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The purpose of this paper is to prove a uniform convergence rate of thesolutions of the $p$-Laplace equation $\Delta_p u = 0$ with Dirichlet boundaryconditions 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 ofthe 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 ofthis result solely relies on the comparison principle with the fundamentalsolutions of the $p$-Laplace and the infinity-Laplace equation, respectively.Our argument does not use viscosity solutions, is purely metric, and istherefore generalizable to more general settings where a comparison principlewith H\"older cones and H\"older regularity is available.

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