Matzoh ball soup revisited: the boundary regularity issue

Abstract

We consider nonlinear diffusion equations of the form ${\partial }_{t}u=\Delta \varphi (u)$ in ${\mathbb{R}}^N$ with $N\ge 2.$ When $\varphi (s)\equiv s$, this is just the heat equation. Let $\Omega$ be a domain in ${\mathbb{R}}^N$, where $\partial \Omega$ is bounded and $\partial \Omega =\partial ({\mathbb{R}}^N\setminus \bar{\Omega })$. We consider the initial-boundary value problem, where the initial value equals zero and the boundary value equals 1, and the Cauchy problem where the initial data is the characteristic function of the set ${\Omega }^c={\mathbb{R}}^N\setminus \Omega$. We settle the boundary regularity issue for the characterization of the sphere as a stationary level surface of the solution $u:$ no regularity assumption is needed for $\partial \Omega .$

16 pages; no figures. Added an appendix with the proof of Theorem B, to make the paper self-contained. Some other minor modifications

References in corpus (1)

• Interaction between nonlinear diffusion and geometry of domain