The convergence rate of $p$-harmonic to infinity-harmonic functions

TL;DR

This paper establishes explicit uniform convergence rates for solutions of the p-Laplace equation to the infinity-Laplace equation as p approaches infinity, using a comparison principle approach that is broadly applicable.

Contribution

It provides the first explicit convergence rate estimates for p-harmonic to infinity-harmonic functions without relying on viscosity solutions.

Findings

Convergence rate scales like p^{-1/4} for general solutions.

Convergence rate scales like p^{-1/2} for solutions with positive gradient.

An example shows the rate cannot be better than p^{-1}.

Abstract

The purpose of this paper is to prove a uniform convergence rate of the solutions of the $p$-Laplace equation $\Delta_p u = 0$ with Dirichlet boundary conditions to the solution of the infinity-Laplace equation $\Delta_\infty u = 0$ as $p\to\infty$. The rate scales like $p^{-1/4}$ for general solutions of the Dirichlet problem and like $p^{-1/2}$ for solutions with positive gradient. An explicit example shows that it cannot be better than $p^{-1}$. The proof of this result solely relies on the comparison principle with the fundamental solutions of the $p$-Laplace and the infinity-Laplace equation, respectively. Our argument does not use viscosity solutions, is purely metric, and is therefore generalizable to more general settings where a comparison principle with H\"older cones and H\"older regularity is available.