An anisotropic Serrin's problem in general domains

Abstract

Serrin's symmetry theorem shows that the classical overdetermined torsion problem forces the domain to be a ball. Extending this rigidity statement to merely Lipschitz (and more generally rough) domains in the weak formulation has been a long-standing and challenging problem, recently resolved by the authors in~\cite{FZ2025}. In this paper we address the corresponding question in the anisotropic setting: Given a uniformly convex $C^{2,\gamma}$ anisotropy $H$, we study the overdetermined problem for the anisotropic Laplacian $\Delta_H u={\rm div}\big(H(\nabla u)\,DH(\nabla u)\big)$ on a bounded indecomposable set of finite perimeter $\Omega$. Assuming the Ahlfors--David regularity of $\partial^*\Omega$ and a global $\beta$-number square-function bound (a weak uniform rectifiability hypothesis), we prove that a weak solution exists if and only if $\Omega$ is a translate and dilation of {the reflected Wulff shape $-K$}, in which case the solution is unique and explicit. In particular, the result applies to Lipschitz domains. While our approach follows the rough-domain strategy of~\cite{FZ2025} at a high level, the key Laplacian-specific ingredients exploited there have no direct analog for $\Delta_H$, necessitating the development of new ideas and techniques.