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A new, self-contained proof of Shahgholian's theorem using the thickness function

Mohammed Barkatou

Abstract

This note presents a new, self-contained proof of Shahgholian's geometric theorem on quadrature surfaces using the thickness function and level set methods. By relying on a radial parametrisation and fundamental maximum principles, the proof avoids the technical complexity of the moving plane method. It provides a more conceptual view, revealing that the overdetermined condition forces all level sets to be parallel to the convex hull of the support of the measure.

A new, self-contained proof of Shahgholian's theorem using the thickness function

Abstract

This note presents a new, self-contained proof of Shahgholian's geometric theorem on quadrature surfaces using the thickness function and level set methods. By relying on a radial parametrisation and fundamental maximum principles, the proof avoids the technical complexity of the moving plane method. It provides a more conceptual view, revealing that the overdetermined condition forces all level sets to be parallel to the convex hull of the support of the measure.

Paper Structure

This paper contains 10 sections, 9 theorems, 10 equations.

Table of Contents

  1. Introduction and Shahgholian's theorem
  2. The thickness function and its properties
  3. Radial parametrisation of $\Omega$
  4. The key rigidity lemma
  5. Explicit form of the thickness function
  6. Proof of Shahgholian's theorem
  7. Comparison with the original proof
  8. Extensions and generalisations
  9. Conclusion
  10. Proof of the rigidity lemma (detailed)

Key Result

Theorem 1.1

Let $\Omega \subset \mathbb{R}^N$ be a bounded $C^2$ domain and let $\mu$ be a positive measure compactly supported in $\Omega$. If $\partial\Omega$ is a quadrature surface for $\mu$, i.e. there exists a solution $u$ of the overdetermined problem then for every $x\in\partial\Omega$ the inward normal ray to $\partial\Omega$ at $x$ meets the convex hull of $\text{supp}\mu$.

Theorems & Definitions (18)

  • Theorem 1.1: Shahgholian, 1994
  • Proposition 2.1: Strict radial decrease
  • proof
  • Proposition 2.2: Thickness function
  • proof
  • Proposition 2.3: Differential relation
  • proof
  • Proposition 3.1
  • proof
  • Lemma 4.1: Rigidity
  • ...and 8 more