Serrin's overdetermined theorem within Lipschitz domains

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