Asymptotics for the best Sobolev constants and their extremal functions

TL;DR

This paper investigates the asymptotic behavior of Sobolev constants and extremal functions in bounded domains as p approaches infinity, revealing their convergence to solutions of an infinity Laplacian problem.

Contribution

It establishes the limit of Sobolev constants as p approaches infinity and characterizes the convergence of extremal functions to viscosity solutions of a related PDE.

Findings

Limit of Sobolev constants as p→∞ is 1/||ρ||_∞.

Extremal functions converge to viscosity solutions of an infinity Laplacian problem.

Provides a link between Sobolev constants and geometric properties of the domain.

Abstract

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