Matzoh ball soup revisited: the boundary regularity issue

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 $\mathrm{\Omega }$ be a domain in ${\mathbb{R}}^N$, where $\mathrm{\partial }\mathrm{\Omega }$ 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 ${\mathrm{\Omega }}^c={\mathbb{R}}^N\setminus \mathrm{\Omega }$. 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 $\mathrm{\partial }\mathrm{\Omega }\mathrm{.}$