Data: 22/Fev/24
Local: Sala 03, Bloco 914
Palestrantes:
João Vítor da Silva (UNICAMP)
Horário: 10h
Title: Optimal regularity for a quasi-linear obstacle problem and applications.
Abstract: In this Lecture we show existence/uniqueness of weak solutions of an obstacle problem for a quasi-linear operator with unbounded source terms. In our results, we obtain sharp gradient estimates, namely, $C^{1, \alpha}_{loc}(B_1)$ for the solution to an explicit and universal regularity exponent. Our results are relevant even for the simplest model case governed by the $p-$Laplacian with H\"{o}lder continuous coefficients
$$
\left\{
\begin{array}{rclcl}
\text{div}\left( |\nabla u|^{p-2}\mathfrak{A}(x) \nabla u\right) & = & f(x) & \text{in} & \{u> \varphi\}\cap B_1 \\
\text{div}\left( |\nabla u|^{p-2}\mathfrak{A}(x) \nabla u\right) & \le & f(x) & \text{in} & B_1\\
u(x)& \ge & \varphi(x) &\text{in} & B_1 \\
u(x) & = & 0 & \text{on} & \partial B_1,
\end{array}
\right.
$$
where $f \in L^q(\Omega)$ for $q>n$ and $q \geq \frac{p}{p-1}$ % $q>\dfrac{np}{p-1}$
($1 < p<\infty$), $\mathfrak{A} \in C^{0, \sigma}(\Omega, \mathbb{R}^{n \times n})$ (for some $\sigma \in (0, 1]$) with $\mathfrak{A}$ a $(\lambda, \Lambda)-$uniformly elliptic matrix, and $\varphi \in C^{1,\beta}(\Omega) \cap \mathfrak{X}_{p, q}$, for some $\beta \in (0, 1]$ where
$$
\mathfrak{X}_{p, q}:=\left\{v\in W^{1,p}(\Omega); \quad \text{div} \ \mathfrak{a}(x,\nabla v) \in L^q(\Omega) \right\}.
$$
For some specific scenarios, we show the non-degeneracy of solutions, which provides crucial information about the free boundary of solutions. Our regularity estimates improve and extend, to a certain extent, results previously obtained for the obstacle problem governed by the $p-$Laplacian with bounded source term (cf. [1] and [3]). Furthermore, we gave special emphasis to the study of the linear and non-homogeneous case, i.e., $p=2$ and $f \neq 0$, which was not available in the literature and it plays a decisive role in analysing the non-linear case (cf. [2]).
This is a joint work with Elzon C. Bezerra Júnior (UFCA-Brazil) and Romário T. Frias (Unicamp-Brazil).
Bibliografia
[1] Andersson, J., Lindgren, E. and Shahgholian, H. Optimal regularity for the obstacle problem for the $p$-Laplacian. J. Differential Equations. 259 (2015), no. 6, 2167-2179.
[2] Caffareli, L. A., and Kinderlehrer, D. Potential methods in variational inequalities. J. Analyse Math}. 37 (1980), 285-295.
[3] Rodrigues, J.F. Stability remarks to the obstacle problem for $p$-Laplacian type equations. Calc. Var. Partial Differential Equations. 23 (2005), no. 1, 51-65.
Cleon da Silva Barroso (UFC)
Horário: 10:50h
Title: Existence of fixed-point free Hölder-Lipschitz maps with null minimal displacement on unit balls.
Abstract: It is well-known that if X is a Banach space, then norm-compactness of its closed unit ball characterizes the fixed point property for the class of Lipschitz self-mapping T. In fact, if X is infinite-dimensional then one can build a Lipschitz mapping T from the unit ball of X into itself such that its {\it minimal displacement} is positive. In this talk, we will discuss some interesting and hard solving problems about the existence of Hölder-Lipschitz maps that leave invariant the unit ball of X, are fixed-point free but have null minimal displacement. In the first part we will set up the problem and highlight our main contributions. In the last part we will liist some open problems. The talk meets a general audience, at master's level, but with very basic requirements. The content of this talk is part of a joint work with Professor Valdir Ferreira (UFCA).
Disson dos Prazeres (UFS)
Horário: 14h
Título: Propriedade de Liouville para equações não lineares degeneradas.
Resumo: Nesta palestra, apresentaremos uma nova estratégia para obtermos a propriedade de Liouville para equações não-lineares degeneradas. Como aplicação iremos obter resultados de existência para problemas de Dirichlet.
José Ederson Melo Braga (UFC)
Horário: 14:50h
Título: Estimativa de Carleson não-homogênea para o g-Laplaciano.
Resumo: O objetivo desta palestra é apresentar a estimativa de Carleson não-homogênea para o operador g-Laplaciano assim como algumas aplicações desta estimativa. A estimativa diz respeito a uma desigualdade do supremo da função até a fronteira do domínio. Uma vez obtida tal estimativa, apresentamos um controle de crescimento esponencial de soluções do g-Laplaciano em cilindos não limitados. Em seguida, provamos a estimativa de Harnack até a fronteira nesse contexto.
Marcelo Dário dos Santos Amaral (UNILAB)
Horário: 15:40h
Título: Optimal regularity estimates for doubly nonlinear equations.
Abstract: We establish sharp gradient regularity estimates for solutions to the zero-obstacle ruled by inhomogeneous doubly nonlinear equations, which are universally close to the evolution p-Laplacian equation. More specifically, we derive optimal C^{1,α} regularity estimates at free boundary points, i.e. boundary points of the positivity set of a given solution. As a result, we prove sharp local C^{1,α} regularity estimates in space and time. Our results hold particular relevance in notable contexts, as in applications to the porous medium equation.