Characterization of stadium-like domains via boundary value problems for the infinity Laplacian

TL;DR

This paper characterizes convex domains called stadium-like domains where solutions to boundary value problems involving the infinity Laplacian exist, extending previous results and establishing uniqueness and regularity conditions.

Contribution

It provides a complete characterization of stadium-like domains for the infinity Laplacian, including new results for the normalized operator and regularity of solutions.

Findings

Stadium-like domains are precisely the convex sets with solutions to the boundary problems.

Extended previous results to more general convex domains.

Established uniqueness and regularity of solutions in these domains.

Abstract

We give a complete characterization, as "stadium-like domains", of convex subsets $\Omega$ of $\mathbb{R}^n$ where a solution exists to Serrin-type overdetermined boundary value problems in which the operator is either the infinity Laplacian or its normalized version. In case of the not-normalized operator, our results extend those obtained in a previous work, where the problem was solved under some geometrical restrictions on $\Omega$. In case of the normalized operator, we also show that stadium-like domains are precisely the unique convex sets in $\mathbb{R}^n$ where the solution to a Dirichlet problem is of class $C^{1,1} (\Omega)$.