Matzoh ball soup revisited: the boundary regularity issue

TL;DR

This paper investigates boundary regularity for nonlinear diffusion equations, establishing that spheres can be characterized as stationary level surfaces without regularity assumptions on the boundary.

Contribution

It proves that the sphere's characterization as a stationary level surface holds for nonlinear diffusion equations without requiring boundary regularity.

Findings

Sphere as stationary level surface characterized without boundary regularity assumptions

Results apply to nonlinear diffusion equations including the heat equation

Boundary regularity issue resolved for boundary characterization

Abstract

We consider nonlinear diffusion equations of the form $\partial_t u= \Delta \phi(u)$ in $\mathbb R^N$ with $N \ge 2.$ When $\phi(s) \equiv s$, 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 $\partial\Omega.$