An anisotropic Serrin's problem in general domains

TL;DR

This paper extends Serrin's symmetry theorem to anisotropic settings with rough domains, proving that solutions exist only for Wulff shapes under certain regularity conditions, and introduces new techniques for anisotropic Laplacians.

Contribution

The authors generalize the anisotropic Serrin's problem to Lipschitz and rough domains, establishing existence and uniqueness of solutions only for Wulff shapes, with new methods for anisotropic Laplacians.

Findings

Solutions exist only for Wulff shapes under given conditions.

The result applies to Lipschitz domains.

New techniques developed for anisotropic Laplacian analysis.

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 [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.