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Gilberto de Assis Pereira

Publications and source records attributed to Gilberto de Assis Pereira.

5 recordsLinked to original sources
arXiv2020

An optimal pointwise Morrey-Sobolev inequality

Let $Ω$ be a bounded, smooth domain of $\mathbb{R}^{N},$ $N\geq1.$ For each $p>N$ we study the optimal function $s=s_{p}$ in the pointwise inequality \[ \left\vert v(x)\right\vert \leq s(x)\left\Vert \nabla v\right\Vert _{L^{p}(Ω)},\quad\forall\,(x,v)\in\overlineΩ\times W_{0}% ^{1,p}(Ω). \] We show that $s_{p}\in C_{0}^{0,1-(N/p)}(\overlineΩ)$ and that $s_{p}$ converges pointwise to the distance function to the boundary, as $p\rightarrow\infty.$ Moreover, we prove that if $Ω$ is convex, then $s_{p}$ is concave and has a unique maximum point.

math.AP↗
arXiv2019

Asymptotic behavior as $p\rightarrow\infty$ of least energy solutions of a $(p,q(p))$-Laplacian problem

\[ \left\{ \begin{array} [c]{lll} -\left( Δ_{p}+Δ_{q(p)}\right) u=λ_{p}\left\vert u(x_{u})\right\vert ^{p-2}u(x_{u})δ_{x_{u}} & \mathrm{in} & Ω\\ u=0 & \mathrm{on} & \partialΩ, \end{array} \right. \] where $x_{u}$ is the (unique) maximum point of $\left\vert u\right\vert ,$ $δ_{x_{u}}$ is the Dirac delta distribution supported at $x_{u},$ \[ \lim_{p\rightarrow\infty}\frac{q(p)}{p}=Q\in\left\{ \begin{array} [c]{lll} (0,1) & \mathrm{if} & N 0$ is such that \[ \min\left\{ \frac{\left\Vert \nabla u\right\Vert _{\infty}}{\left\Vert u\right\Vert _{\infty}}:0\not \equiv u\in W^{1,\infty}(Ω)\cap C_{0}(\overlineΩ)\right\} \leq\lim_{p\rightarrow\infty}(λ_{p})^{\frac{1}{p}}<\infty. \]

math.AP↗
arXiv2016

On a singular minimizing problem

We study a minimizing problem associated with the singular problem \[ \left\{ \begin{array} [c]{ll} -\operatorname{div}\left( \left\vert \nabla u\right\vert ^{p-2}\nabla u\right) =λu^{-1} & \mathrm{in\ }Ω\\ u>0 & \mathrm{in\ }Ω\\ u=0 & \mathrm{on\ }\partialΩ, \end{array} \right. \] where $p>1$, $λ>0$ and $Ω$ is a bounded and smooth domain of $\mathbb{R}^{N}$, $N\geq2.$ A new log-Sobolev type inequality is proved and the corresponding best constant is identifyied.

math.AP↗
arXiv2015

Asymptotics for the best Sobolev constants and their extremal functions

Let $Ω$ be a bounded domain of $\mathbf{R}^{N},$ $N\geq2.$ Let, for $p>N,$ \[ Λ_{p}(Ω):=\inf\left\{ \left\Vert \nabla u\right\Vert _{p}^{p}:u\in W_{0}^{1,p}(Ω)\quad and\quad\left\Vert u\right\Vert _{\infty}=1\right\} . \] We first prove that \[ \lim_{p\rightarrow\infty}Λ_{p}(Ω)^{\frac{1}{p}}=\frac{1}{\left\Vert ρ\right\Vert _{\infty}}, \] where $ρ$ denotes the distance function to the boundary. Then, we show that, up to subsequences, the extremal functions of $Λ_{p}(Ω)$ converge (as $p\rightarrow\infty$) to the viscosity solutions of a specific Dirichlet problem involving the infinity Laplacian in the punctured $Ω.$

math.AP↗