Continuity of derivatives of a convex solution to a perturbed one-Laplace equation by $p$-Laplacian

TL;DR

This paper proves that convex solutions to a perturbed one-Laplace equation with p-Laplacian are continuously differentiable, addressing boundary regularity issues through blow-up analysis and Liouville-type results, which was not possible for the unperturbed case.

Contribution

It establishes the $C^{1}$ regularity of convex solutions to a perturbed one-Laplace equation, a novel result not valid for the unperturbed equation, using elementary convex analysis.

Findings

Convex solutions are $C^{1}$ when perturbed by p-Laplacian.

Boundary regularity is achieved via blow-up and Liouville-type theorems.

The approach is elementary due to convexity assumptions.

Abstract

We consider a one-Laplace equation perturbed by $p$-Laplacian with $1<p<\infty$. We prove that a weak solution is continuously differentiable ($C^{1}$) if it is convex. Note that similar result fails to hold for the unperturbed one-Laplace equation. The main difficulty is to show $C^{1}$-regularity of the solution at the boundary of a facet where the gradient of the solution vanishes. For this purpose we blow-up the solution and prove that its limit is a constant function by establishing a Liouville-type result, which is proved by showing a strong maximum principle. Our argument is rather elementary since we assume that the solution is convex. A few generalization is also discussed.