Serrin's overdetermined theorem within Lipschitz domains

Abstract: Let $\Omega\subset\mathbb R^n$ be a Lipschitz domain, $K$ be a (bounded) ellipsoid centered at the origin and $H$ be the associated Wulff potential. We prove that, % under a $L^2$-Dini-VMO condition and a non-degeneracy condition on $|Du|$ near the boundary, $\Omega$ satisfies the following Serrin-type overdetermined system $$u \in W^{1,2}(\mathbb R^n), \quad u=0\ \text{ a.e. in }\R^n\setminus \Omega,\quad \Delta_H u=\mathbf{c}\mathscr{H}^{n-1}|_{\partial^*\Omega} - \mathbf{1}_{\Omega}\,dx,$$ in the weak sense if and only if $\Omega$ is homothetic to $K$. Here $\Delta_H$ denotes the anisotropic Laplacian associated to $H$, and $\mathscr H^{n-1}$ denotes the $(n-1)$-dimensional Hausdorff measure. % Furthermore, the two conditions on $|Du|$ become redundant when $\Delta_H$ reduces to the standard Euclidean Laplacian. Our approach offers an alternative proof to [11] in the case of Lipschitz domains, introducing a novel viewpoint to settle [13, Question 7.1].