C1,α regularity for the normalized p-Poisson problem

Abstract

We consider the normalized p-Poisson problem−ΔpNu=finΩ⊂Rn. The normalized p-Laplacian ΔpNu:=|Du|2−pΔpu is in non-divergence form and arises for example from stochastic games. We prove Cloc1,α regularity with nearly optimal α for viscosity solutions of this problem. In the case f∈L∞∩C and p>1 we use methods both from viscosity and weak theory, whereas in the case f∈Lq∩C, q>max⁡(n,p2,2), and p>2 we rely on the tools of nonlinear potential theory.