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].
Categories: math.AP