Asymptotics for Sobolev extremals: the hyperdiffusive case

TL;DR

This paper investigates the asymptotic behavior of Sobolev extremals in the hyperdiffusive case, showing convergence to solutions of a specific infinity Laplacian problem and relating extremal values to the domain's boundary distance.

Contribution

It establishes the limit of Sobolev extremals as p approaches infinity in the hyperdiffusive regime and characterizes their uniform convergence to viscosity solutions of a boundary value problem.

Findings

Limit of λ_{p,q(p)}^{1/p} equals inverse of the boundary distance function norm.

Sequences of extremals converge uniformly to solutions of an infinity Laplacian problem.

Identifies the set where the limit function attains its maximum as a subset of maximum points of the distance function.

Abstract

Let $\Omega$ be a bounded, smooth domain of $\mathbb{R}^{N},$ $N\geq2.$ For $p>N$ and $1\leq q(p)<\infty$ set \[ \lambda_{p,q(p)}:=\inf\left\{ \int_{\Omega}\left\vert \nabla u\right\vert ^{p}\mathrm{d}x:u\in W_{0}^{1,p}(\Omega)\text{ \ and \ }\int_{\Omega }\left\vert u\right\vert ^{q(p)}\mathrm{d}x=1\right\} \] and let $u_{p,q(p)}$ denote a corresponding positive extremal function. We show that if $\lim\limits_{p\rightarrow\infty}q(p)=\infty$, then $\lim\limits_{p\rightarrow\infty}\lambda_{p,q(p)}^{1/p}=\left\Vert d_{\Omega }\right\Vert _{\infty}^{-1}$, where $d_{\Omega}$ denotes the distance function to the boundary of $\Omega.$ Moreover, in the hyperdiffusive case: $\lim\limits_{p\rightarrow\infty}\frac{q(p)}{p}=\infty,$ we prove that each sequence $u_{p_{n},q(p_{n})},$ with $p_{n}\rightarrow\infty,$ admits a subsequence converging uniformly in $\overline{\Omega}$ to a viscosity solution to the problem \[ \left\{ \begin{array} [c]{lll} -\Delta_{\infty}u=0 & \text{in} & \Omega\setminus M\\ u=0 & \text{on} & \partial\Omega\\ u=1 & \text{in} & M, \end{array} \right. \] where $M$ is a closed subset of the set of all maximum points of $d_{\Omega}.$