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An anisotropic Serrin's problem in general domains

Alessio Figalli, Yi Ru-Ya Zhang

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 $C^{2,γ}$ anisotropy $H$, we study the overdetermined problem for the anisotropic Laplacian $Δ_H u={\rm div}\big(H(\nabla u)\,DH(\nabla u)\big)$ on a bounded indecomposable set of finite perimeter $Ω$. Assuming the Ahlfors--David regularity of $\partial^*Ω$ and a global $β$-number square-function bound (a weak uniform rectifiability hypothesis), we prove that a weak solution exists if and only if $Ω$ 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 $Δ_H$, necessitating the development of new ideas and techniques.

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 [12]. In this paper we address the corresponding question in the anisotropic setting: Given a uniformly convex anisotropy , we study the overdetermined problem for the anisotropic Laplacian on a bounded indecomposable set of finite perimeter . Assuming the Ahlfors--David regularity of and a global -number square-function bound (a weak uniform rectifiability hypothesis), we prove that a weak solution exists if and only if 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 , necessitating the development of new ideas and techniques.
Paper Structure (9 sections, 8 theorems, 201 equations)

This paper contains 9 sections, 8 theorems, 201 equations.

Table of Contents

  1. Introduction
  2. Serrin's theorem in general domains
  3. Main result
  4. Preliminary results
  5. Proof of Theorem \ref{['main thm']}
  6. Green function and harmonic measure for $L_{\mathcal{A}}$
  7. A maximum principle for $H(\nabla u)$
  8. Conclusion via the $P$-function
  9. A covering lemma under ADR with a $1/|\log r|$ gain at selected scales

Key Result

Theorem 1.1

Let $\Omega\subset \mathbb R^n$ be a bounded indecomposable set of finite perimeter and assume there exist constant $A_1,A_2\geq 1$ such that and that, for $\mathscr H^{n-1}$-a.e. $x\in \partial^* \Omega$ and every $r \in (0,1)$, Let $K$ be a bounded uniformly convex body whose boundary is of class $C^{2,\gamma}$ for some $\gamma>0$ small, and let $H$ be the corresponding Wulff potential. Then $

Theorems & Definitions (21)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • ...and 11 more