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

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 Hölder cones and Hölder regularity is available.

Expanded examples and corrected some typos

References in corpus (3)

• Uniform Convergence Rates for Lipschitz Learning on Graphs
• Eigenvalue Problems in ${\mathrm{L}}^{\infty }$: Optimality Conditions, Duality, and Relations with Optimal Transport
• Evolution Driven by the Infinity Fractional Laplacian