Abstract
The purpose of this paper is to prove a uniform convergence rate of the solutions of the p-Laplace equation with Dirichlet boundary conditions to the solution of the infinity-Laplace equation
as
. The rate scales like
for general solutions of the Dirichlet problem and like
for solutions with positive gradient. An explicit example shows that it cannot be better than
. 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.
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.