$C^{1,\alpha }$ regularity for the normalized $p$-Poisson problem

Abstract

We consider the normalized $p$-Poisson problem $$-{\Delta }_p^{N}u=f\text{in}\Omega .$$ The normalized $p$-Laplacian ${\Delta }_p^{N}u:=|Du{|}^{2-p}{\Delta }_{p}u$ is in non-divergence form and arises for example from stochastic games. We prove $C_{loc}^{1,\alpha }$ regularity with nearly optimal $\alpha$ for viscosity solutions of this problem. In the case $f\in L^{\infty }\cap C$ and $p>1$ we use methods both from viscosity and weak theory, whereas in the case $f\in L^q\cap C$, $q>\max (n,\frac{p}{2},2)$, and $p>2$ we rely on the tools of nonlinear potential theory.

41 pages