C1,α regularity for the normalized p-Poisson problem

Abstract

We consider the normalized p -Poisson problem − Δ N p u = f in Ω ⊂ R n . The normalized p -Laplacian Δ N p u := | Du | 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 ∈ L ∞ ∩ C and p> 1 we use methods both from viscosity and weak theory, whereas in the case f ∈ L q ∩ C , q> max( n, p 2 , 2), and p> 2 we rely on the tools of nonlinear potential theory