Continuity of a spatial derivative for a perturbed one-Laplace equation

This talk is concerned with continuity of a spatial derivative for a weak solution to a one-Laplace elliptic equation perturbed by the $p$-Laplace operator with $p\in(1,\,\infty)$. The main difficulty herein is that the one-Laplace operator has both singular and degenerate ellipticity. Due to this anisotropic diffusivity, the equation becomes no longer uniformly elliptic near a facet, the place where a gradient degenerates. Since the one-Laplace operator seems difficult to handle in existing regularity theory, continuity of a spatial gradient has been a long-standing open problem. Recently, the lecturer, inspired by a regularity result on a very degenerate elliptic problem, has found it possible to prove continuous differentiability of a weak solution. The aim of this talk is to explain how continuity of a spatial gradient is shown. .