Continuous differentiability of a weak solution to very singular elliptic equations involving anisotropic diffusivity

TL;DR

This paper proves that weak solutions to a complex, anisotropic elliptic equation involving singular and degenerate operators are continuously differentiable across facets, using approximation and classical PDE techniques.

Contribution

It establishes the continuous differentiability of weak solutions to a very singular anisotropic elliptic equation, including the one-Laplacian and p-Laplacian operators.

Findings

Weak solutions are continuously differentiable across facets.

Approximation methods effectively handle singularities and anisotropy.

Standard PDE techniques are applicable to complex anisotropic equations.

Abstract

In this paper we consider a very singular elliptic equation that involves an anisotropic diffusion operator, including one-Laplacian, and is perturbed by a $p$-Laplacian-type diffusion operator with $1<p<\infty$. This equation seems analytically difficult to handle near a facet, the place where the gradient vanishes. Our main purpose is to prove that weak solutions are continuously differentiable even across the facet. Here it is of interest to know whether a gradient is continuous when it is truncated near a facet. To answer this affirmatively, we consider an approximation problem, and use standard methods including De Giorgi's truncation and freezing coefficient methods.