Remarks on the validity on the maximum principle for the $\infty$-Laplacian

TL;DR

This paper presents counter-examples demonstrating that the maximum principle does not generally hold for classical solutions of certain equations related to the -Laplacian, challenging assumptions about their behavior.

Contribution

It provides three specific counter-examples showing the failure of the maximum principle for the -Laplacian system and scalar equations, clarifying limitations of existing theoretical expectations.

Findings

Maximum principle fails for the -Laplacian system.

Maximum principle fails for scalar -Laplacian with gradient perturbation.

Counter-examples challenge previous assumptions about solution behavior.

Abstract

In this note we give three counter-examples which show that the Maximum Principle generally fails for classical solutions of a system and a single equation related to the $\infty$-Laplacian. The first is the tangential part of the $\infty$-Laplace system and the second is the scalar $\infty$-Laplace equation perturbed by a linear gradient term. The interpretations of the Maximum Principle for the system are that of the Convex Hull Property and also of the Maximum Principle of the modulus of the solution.