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

TL;DR

This paper proves local $C^{1,eta}$ regularity for viscosity solutions of the normalized $p$-Poisson equation, combining viscosity and weak theories for different data regularities and extending regularity results.

Contribution

It establishes nearly optimal regularity results for the normalized $p$-Poisson problem, using novel methods that blend viscosity and nonlinear potential theory.

Findings

Proved $C^{1,eta}_{loc}$ regularity for solutions.

Extended regularity results to broader function spaces.

Combined viscosity and potential theory techniques.

Abstract

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