A symmetry problem for the infinity Laplacian

TL;DR

This paper characterizes the geometric conditions under which the infinity Laplacian's Dirichlet problem admits solutions depending solely on the distance to the boundary, removing regularity assumptions through advanced viscosity methods.

Contribution

It provides strengthened geometric characterizations for the infinity Laplacian problem, eliminating regularity assumptions and introducing new estimates for distance functions near singularities.

Findings

Necessary and sufficient geometric conditions identified

Regularity assumptions on domains and solutions removed

New estimates for distance functions near singular points

Abstract

Aim of this paper is to prove necessary and sufficient conditions on the geometry of a domain $\Omega \subset \mathbb{R}^n$ in order that the homogeneous Dirichlet problem for the infinity-Laplace equation in $\Omega$ with constant source term admits a viscosity solution depending only on the distance from $\partial \Omega$. This problem was previously addressed and studied by Buttazzo and Kawohl. In the light of some geometrical achievements reached in our recent paper "On the characterization of some classes of proximally smooth sets", we revisit the results obtained by Buttazzo and Kawohl and we prove strengthened versions of them, where any regularity assumption on the domain and on the solution is removed. Our results require a delicate analysis based on viscosity methods. In particular, we need to build suitable viscosity test functions, whose construction involves a new estimate of the distance function near singular points.