An anisotropic Serrin 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 [J. Eur. Math. Soc., DOI 10.4171/JEMS/1726]. 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 ${\Delta }_{H}u=\mathrm{div}(H(\nabla u)DH(\nabla u))$ on a bounded indecomposable set of finite perimeter $\Omega$. Assuming the Ahlfors–David regularity of ${\partial }^{\ast }\Omega$ 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 [J. Eur. Math. Soc., DOI 10.4171/JEMS/1726] 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.