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.

KEYWORDS:

• p-Laplacian
• infinity-Laplacian
• convergence rates
• comparison principle
• 26A16
• 35B51
• 35D30
• 35D40
• 35J92
• 35J94

Acknowledgments

The Vetenskapsrådet; author would also like to thank Mikko Parviainen and Tim Roith for interesting discussions on the topic of this paper as well as Peter Lindqvist and Antoni Kijowski for helpful remarks on the first version of the preprint.

Additional information

Funding

Most of this work was done while the author was affiliated with the University of Bonn, supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy - GZ 2047/1, Projekt-ID 390685813. Parts of this work were also done while the author was in residence at Institut Mittag-Leffler in Djursholm, Sweden during the semester on Geometric Aspects of Nonlinear Partial Differential Equations in 2022, supported by the Swedish Research Council under grant no. 2016-06596.