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 [12].
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 Wulff shape, 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 [12] 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.
