Serrin's overdetermined theorem within Lipschitz domains

Let $\Omega\subset\mathbb R^n$ be a Lipschitz domain. We prove that, $\Omega$ satisfies the following Serrin-type overdetermined system $$u \in W^{1,2}(\mathbb R^n), \quad u=0\ \text{ a.e. in }\mathbb R^n\setminus \Omega,\quad \Delta u=\mathbf{c}\mathscr{H}^{n-1}|_{\partial^*\Omega} - \mathbf{1}_{\Omega}\,dx,$$ in the weak sense if and only if $\Omega$ 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].