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

The purpose of this paper is to prove a uniform convergence rate of the solutions of the p-Laplace equation Apu = 0 with Dirichlet boundary conditions to the solution of the infinity-Laplace equation Delta(infinity)u = 0 as p -> infinity. 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 & ouml;lder cones and H & ouml;lder regularity is available.