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Matzoh ball soup revisited: the boundary regularity issue

We consider nonlinear diffusion equations of the form $\partial_t u= Δϕ(u)$ in $\mathbb R^N$ with $N \ge 2.$ When $ϕ(s) \equiv s$, this is just the heat equation. Let $Ω$ be a domain in $\mathbb R^N$, where $\partialΩ$ is bounded and $\partialΩ= \partial (\mathbb R^N\setminus \bar Ω)$. 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 $Ω^c = \mathbb R^N\setminus Ω$. 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 $\partialΩ.$