The Infinity-Laplacian in Smooth Convex Domains and in a Square

TL;DR

This paper extends theorems related to the Infinity-Ground State and Infinity-Potential from convex polygons to smooth convex domains and squares, using Alexandroff's method, and discusses a recent explicit solution that challenges a previous conjecture.

Contribution

It generalizes existing results for the Infinity-Laplacian to new domain types and addresses a disproving explicit solution, advancing understanding of the problem.

Findings

Theorems for the Infinity-Ground State and Infinity-Potential are extended to smooth convex domains.

A recent explicit solution disproves a longstanding conjecture.

Application of Alexandroff's method to curved boundaries enhances analytical techniques.

Abstract

We extend some theorems for the Infinity-Ground State and for the Infinity-Potential, known for convex polygons, to other domains in the plane, by applying Alexandroff's method to the curved boundary. A recent explicit solution disproves a conjecture.