Alexandre Nolasco de Carvalho (ICMC-USP)
Title: Structural stability of uniform attractors: topological and geometrical
Abstract: We present a careful description of the relationship between pullback and uniform attractors, leading to a detailed description of the uniform attractor and providing the understanding of its dynamical structures. That will enable us to talk about upper and lower semicontinuity, topological and geometrical structural stability of uniform attractors, at least for a non-autonomous perturbation of a semigroup.
Athanasios E. Tzavaras (KAUST - Saudi Arabia)
Title: Singular Limiting Induced from Continuum Solutions and the Problem of Dynamic Cavitation
Abstract (Minicourse): In the works of Pericak-Spector and Spector (Arch Rational Mech Anal. 101:293-317, 1988, Proc. Royal Soc. Edinburgh Sect A 127:837-857, 1997) a class of self-similar solutions are constructed for the equations of radial isotropic elastodynamics that describe cavitating solutions. Cavitating solutions decrease the total mechanical energy and provide a striking example of non-uniqueness of entropy weak solutions (for polyconvex energies) due to point-singularities at the cavity. To resolve this paradox, we introduce the concept of singular limiting induced from continuum solution (or slic-solution), according to which a discontinuous motion is a slic-solution if its averages form a family of smooth approximate solutions to the problem. It turns out that there is an energetic cost for creating the cavity, which is captured by the notion of slic-solution but neglected by the usual entropic weak solutions. Once this cost is accounted for, the total mechanical energy of the cavitating solution is in fact larger than that of the homogeneously deformed state. We also apply the notion of slic-solutions to a one-dimensional example describing the onset of fracture, and to gas dynamics in Langrangean coordinates with Riemann data inducing vacuum in the wave fan.
Boyan Sirakov (PUC-Rio)
Title: A priori bounds for elliptic inequalities via regularity estimates
Abstract: We show how basic estimates from elliptic regularity theory,such as growth lemmas and half-Harnack inequalities, can be used to obtain new and optimal a priori bounds for positive sub- and super-solutions of nonlinear elliptic equations.
We prove new boundary versions of these regularity estimates, which play an important role in the proofs of the a priori bounds, and are of importance in themselves.
We apply the a priori bounds in order to study the existence and multiplicity of solutions of the Dirichlet problem for a general class of elliptic operators in which the first and the second order terms have the same scaling with respect to dilations.
Daniel Pellegrino (UFPB)
Title: Applications of the Hölder inequality for mixed sums in Functional Analysis
Abstract: The Hölder inequality for mixed Lp spaces seems to be well known in PDEs but, curiously, it seems to have been overlooked in Functional Analysis for a long time and just very recently this beautiful inequality was rediscovered in this field. In this talk we present some applications of this inequality in Functional Analysis: improvement of the constants of the Bohnenblust-Hille and Hardy-Littlewood inequalities and the final solution to the Bohr radius problem.
Djairo Guedes de Figueiredo (UNICAMP)
Title: The role of symmetry in changing critical exponents.
Abstract: Nonlinear equations and systems in domains with symmetry have the usual critical exponents changed.
Eduardo Teixeira (UFC)
Title: Geometric regularity estimates for degenerate elliptic problems
Abstract: The study of smoothness properties for weak solutions of degenerate elliptic operators has been a major line of research since the foundation of the modern theory of PDEs in the 19th century. Most of the well known results pertaining to this line of research are of qualitative nature, i.e., solutions (or their gradients or even their hessians) are of class \(C^{0,\alpha}\) for some universal, but a priori unknown exponent \(0<\alpha<1\). It has been observed, however, that finding the precise sharp, optimal regularity estimate available for a given elliptic operator, not only reveals important analytic and geometric information about the problem, but actually has a decisive role in finer geometric-measure analysis of the model. In this talk I will discuss some contemporary developments on this line of research and, in particular, I'll report our recent solution (jointly with Araújo and Urbano) of the \(C^{p'}\)-regularity conjecture in 2D, which regards the best regularity possible for a function whose \(p\)-laplacian is bounded.
Fágner Dias Araruna (UFC)
Title: Control and stability to some beams and plates systems
Abstract: In this talk, I will discuss recent results on asymptotic properties (when the modulus of elasticity in torsion tends to infinity), controllability and stability to Mindlin-Timoshenko systems, which describe vibration of beams and plates, in their semilinear and nonlinear formulations.
Héctor Sánchez-Morgado (México)
Title: Hamilton Jacobi equations for periodic Hamiltonians
Abstract: We consider space-time 1-periodic Hamiltonians under the assumption that the Aubry set consists in a finite number of hyperbolic periodic orbits and study two limit behaviours.
1. The long time limit for the Cauchy problem: We prove that the viscosity solution converges exponentially to an N-time-periodic solution of the Hamilton Jacobi equation.
2. The viscosity limit of the viscous Hamilton Jacobi equation: We identify the limit set and under a further hypothesis, the uniqueness of the limit.
Hugo Tavares (IST - Portugal)
Click here for the slides of the lectures
Title: Asymptotic Study of Nonlinear Elliptic Systems with Strong Competition
Abstract (Minicourse): In the last decade, a lot of attention has been drawn to the elliptic system
\[ -\Delta u + \lambda_1 u = \mu_1 u^3 - \beta u v^2 \]
\[-\Delta v + \lambda_2 v = \mu_2 v^3 - \beta u^2 v \]
\[u, v \in H_0^1(\Omega), \]
where \(\Omega \subseteq \mathbb{R}^N \) \((N\leq2,3)\), \(\lambda_1,\lambda_2, \mu_1,\mu_2>0\) and \(\beta<0\). The main motivation for its study appears when searching for standing wave solutions of the corresponding nonlinear Schrödinger system
\[ i \partial_t \Phi-\Delta \Phi=\mu_1 \Phi|\Phi|^2-\beta\Phi|\Psi|^2,\quad i \partial_t \Psi-\Delta \Psi=\mu_2 \Psi|\Psi|^2-\beta\Psi|\Phi|^2 \]
which models phenomena appearing in the theory of Bose-Einstein condensation. The parameters \(\mu_1,\mu_2\) represent self-interactions within the same component, while \(\beta\) express the strength and type of interaction between the different components \(u\) and \(v\). We take \(\beta>0\), which means that different components compete. So, from a more general perspective, we can think of this system as a good prototype of a nonlinear elliptic system with competitive interaction between components. Existence of solutions can be obtained via variational methods, as the system is of gradient type.
One of the interesting questions for these problems is to understand what happens when the competition becomes the prevalent phenomenon, that is, when \(\beta\) becomes larger and larger. Taking a uniformly bounded family of solutions \((u_\beta,v_\beta)\), as it is reasonable to expect, they will convergence (as \(\beta\to +\infty)\) to some limiting profiles \(u\) and \(v\) that cannot coexist and tend to segregate, that is, \(u\cdot v\equiv 0\) in \(\Omega\). Moreover, the nodal sets \(\{u\neq 0\}\) and \(\{v\neq 0\}\) form a partition of the domain \(\Omega\).
In this mini-course, after establishing some existence results for the system, we will perform its asymptotic study as \(\beta\to +\infty\), proving uniform estimates as well as the regularity of the limiting profiles. We will then study the limiting partition, studying the regularity of the associated interfaces (a free boundary). If time allows, we will then give some applications to the study of problems appearing in shape optimization.
The study of these questions involves some tools coming from the theory of Variational Methods, Elliptic Regularity, Geometric Measure Theory and Free Boundary Problems. In particular, we will discuss some monotonicity formulas.
Ilya Kossovskiy (Viena, Austria)
Title: On the equivalence of real-analytic Cauchy-Riemann manifolds
Abstract: The art Interplay between different types of equivalence in CR-Geometry has attracted considerable attention in the last 20 years. Usually on considers the following 3 types of equivalence for CR-manifolds: biholomorphic equivalence, smooth CR equivalence, and formal equivalence. Using our recently developed theory on connecting degenerate CR-manifolds and Dynamical Systems, we showed that these 3 shorts of equivalence are significantly different from each other in general. In connection with that, I will concern in this lecture our recent results with Lamel and Stolovitch showing that for real-analytic hypersurfaces in \(\mathbb C^2\) their equivalence in the formal category implies that in the smooth one.
Irina Mitrea (Philadelphia, USA)
Title: The art of integration by
Abstract: The Integration by Parts
Luiz Gustavo Farah (UFMG)
Title: Nonlinear Profile Decomposition and the Concentration Phenomenon for Supercritical Generalized KdV Equations
Abstract: A nonlinear profile decomposition is established for solutions of supercritical generalized Korteweg-de Vries equations. As a consequence, we obtain a concentration result for finite time blow-up solutions that are of Type II. This is a joint work with Brian Pigott (Wofford College - EUA)
Mariana Smit Vega Garcia (Alemanha)
Title: The obstacle problem for the fractional Laplacian with drift
Abstract: We present the \(C^{1,\alpha}\) regularity of the regular part of the free boundary in the obstacle problem defined by the fractional Laplacian operator with gradient perturbation, in the subcritical regime (\(s\in (1/2,1)\)). More specifically, we consider
\[
\min\{L u, u-\varphi\}=0,
\]
where we denote
\[
Lu :=(-\Delta )^s u + \langle b(x),\nabla u \rangle +c(x)u.
\]
Our proof relies on a new Weiss-type monotonicity formula and an epiperimetric inequality. Both are generalizations of the ideas of G. Weiss, used in the classical obstacle problem for the Laplace operator, to our framework of fractional powers of the Laplace operator with drift.
This is joint work with Nicola Garofalo, Arshak Petrosyan and Camelia Pop.
Ma To Fu (ICMC-USP)
Title: On nonuniform Timoshenko systems
Abstract: The Timoshenko system is a pair of wave equations with a distinguished coupling. Here we discuss the stability of the partially damped system with respect to the so called equal wave speeds condition in a context of non constant coefficients.
Olímpio Myiagaki (Dallas, EUA)
Title: Nonlocal scalar field equations with Trudinger-Moser critical nonlinearity: Ground states and vanishing potential
Abstract: We investigate the existence of ground state solutions for a class of nonlinear scalar field equations defined on whole real line, involving a fractional Laplacian and nonlinearities with Trudinger-Moser critical growth. We also treat a class of nonlinear nonautonomous scalar field equations with fractional diffusion, exponential critical nonlinearity and a subcritical term. The involved potentials are allowed for vanishing behavior at infinity. We handle the lack of compactness of the associated energy functional due to the unboundedness of the domain and the presence of a limiting case embedding.
Olivâine Santana de Queiroz (UNICAMP)
Title: Some classical inequalities revisited for the fractiona Laplacian
Philip Jameson Graber (UT-Dallas)
Title: Nonlinear PDE and mean field games
Abstract: Mean field game theory is a rich new area of mathematics which has attracted a lot of attention from researchers in multiple disciplines. There are now many applications in finance, economics, pedestrian dynamics and even biology. From the point of view of PDE analysis, the center of mean field game theory is a system of Hamilton-Jacobi-Bellman/Fokker-Planck equations which are strongly coupled. In this short course we will discuss some of the latest techniques showing existence, uniqueness, and regularity of solutions to PDE systems of this type. For a certain class of mean field games, we find that the calculus of variations permits us to define a large class of weak solutions. However, in most cases, including those of interest in economics, variational methods will not work. In these cases, we will see how to construct a fixed point by deriving new a priori estimates.
Ricardo Alonso (PUC-Rio)
Title: A fractional Laplace-Beltrami operator arising in radiated transfer
Abstract: In this lecture we present the radiative transfer equation in the forward-peaked regime in free space. We show instantaneous regularization of solution using hypo-elliptic techniques, convergence of the Henyey-Greenstein scattering models towards the peaked regime and time vanishing of solutions due to scattering. The analysis of the scattering operator is performed through elementary use of the stereographic projection, which renders a precise representation of the scattering mechanism in terms of a fractional Laplace-Beltrami operator on the sphere.
Tiago Picon (FFCLRP/USP)
Title: Decay estimates in real Hardy spaces for dissipative evolution equations
Abstract: In this lecture we present asymptotic-in-time linear estimates in Hardy spaces \(H^p(\mathbb{R}^{n})\) for the Cauchy problem for evolution operators with structural dissipation. The obtained estimates are a natural extension of the known \(L^p-L^q\) estimates, \(1\leq p\leq q\leq\infty\), for these models. Different, standard, tools to work in Hardy spaces are used to derive optimal estimates.
This is joint work with Marcelo Ebert and Marcello D'Abbicco from FFCLRP (Faculdade de Filosofia, Ciências e Letras de Ribeirão Preto/USP).
Wladimir Neves (UFRJ)
Title: Strong Traces for Conservations Laws
Abstract: In this talk, we discuss about the important issue of strong traces for scalar conservation laws. Moreover, we present some results about strong traces, where the flux function is assumed non-homogeneous with low regularity in the spatial variable. This is a joint work (in progress) with Evgeniy Panov, and Jean Silva.
