A weak solution to a perturbed one-Laplace system by $p$-Laplacian is continuously differentiable

TL;DR

This paper proves that weak solutions to a perturbed one-Laplace system involving the p-Laplacian are continuously differentiable, even across facets where the gradient vanishes, by establishing H"{o}lder continuity of derivatives.

Contribution

It demonstrates continuous differentiability of solutions to a complex perturbed system, overcoming challenges posed by degeneracy near facets.

Findings

Weak solutions are continuously differentiable across facets.

H"{o}lder continuity of derivatives is established.

Method involves approximations and standard regularity techniques.

Abstract

In this paper we aim to show continuous differentiability of weak solutions to a one-Laplace system perturbed by $p$-Laplacian with $1<p<\infty$. The main difficulty on this equation is that uniform ellipticity breaks near a facet, the place where a gradient vanishes. We would like to prove that derivatives of weak solutions are continuous even across the facets. This is possible by estimating H\"{o}lder continuity of Jacobian matrices multiplied with its modulus truncated near zero. To show this estimate, we consider an approximated system, and use standard methods including De Giorgi's truncation and freezing coefficient arguments.