Overdetermined boundary value problems for the $\infty$-Laplacian

TL;DR

This paper investigates overdetermined boundary value problems for the $ abla_ ext{infty}$-Laplacian, exploring how solutions influence the geometric shape of the domain across different $ ext{L}_ ext{infty}$ variants.

Contribution

It analyzes implications of solutions to overdetermined problems for various $ ext{L}_ extinfty$-Laplacian formulations on domain geometry.

Findings

Different $ ext{L}_ extinfty$-Laplacian variants yield distinct geometric implications.

The existence of solutions constrains the shape of the domain in specific ways.

Results extend classical overdetermined problem theory to $ ext{L}_ extinfty$ contexts.

Abstract

We consider overdetermined boundary value problems for the $\infty$-Laplacian in a domain $\Omega$ of $\R^n$ and discuss what kind of implications on the geometry of $\Omega$ the existence of a solution may have. The classical $\infty$-Laplacian, the normalized or game-theoretic $\infty$-Laplacian and the limit of the $p$-Laplacian as $p\to \infty$ are considered and provide different answers.