| Abstract: |
| This talk is concerned with gradient continuity for a weak solution to a very singular elliptic problem involving both one-Laplacian and $p$-Laplacian with $p\in(1,\,\infty)$. The main difficulty herein is that this problem becomes no longer uniformly elliptic near a facet, the degenerate region of a gradient. This fact prevents us from using standard methods from the De Giorgi--Nash--Moser theory to prove H\{o}lder continuity of a gradient, especially near the facet. It is substantially because the one-Laplace operator is degenerate elliptic in a direction of a gradient, while this operator becomes singular elliptic in other directions. Such anisotropic diffusivity appears difficult to handle in existing regularity theory. Continuity of a derivative was first established by Yoshikazu Giga and the speaker, in a special case where a weak solution is both scalar-valued and convex. After that work was completed, the speaker, inspired by a recent work on higher regularity for a very degenerate elliptic problem, has found it possible to show gradient continuity in general cases. The aim of this talk is to explain briefly how to prove continuous differentiability of general weak solutions, even across facets. |
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