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

We consider the normalized $p$-Poisson problem $-Δ^N_p u=f \qquad \text{in}\quad Ω.$ The normalized $p$-Laplacian $Δ_p^{N}u:=|D u|^{2-p}Δ_p u$ is in non-divergence form and arises for example from stochastic games. We prove $C^{1,α}_{loc}$ regularity with nearly optimal $α$ 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 $f\in L^q\cap C$, $q>\max(n,\frac p2,2)$, and $p>2$ we rely on the tools of nonlinear potential theory.

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