Serrin's overdetermined theorem within Lipschitz domains
Hongjie Dong, Yi Ru-Ya Zhang
TL;DR
This work proves that, for a Lipschitz domain Ω, the Serrin-type overdetermined system $u\in W^{1,2}(\mathbb{R}^n)$ with $u=0$ outside Ω and $Δu=\mathbf{c}\mathscr{H}^{n-1}|_{∂^*Ω}-\mathbf{1}_Ω\,dx$ holds, if and only if Ω is a Euclidean ball. The authors adapt Weinberger’s interior-approximation strategy to Lipschitz domains, employing non-tangential boundary controls and a Pohozaev identity to show that the gradient and potential satisfy rigid constraints, forcing $u(x)=\frac{r^2-|x|^2}{2n}$ and hence spherical symmetry. They extend the framework to anisotropic settings using Wulff potentials, define the Δ_H operator, and show that under small Lipschitz constants and $Du\in VMO$, the same rigidity result holds with $u=\frac{r^2-H_*^2(x)}{2n}$ and Ω, K being homothetic. The paper further situates these results in the context of Alexandrov-type theorems and Maggi’s conjecture, proposing an anisotropic Serrin-type conjecture for rough domains and highlighting the role of DKP conditions and non-tangential estimates in obtaining boundary regularity and convergence without requiring Neumann data boundedness.
Abstract
Let $Ω\subset\mathbb R^n$ be a Lipschitz domain. We prove that, $Ω$ satisfies the following Serrin-type overdetermined system $$u \in W^{1,2}(\mathbb R^n), \quad u=0\ \text{ a.e. in }\mathbb R^n\setminus Ω,\quad Δu=\mathbf{c}\mathscr{H}^{n-1}|_{\partial^*Ω} - \mathbf{1}_Ω\,dx,$$ in the weak sense if and only if $Ω$ is a ball. Here $\mathscr H^{n-1}$ denotes the $(n-1)$-dimensional Hausdorff measure. Moreover, a generalization of our method in the anisotropic setting is discussed. Our approach offers an alternative proof to [15] in the case of Lipschitz domains, introducing a novel viewpoint to settle [18, Question 7.1].