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On the Serrin problem for ring-shaped domains: the $n$-dimensional case

Virginia Agostiniani, Chiara Bernardini, Stefano Borghini, Lorenzo Mazzieri

TL;DR

The paper develops a framework to characterize ring-shaped Serrin-type solutions in $\mathbb{R}^n$ by introducing an expected core radius and a normalization of boundary shear through NWSS. A gradient comparison principle $W\le W_R$ between a general solution and its associated ring-model with the same core radius yields rigidity when equality occurs, leading to rotational symmetry characterizations. It proves well-posedness and non-negativity of the core radius, shows that equal inner and outer radii imply the solution is the ring-model, and derives mean curvature and area bounds with rigidity statements. Extending previous planar results to arbitrary dimensions, the work links domain geometry to PDE overdetermination via comparison geometry, establishing a robust set of a priori and rigidity results for ring-shaped domains.

Abstract

We provide several characterizations of ring-shaped rotationally symmetric solutions to the Serrin problem in arbitrary dimensions.

On the Serrin problem for ring-shaped domains: the $n$-dimensional case

TL;DR

The paper develops a framework to characterize ring-shaped Serrin-type solutions in by introducing an expected core radius and a normalization of boundary shear through NWSS. A gradient comparison principle between a general solution and its associated ring-model with the same core radius yields rigidity when equality occurs, leading to rotational symmetry characterizations. It proves well-posedness and non-negativity of the core radius, shows that equal inner and outer radii imply the solution is the ring-model, and derives mean curvature and area bounds with rigidity statements. Extending previous planar results to arbitrary dimensions, the work links domain geometry to PDE overdetermination via comparison geometry, establishing a robust set of a priori and rigidity results for ring-shaped domains.

Abstract

We provide several characterizations of ring-shaped rotationally symmetric solutions to the Serrin problem in arbitrary dimensions.
Paper Structure (7 sections, 15 theorems, 88 equations)

This paper contains 7 sections, 15 theorems, 88 equations.

Table of Contents

  1. Introduction
  2. Statement of the main results
  3. Well-posedness and non-negativity of the expected core radius
  4. Gradient estimates
  5. Mean curvature bounds
  6. Proof of Theorem \ref{['theo d new']} and Theorem \ref{['theo b new']}
  7. Proof of the area bounds

Key Result

Theorem 2.1

Let $(\Omega,u)$ be a solution to problem 1 where $\Omega$ bounded domain with smooth boundary, and let $N$ be a connected component of $\Omega\setminus\mathrm{MAX}(u)$. Then, Moreover, $R(u,N)=0$ if and only if $(\Omega,u)$ is equivalent to the Serrin solution serrin.

Theorems & Definitions (25)

  • Definition 1.1: Equivalent solutions
  • Definition 1.2: Normalized Wall Shear Stress
  • Remark 1
  • Definition 1.3: Inner and Outer NWSS
  • Definition 1.4: Expected core radius
  • Remark 2
  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Proposition 2.4
  • ...and 15 more