounded branch of positive solutions, which is bounded in positive lambda-direction and bifurcates from infinity at lambda=0. As consequence of the bifurcation result, we determine intervals of existence, nonexistence and, in particular cases, global multiplicity. This is s joint work with Professors Laís Santos - UFV and Jacques Giacomoni - Université de Pau et des Pays de l'Adour. ----------------------------------------------------------------- 20/04/2021 - Marcos Montenegro (UFMG - Belo Horizonte - Brazil) Title: The effect of diffusions and sources on semilinear elliptic problems Abstract: This talk concerns existence of positive solution and extremal solution as well as dependence with respect to some involved parameters related to certain elliptic problems in the presence of diffusion and source terms. ----------------------------------------------------------------- 27/04/2021 - Ana Menezes (Princeton University - USA) Title: A two-piece problem for free boundary minimal surfaces in the 3-dimensional ball Abstract: In this talk we will prove that every plane passing through the origin divides an embedded compact free boundary minimal surface of the euclidean 3-ball in exactly two connected surfaces. This result gives evidence to a conjecture by Fraser and Li. This is a joint work with Vanderson Lima from UFRGS. ----------------------------------------------------------------- 04/05/2021 - Rafayel Teymurazyan (University of Coimbra - Portugal) Title: Fully nonlinear dead-core systems Abstract: We study fully nonlinear dead-core systems coupled with strong absorption terms. We discover a chain reaction, exploiting properties of an equation along the system. The lack of both the classical Perron's method and comparison principle for the systems requires new tools for tackling the problem. By means of a fixed point argument, we prove existence of solutions, and obtain higher sharp regularity across the free boundary. Additionally, we prove a variant of a weak comparison principle and derive several geometric measure estimates for the free boundary, as well as a Liouville type theorems for entire solutions. These results are new even for linear dead-core systems. This is a joint work with D.J. Araújo ----------------------------------------------------------------- 18/05/2021 - Almir Santos (UFS - Aracaju - Brazil) Title: Prescribing Curvature of Metrics in Manifolds with Boundary Abstract: In this talk we aim to discuss the problem of prescribe curvature of metrics in manifolds with boundary. We start to talk about the well known Yamabe problem. We are interesting to investigate the influence of the volume and area of the boundary on the scalar and mean curvature. We will show how to use the Yamabe invariant, in the boundary setting, to study a type of Kazdan-Warner-Kobayashi problem in a compact manifold with boundary. This is a joint work with Tiarlos Cruz (UFAL). ----------------------------------------------------------------- 25/05/2021 - Mateus Sousa (BCAM - Spain) Title: Some sharp inequalities in Fourier restriction theory Abstract: In this talk we will discuss some sharp inequalities in Fourier restriction theory, their connections with partial differential equations, some open problems and also some recent developments. The talk is meant for a broad audience in analysis. ----------------------------------------------------------------- 01/06/2021 - Lucas Catão Ferreira (UNICAMP - Campinas - Brazil) Title: On nonhomogeneous elliptic problems in the half-space with nonlinear and singular boundary conditions Abstract: We consider a class of elliptic problems in the half-space $\mathbb{R}^{n}_+$ with nonhomogeneous boundary conditions containing nonlinearities and critical singular potentials. We obtain existence and regularity results by means of a harmonic analysis approach based on a framework of weighted spaces in Fourier variables. This framework seems to be new in the context of elliptic boundary value problems and allows us to consider Hardy's potential $\lambda_{1} /|x|^{2}$ in $\mathbb{R}^{n}_+$ and Kato's potential $\lambda_{2}/|x^{\prime }|$ on the boundary $\partial\mathbb{R}^{n}_+$, as well as their versions with multiple poles, without using the so-called Kato and Hardy inequalities. Singular boundary forcing terms can also be addressed. Moreover, our results cover supercritical nonlinearities, such as $\pm u^p$ in $\mathbb{R}^{n}_+$ and $\pm u^q$ on $\partial\mathbb{R}^{n}_+$ with integers $p>2^{*}-1$ and $q>2_{*}-1$. Joint work with Nestor F. Castaneda-Centurion (UESC, BR) ----------------------------------------------------------------- 08/06/2021 - Marcelo Furtado (UnB - Brasília - Brazil) Title: Multiplicity of solutions for a nonlinear boundary value problem in the upper half-space Abstract: We obtain multiple solutions for the nonlinear boundary value problem $$ -\Delta u-\dfrac{1}{2}\left( x\cdot\nabla u\right) = f(\lambda,x,u), \mbox{ in }\mathbb{R}_{+}^{N}, \qquad \dfrac{\partial u}{\partial\nu}=g(\mu,x',u), \mbox{ on } \partial \mathbb{R}_{+}^{N}, $$ where $\mathbb{R}^N_+$ is the upper half-space and $\lambda, \mu>0$ are parameters. We consider sublinear, linear and superlinear cases and the function g has critical growth. The talk is based in some works done in collaboration with Karla Sousa (UnB) and João Pablo Silva (UFPA). ----------------------------------------------------------------- 22/06/2021 - Maria Andrade (UFS - Brazil) Title: Resultados de classificação para superfícies CMC de bordo livre Abstract: Em 1995, A. Ros e E. Vergasta provaram que uma superfície Sigma compacta, orientável, estável com bordo livre na bola fechada B do R3 deve ser um equador planar, uma calota esférica ou uma superfície de gênero 1 com no máximo duas componentes no bordo. I. Nunes, usando um balanceamento modificado do tipo Hersch, provou que Sigma não pode ter gênero 1. Podemos citar ainda trabalhos de classificação via existência de uma função convexa como por exemplo o de L. Ambrozio e I. Nunes no caso de superfície mínima em B ou ainda, E. Barbosa, M. Cavalcante e E. Pereira no caso de superfície CMC em B. Consideramos M=({B}^3_r,\bar{g}) como a bola Euclidiana de dimensão 3 com raio r, equipada com uma métrica \bar{g}=e^{2h}< , > conforme à métrica Euclidiana, onde a função h=h(x) depende somente da distância de x ao centro da bola {B}^3_r. Mostramos que se uma superfície CMC Sigma em M satisfaz uma condição de pinching do comprimento do tensor da segunda forma sem traço, o qual envolve a função de suporte de Sigma, o campo vetorial conforme e sua função potencial, então Sigma é um disco ou um anel rotacionalmente simétrico. Este é um trabalho em parceria com Ezequiel Barbosa (UFMG) e Edno Pereira (UFMG). ----------------------------------------------------------------- 29/06/2021 - Diego Marcon (UFRGS - Brazil) Title: A general obstacle problem in American option pricing: regularity of solutions Abstract: In this talk, we consider a general parabolic obstacle problem that naturally arises in American options pricing. We introduce obstacle problems in somewhat elementary words and then proceed to present our model in some detail. The parabolic operator that shows up is a combination of nonlocal diffusion terms and a drift term. We prove optimal regularity of solutions in space and almost optimal regularity in time. This is a joint work with Henrique Borrin (UNICAMP). link to the seminar recording: https://youtu.be/cuY3nffPuHI ----------------------------------------------------------------- 06/07/2021 - Levi Lopes de Lima (UFC - Brazil) Title: Operadores elípticos em variedades cônicas: exemplos e aplicações Abstract: Revisaremos a teoria de operadores elípticos em variedades com singularidades cônicas isoladas (como desenvolvida por Schulze, Lesch, Schrohe, entre outros), com ênfase naqueles operadores que frequentemente aparecem em aplicações geométricas (Laplaciano, Dirac, etc.) Em seguida, discutiremos algumas aplicações deste formalismo, incluindo um resultado de prescrição da curvatura escalar em tais variedades (baseado em arXiv:2104.13882) link to the seminar recording: https://youtu.be/b3ggndFsENY ----------------------------------------------------------------- 20/07/2021 - Hugo Tavares (Inst. Superior Técnico - Portugal) Title: Free boundary problems with long-range interaction Abstract: In this talk we consider a class of variational shape optimisation problems for densities that repel each other at a certain distance. Typical examples are given by the Dirichlet functional and the Rayleigh functional minimised in the class of functions that attain some H^1 boundary conditions, subject to the constraint that the supports of different densities are at a certain fixed distance. We show a connection with solutions to variational elliptic systems with nonlocal competing interactions, investigate the optimal regularity of the solutions (and prove uniform estimates with respect to the distance parameter), prove a free-boundary condition and derive some preliminary results characterising the free boundary. The talk is based in the following works: [1] N. Soave, H. Tavares, S. Terracini, A. Zilio, Variational problems with long-range interaction, Arch. Rational Mech. Anal.228(2018), 743–772. [2]Nicola Soave, Hugo Tavares and Alessandro Zilio, Free boundary problems with long-range interactions: uniform Lipschitz estimates in the radius, arXiv:2106.03661. link to the seminar recording: https://youtu.be/6MZyEtVyxuY ----------------------------------------------------------------- 27/07/2021 - David Costa (University of Nevada Las Vegas - EUA) Title: Multiple Solutions for Heterogeneous Semilinear Equations in Exterior Domains of R^2 Abstract: We prove some multiplicity results, in the Beppo-Levi space, for solutions of a class of semilinear boundary value problems defined in exterior domains of the two-dimensional Euclidean space. We deal with the existence of extremal constant-sign solutions and nodal solutions for this problem. The main tools in the arguments developed in this paper are the Kelvin transform, the duality method, the Brezis-Nirenberg theorem, and related variational tools. link to the seminar recording: https://youtu.be/1oZqd0YBkNI ----------------------------------------------------------------- 17/08/2021 - Ederson Santos (USP-São Carlos - Brazil) Title: Unique continuation principles for systems Abstract: In this talk I will discuss on some recent unique continuation results for systems of elliptic partial differential equations satisfying a suitable superlinearity condition. As an application, we obtain nonexistence of nontrivial (not necessarily positive) radial solutions, in the critical and supercritical regimes, for the Lane-Emden posed in a ball. Some of our results also apply to general fully nonlinear operators, such as Pucci's extremal operators, being new even for scalar equations. This is a joint work with Gabrielle Nornberg and Nicola Soave. ----------------------------------------------------------------- 17/08/2021 - Boyan Sirakov (PUC - Rio - Brazil) Title: Regularidade global para soluções fracas de EDPs elípticas degeneradas. Abstract: Vamos apresentar um panorama geral da teoria da regularidade global para soluções fracas de viscosidade de EDP elípticas, tanto uniformemente elípticas quanto com elipticidade degenerada, desenvolvida durante as últimas décadas. Vamos detalhar alguns resultados recentes para equações degeneradas na forma não divergente. link to the seminar recording: https://youtu.be/IeR4jv2q0ic ----------------------------------------------------------------- 24/08/2021 - Mónica Clapp (Universidad Nacional Autónoma de México) Title: Optimal partitions for the Yamabe equation Abstract: The Yamabe equation on a Riemannian manifold (M,g) is of relevance in differential geometry. A positive solution to it gives rise to a metric on M which has constant scalar curvature and is conformally equivalent to the given metric g. An optimal l-partition for the Yamabe equation is a cover of M by l pairwise disjoint open subsets such that the Yamabe equation with Dirichlet boundary condition has a least energy solution on each one of these sets, and the sum of the energies of these solutions is minimal. We will present some recent results obtained in collaboration with Angela Pistoia (La Sapienza Università di Roma) and Hugo Tavares (Universidade de Lisboa) that establish the existence and qualitative properties of such partitions. We will also present some results on symmetric optimal partitions obtained in collaboration with Angela Pistoia, and with Alberto Saldaña (UNAM) and Andrzej Szulkin (Stockholm University). ----------------------------------------------------------------- 31/08/2021 - João Henrique Andrade (USP - São Paulo - Brazil) Title: Compactness within the space of complete, constant Q-curvature metrics on the sphere with isolated singularities Abstract: This talk addresses the moduli space of complete, conformally flat metrics metrics on a sphere with k punctures having constant positive Q-curvature and positive scalar curvature. Previous work has shown that such metrics admit an asymptotic expansion near each puncture, allowing one to define an asymptotic necksize of each singular point. We prove that any set in the moduli space such that the distances between distinct punctures and the asymptotic necksizes all remain bounded away from zero is sequentially compact, mirroring a theorem of D. Pollack about singular Yamabe metrics. Along the way, we define a radial Pohozaev invariant at each puncture and refine some a priori bounds of the conformal factor, which may be of independent interest. ----------------------------------------------------------------- 14/09/2021 - Evelina Shamarova (UFPB - João Pessoa - Brazil) Title: Singular solutions to k-Hessian equations with fast-growing nonlinearities Abstract: In this talk, we deal with a Dirichlet problem for a k-Hessian equation with a quite general fast-growing nonlinearity on a unit ball. We prove the existence of a radial singular solution and obtain its exact asymptotic behavior in a neighborhood of the origin. We furthermore demonstrate that this specific singular solution allows to study properties of radial regular solutions such as bifurcation diagrams and multiplicity, where an important tool is the number of intersection points between the singular and regular solutions for rescaled problems. In the particular case of the exponential nonlinearity, we obtain the convergence of regular solutions to the singular. This talk is based on a joint work with João Marcos do Ó and Esteban da Silva. link to the seminar recording: https://youtu.be/mG_6phpi-cA ----------------------------------------------------------------- 28/09/2021 - Ederson Braga (UFC - Fortaleza - Brazil) Title: Sobre o Teorema de Regularidade de Fronteira de Krylov Abstract: Nesta palestra apresentamos alguns avanços sobre o Teorema de Regularidade 𝐶1,𝛼 de fronteira devido a N. Krylov originalmente provado na década de 80. Os últimos avanços que tenho conhecimento datam de 2020. A fim de apresentar resultados, demonstrações e técnicas usadas neste tipo de resultado, tomamos um caso particular coberto pelo Teorema de Krylov para expô-lo de modo formal. This webinar was recorded and will be available soon ----------------------------------------------------------------- 05/10/2021 - Ezequiel Barbosa (UFMG - Belo Horizonte - Brazil) Title: On non-compact free boundary minimal hypersurfaces in the Riemannian Schwarzschild Spaces Abstract: In this talk we will discuss on the existence and the Morse index of an unbounded free boundary minimal hypersurface of the Riemannnian Schwarzschild This webinar was recorded and will be available soon ----------------------------------------------------------------- 12/10/2021 - Stefano Nardulli (UFABC - São Paulo - Brazil) Title: On non-compact free boundary minimal hypersurfaces in the Riemannian Schwarzschild Spaces Abstract: In this paper we want to give a result on the existence and the number of solutions of the Van der Waals-Cahn-Hilliard two phase transition equation with a small volume constraint inside a compact Riemannian manifold M^n without boundary. We assume that the double well potential is regular enough and satisfies suitable growth conditions at infinity to apply the abstract topological method of photography for small temperatures close to the 0º Kelvin and small enclosed volume. Finally we send the temperature parameter to zero for a fixed small enclosed volume to find a lower bound on the number of solutions that depends just on topological invariants of the manifold M. This is a joint work with Vieri Benci (Univ. Pisa), Luis Eduardo Osorio Acevedo (IME-USP), Paolo Piccione (IME-USP). This webinar was recorded and will be available soon ----------------------------------------------------------------- 19/10/2021 - Tiarlos Cruz (UFAL - Alagoas - Brazil) Title: Deformations and prescriptions of scalar curvature and mean curvature Abstract: I will talk about a map involving the scalar curvature and mean curvature on the space of Riemannian metrics of a compact manifold with non empty boundary. We will include some prescribing curvature results. Moreover, I will discuss this map under the effect of the volume and the area of the boundary through two approaches: on the one hand, we vary the volume or area of the boundary and, on the other hand, we restrict it to metrics of unit volume or unit area of the boundary. This talk is based on joint works with Almir Santos (UFS) and Feliciano Vitório (UFAL). This webinar was recorded and will be available soon ----------------------------------------------------------------- 09/11/2021 - Anna Maria Candela (Università degli Studi di Bari - ITA) Title: A quasilinear modified Schrodinger equation: from bounded domains to the Euclidean space This webinar was recorded and will be available soon ----------------------------------------------------------------- 23/11/2021 - José Nazareno Gomes (UFSCAR -São Carlos - Brazil) Title: Eigenvalue estimates of the drifted Cheng-Yau operator on bounded domains in pinched Cartan-Hadamard manifolds Abstract: In this talk, we will see how a Bochner type formula can be used to establish universal inequalities for the eigenvalues of the drifted Cheng-Yau operator on a bounded domain in a pinched Cartan-Hadamard manifold with the Dirichlet boundary condition. In the first theorem, the hyperbolic space case is treated in an independent way. For the more general setting, we first establish a Rauch comparison theorem for the Cheng-Yau operator and two estimates associated with the Bochner type formula for this operator. Next, we get some integral estimates of independent interest. As an application, we compute our universal inequalities. In particular, we obtain the corresponding inequalities for both Cheng-Yau operator and drifted Laplacian cases, and we recover the known inequalities for the Laplacian case. We also obtain a rigidity result for a Cheng-Yau operator on a class of bounded annular domains in a pinched Cartan-Hadamard manifold. In particular, we can use, e.g., the potential function of the Gaussian shrinking soliton to obtain such a rigidity for the Euclidean space case. This is a joint work with professor Júlio C. M. da Fonseca of the Centro de Estudos Superiores de Parintins, Universidade do Estado do Amazonas, Parintins, Amazonas, Brazil. ----------------------------------------------------------------- 30/11/2021 - Pierluigi Benevieri (USP -São Paulo - Brazil) Title: Periodic positive solutions for delay equations and the use of the topological degree Abstract: We present a brief summary of the literature concerning periodic positive solutions for differential equations and, then, a joint work with with Pablo Amster (Universidad de Buenos Aires, Argentina) and Julián Haddad (Universidade Federal de Minas Gerais, Brazil), which deals with an extension to delay problems of some recent results by G. Feltrin and F. Zanolin. They obtained existence and multiplicity results for positive periodic solutions to nonlinear e nondelayed differential equations of second order with periodic or Newmann boundary conditions and special assumptions on nonlinear part of the equations. We studied analogous problems with dependence of a constant delay. Our approach, as well as that of Feltrin and Zanolin, is topological and based on the coincidence degree introduced by J. Mawhin. This webinar was recorded and will be available soon ----------------------------------------------------------------- 08/03/2022 - Kanishka Perera (Florida Institute of Technology - USA) Title: New multiplicity results for critical p-Laplacian problems Abstract: We prove new multiplicity results for the Brezis-Nirenberg problem for the p-Laplacian. Our proofs are based on a new abstract critical point theorem involving the Z_2-cohomological index that requires less compactness than the (PS) condition. This webinar was recorded and will be available soon ----------------------------------------------------------------- 22/03/2022 - Igor Verbitsky (University of Missouri System - USA) Title: Some classes of solutions of quasilinear elliptic equations of the p-Laplace type Abstract: We consider positive solutions to quasilinear elliptic equations of the p-Laplace type with sub-natural growth terms, and coefficients and data which are nonnegative measurable functions (or locally finite Radon measures). General solutions, as well as certain classes of solutions, including BMO solutions, will be treated. We will discuss bilateral pointwise estimates of solutions, along with sharp criteria for their existence, based on nonlinear potential theory. Our results hold for more general quasilinear operators, as well as for the fractional Laplace operators. ----------------------------------------------------------------- 29/03/2022 - Otared Kavian (Université Paris–Saclay (site de Versailles) - France) Title: Remarks on the existence of equilibrium in some parabolic evolution equations Abstract: We consider a few examples of parabolic evolution equations in the whole space, linear or nonlinear, for which one can show the existence of a positive equilibrium and then one may prove convergence of positive solutions to a multiple of the equilibrium. We consider a (weekly) nonlinear mutation selection model, known as replicator-mutator equation in evolutionary biology. These models involve a nonlocal mutation kernel and a confining fitness potential. We prove that the long time behaviour of the Cauchy problem is determined by the underlying principal eigenelement of the underlying linear operator. To do so, through a minimization problem under constraints, we prove first the existence of such an eigenlement. Then we first analyze the linear evolution problem through the proof of the existence of a spectral gap, by taking advantage of the theory of strongly continuous semigroups of positive operators. We conclude with the analysis of the nonlinear problem. ----------------------------------------------------------------- 12/04/2022 - Leonardo Francisco Cavenaghi (University of Fribourg - Switzerland) Title: Old and new connections between elliptic PDE's and symmetries Abstract: It is classical nowadays that the solution to the Yamabe problem took a difficult path due to the lack of compact embeddings of Sobolev spaces for critical exponents. Not that well known, however, are the results firstly appearing in the now classical work of E. Hebey and M. Vaugon, where the presence of symmetries on Riemannian manifolds make it way easier to deal with not only the Yamabe problem, but other critical elliptic PDE's via standard variational methods. In this talk we explore this old feature to show how to completely solve the Kazdan--Warner under the presence of symmetries induced both by group actions and by singular Riemannian foliations (SRF). These results were obtained in collaboration to Prof. João Marcos do Ó and Prof. Llohann Sperança, in the group action case, and to Prof. Marcos Alexandrino in the SRF setting. Instead of focusing on the geometric consequences of these results, we shall mostly emphasize how the same setup can be applied to other problems, such as in the study of optimal constants on Sobolev embeddings, assuming symmetries hypothesis. We also intend to provide a glance about the plausibility of some old conjectures under these symmetric assumptions. ----------------------------------------------------------------- 26/04/2022 - Jianjun Zhang (Chongqing Jiaotong University - China) Title: Another look at planar Schrodinger-Newton systems Abstract: In this talk, we focus on the existence of positive solutions to a planar Schrodinger-Newton system with subcritical or critical growth. A new variational approach is established and enables us to study such problem in the Sobolev space as usual. The analysis developed in this paper also allows to investigate the relation between a Schrodinger-Newton system of Riesz-type and a Schrodinger-Poisson system of logarithmic-type. Furthermore, this new approach can provide a new look at the planar Schrodinger-Newton system and may have some potential applications in various related problems. This is a joint work with Zhisu Liu, Vicent iu D. Radulescu and Chunlei Tang and one with Zhisu Liu, Vicent iu D. Radulescu. ----------------------------------------------------------------- 10/05/2022 - Kaye Silva (UFG - Brazil) Title: The fibering method applied to the level sets of a family of functionals Abstract: Given an one-parameter family of $C^1$-functionals, $\Phi_\mu:X\to \mathbb{R}$, defined on an uniformly convex Banach space $X$, we describe a method that permit us find critical points of $\Phi_\mu$ at some energy level $c\in \mathbb{R}$. In fact, we show the existence of a sequence $\mu(n,c)$, $n\in \mathbb{N}$, such that $\Phi_{\mu(n,c)}$ has a critical level at $c\in \mathbb{R}$, for all $n\in \mathbb{N}$. Moreover, we show some good properties of the curves $\mu(n,c)$, with respect to $c$ (for example, they are Lipschitz), and as a consequence of this analysis, we recover many know results on the literature concerning bifurcations of elliptic partial differential equations. Furthermore we prove new results for a large class of elliptic partial differential equations, which includes, for example, Ouyang, Lane-Enden, Concave-Convex, Kirchhoff and Schrodinger-Bopp-Podolsky type equations. 17/05/2022 - João Henrique de Andrade (USP - Brazil) Title: Multiplicity of solutions to the multiphasic Allen-Cahn-Hilliard system with a small volume constraint on closed parallelizable manifolds Abstract: We prove the existence of multiple solutions to the Allen--Cahn--Hilliard (ACH) vectorial equation involving a multi-well (multiphasic) potential with a small volume constraint on a closed parallelizable Riemannian manifold. More precisely, we find a lower bound for the number of solutions depending on some topological invariants of the underlying manifold. The phase transition potential is considered to have a finite set of global minima, where it also vanishes, and a subcritical growth at infinity. Our strategy is to employ the Lusternik--Schnirelmann and infinite-dimensional Morse theories for the vectorial energy functional. To this end, we exploit that the associated ACH energy $\Gamma$-converges to the vectorial perimeter for clusters, which combined with some deep theorems from isoperimetric theory yields the suitable set up to apply the photography method. Along the way, the lack of a closed analytic expression for the multi-isoperimetric function for clusters imposes a delicate issue. Furthermore, using a transversality theorem, we also show the genericity of the set of metrics for which solutions to our geometric system are nondegenerate. ----------------------------------------------------------------- 31/05/2022 - José Francisco de Oliveira (UFPI - Brazil) Title: On the equivalence of critical and subcritical Trudinger-Moser type inequalities and theirs extremals Abstract: In this talk, we discuss the equivalence for critical and subcritical non-integer dimensional inequalities of Trudinger-Moser type on the entire space. We will show how to get asymptotic lower and upper bounds for the subcritical Trudinger-Moser supremum and how to employ it to obtain the equivalence for critical and subcritical inequalities. In addition, we show how to transform subcritical extremal functions into critical extremal functions. As a by-product of our development, we explicitly calculate the value of the critical supremum in some special situations. ----------------------------------------------------------------- 07/06/2022 - Guozhen Lu (University of Connecticut - USA) 4p.m. local time (-3:00 GMT) Title: Helgason-Fourier analysis and sharp geometric and functional inequalities Abstract: In recent years, we have developed an approach to establish sharp geometric and functional inequalities using the Helgason-Fourier analysis and Kunze-Stein phenomenon on symmetric spaces. Such inequalities include sharp Hardy-Sobolev-Maz'ya and Hardy-Adams inequalities on hyperbolic spaces on all Riemannian symmetric spaces of noncompact type of rank one. Factorization theorems on complex hyperbolic balls and Siegel domains involving CR invariant operators and precise expressions of Green's functions of GJMS operators and their extensions on hyperbolic spaces are of particular interest. As an application, the existence, non-existence and symmetry of solutions to the higher order Brezis-Nirenberg problem on hyperbolic spaces can be established. This is based on a series of joint works with Joshua Flynn, Jungang Li, and Qiaohua Yang. ----------------------------------------------------------------- 09/08/2022 - Levi Lopes de Lima (UFC - Brazil) 4p.m. local time (-3:00 GMT) Title: Prescrevendo a curvatura escalar em espaços com singularidades do tipo "aresta". Abstract: Discutiremos o problema da prescrição da curvatura escalar para espaços com singularidades do tipo "aresta", com ênfase em extensões dos resultados obtidos em arXiv:2104.13882 (AGAG, 2022), onde o caso de singularidades cônicas isoladas é tratado. ----------------------------------------------------------------- 16/08/2022 - Grey Ercole (UFMG - Brazil) Title: Uniform convergence of a family of solutions to problems driven by rapidly growing operators Abstract: We consider a family of Dirichlet problems driven by the equation $-\Phi_p$-Laplacian = f(u) in an N-dimensional, bounded domain $\Omega$ with p>N. Our hypotheses on the N-function $\Phi_p$ allow the operator to be of rapidly growing type so that the corresponding Orlicz-Sobolev space might be non-reflexive. Our goal is to obtain a family (u_p) of nonnegative solutions converging uniformly over the closure of $\Omega$ to the distance function to the boundary as p goes to infinity. For this we impose two simple conditions on the asymptotic behavior of $\Phi_{p}(1)$ as p tends to infinity and consider the following two cases regarding the continuous nonlinearity (M denotes the maximum of the distance function to the boundary): either f has polynomial growth at infinity and its primitive is strictly increasing over [0,M]; or f is strictly positive over [0,M] (which includes Gelfand-type problems). ----------------------------------------------------------------- 13/09/2022 - Julio Rossi (Universidad de Buenos Aires - Argentina) Title: Coupling and mixing local and nonlocal equations Abstract: In this talk we present a simple way of coupling a local and a nonlocal evolution equation in such a way that the usual properties (like existence and uniqueness of solutions, conservation of the total mass, etc) are satisfied. Moreover, we study the limit as we homogenize this setting mixing the regions in which local and nonlocal operators act. (Based on joint works with A. Garriz (Madrid) and F. Quiros (Madrid) and with M. Capanna (L'Aquila)) ----------------------------------------------------------------- 20/09/2022 - Jurandir Ceccon (UFPR - Paraná - Brazil) Title: General entropy inequalities on closed manifolds Abstract: In 2004, Del Pino and Dolbeault and Gentil investigated, independently, best constants and extremals associated to sharp Euclidean entropy inequalities. In this work, we present some important advances in the Riemannian context. Namely, be a compact Riemannian manifold of dimension larger 1. We prove that the sharp Riemannian entropy inequality holds on all functions in the Sobolev space. Moreover, we show that the first best Riemannian constant is equal to the corresponding Euclidean one. Our approach is inspired on the Bakry, Coulhon, Ledoux and Sallof-Coste's idea of getting Euclidean entropy inequalities as a limit case of suitable Gagliardo-Nirenberg inequalities. ----------------------------------------------------------------- 27/09/2022 - Hamilton Bueno (UFMG - Minas Gerais - Brazil) Title: Sobre a convergência do operador relativístico de Schrödinger fracionário Abstract: Um resultado importante sobre o operador Laplaciano fracionário é seu comportamento quando s tende a 1. Se consultarmos o famoso Hitchhiker's guide (Prop. 4.4), veremos que, quando s tende a 1, então o operador Laplaciano fracionário tende ao operador Laplaciano. Esse resultado gera uma segunda pergunta. Soluções de um problema do Laplaciano fracionário com não linearidade f tendem, quando s tende a 1, a uma solução mesmo problema para o Laplaciano? Tanto quanto saibamos, essa segunda pergunta não tem uma resposta positiva até hoje. Em nosso trabalho, lidamos com o operador relativístico de Schrödinger fracionário, que consiste em somar o termo m² ao operador Laplaciano e então passar para a forma fracionária (veja Lieb & Loss, Analysis, capítulo 7, que trata do caso s=1/2). As duas perguntas acima são tratadas neste artigo e respondemos ambas positivamente. A prova de ambos os resultados depende fortemente de resultados da Análise Harmônica. Esse trabalho foi feito em conjunto com V. Ambrosio (Ancona, Itália), Aldo Medeiros (Univ. Federal de Viçosa, MG) e Gilberto Pereira (Univ. Federal de Ouro Preto) |
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| Webinar via Google Meet. Each webinar has its own link. | |
Anyone is invited to attend the meetings.
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| João Marcos do Ó (coordinator - organizer): jmbo@academico.ufpb.br Bruno Ribeiro (organizer): bhcr@academico.ufpb.br
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