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

TL;DR

This paper proves that solutions to a degenerate normalized p-Laplacian equation are continuously differentiable under general degeneracy conditions, using an approximation approach.

Contribution

It establishes interior regularity results for solutions with arbitrary degeneracy laws governed by a Dini continuous modulus.

Findings

Solutions are in C^1 under broad degeneracy conditions

Approximation by hyperplanes is effective for regularity proofs

Regularity holds for general degeneracy laws

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.