Calculus of Variations and Geometric Measure Theory

L. Bungert

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

created by bungert on 17 Feb 2023
modified on 30 Jan 2024


Published Paper

Inserted: 17 feb 2023
Last Updated: 30 jan 2024

Journal: Communications in Partial Differential Equations
Volume: 48
Number: 10-12
Pages: 1323-1339
Year: 2023
Doi: 10.1080/03605302.2023.2283830

ArXiv: 2302.08462 PDF


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.