Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Serrin's overdetermined problem in rough domains

Alessio Figalli, Yi Ru-Ya Zhang

TL;DR

This work proves Serrin's overdetermined problem holds for rough, indecomposable sets of finite perimeter with a uniform boundary-density bound, showing that the only domain admitting a constant-mean-curvature-like solution is a ball and the solution is $u(x)=\frac{R^2-|x|^2}{2n}$. The authors develop a Green-function based maximum principle for $|\nabla u|$ in non-smooth settings and leverage geometric measure theory to obtain a key volume identity, circumventing the standard moving-planes approach. The main theorem extends Serrin's result to Lipschitz domains and even to domains with slit discontinuities, thereby broadening the applicability of the classical symmetry result. An extension to slit domains (Theorem 2) shows how the weak formulation can accommodate an additional boundary set and the corresponding blow-up behavior, preserving the same rigidity conclusion.

Abstract

The classical Serrin's overdetermined theorem states that a $C^2$ bounded domain, which admits a function with constant Laplacian that satisfies both constant Dirichlet and Neumann boundary conditions, must necessarily be a ball. While extensions of this theorem to non-smooth domains have been explored since the 1990s, the applicability of Serrin's theorem to Lipschitz domains remained unresolved. This paper answers this open question affirmatively. Actually, our approach shows that the result holds for domains that are sets of finite perimeter with a uniform upper bound on the density, and it also allows for slit discontinuities.

Serrin's overdetermined problem in rough domains

TL;DR

This work proves Serrin's overdetermined problem holds for rough, indecomposable sets of finite perimeter with a uniform boundary-density bound, showing that the only domain admitting a constant-mean-curvature-like solution is a ball and the solution is . The authors develop a Green-function based maximum principle for in non-smooth settings and leverage geometric measure theory to obtain a key volume identity, circumventing the standard moving-planes approach. The main theorem extends Serrin's result to Lipschitz domains and even to domains with slit discontinuities, thereby broadening the applicability of the classical symmetry result. An extension to slit domains (Theorem 2) shows how the weak formulation can accommodate an additional boundary set and the corresponding blow-up behavior, preserving the same rigidity conclusion.

Abstract

The classical Serrin's overdetermined theorem states that a bounded domain, which admits a function with constant Laplacian that satisfies both constant Dirichlet and Neumann boundary conditions, must necessarily be a ball. While extensions of this theorem to non-smooth domains have been explored since the 1990s, the applicability of Serrin's theorem to Lipschitz domains remained unresolved. This paper answers this open question affirmatively. Actually, our approach shows that the result holds for domains that are sets of finite perimeter with a uniform upper bound on the density, and it also allows for slit discontinuities.
Paper Structure (8 sections, 11 theorems, 86 equations)

This paper contains 8 sections, 11 theorems, 86 equations.

Table of Contents

  1. Introduction
  2. Serrin's Theorem
  3. Generalizations
  4. Serrin's Theorem for more singular domains
  5. Main result
  6. Preliminary results
  7. Proof of Theorem \ref{['main thm']}
  8. Extension to slit domains

Key Result

Theorem 1.1

Let $\Omega\subset \mathbb{R}^n$ be a $C^2$ bounded domain. Then serrin admits a solution $u \in C^2(\Omega)\cap C^1(\overline{\Omega})$ if and only if, up to a translation, $\Omega$ is a ball of radius $R=R(n,\mathbf{c})>0$ and $u$ takes the form

Theorems & Definitions (24)

  • Theorem 1.1: S1971
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • ...and 14 more