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

Abstract

Abstract In this paper we consider a very singular elliptic equation that involves an anisotropic diffusion operator, including the one-Laplacian, and is perturbed by a p-Laplacian-type diffusion operator with 1 < p < ∞ {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.