Regularity for the normalized p-Laplacian equation with an arbitrary degeneracy law

Abstract

We examine the interior regularity of solutions to a degenerate normalized $p$-Laplace equation, where the degeneracy is governed by a modulus of continuity whose inverse satisfies a Dini continuity condition. We prove that under very general assumptions on the degeneracy law, solutions belong to the $C^1$ class. We argue by approximating the solutions by a sequence of hyperplanes, which allows us to prove the desired regularity.