C1,α regularity for the normalized p-Poisson problem

We consider the normalized p-Poisson problem Delta(N)(p)u = f in Omega subset of R-n. The normalized p -Laplacian Delta(N)(p)u := vertical bar Du vertical bar(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 is an element of L-infinity boolean AND C and p > 1 we use methods both from viscosity and weak theory, whereas in the case f is an element of L-q boolean AND C, q > max(n, E/2, 2), and p > 2 we rely on the tools of nonlinear potential theory. (C) 2017 Elsevier Masson SAS. All rights reserved.