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Hausdorff compactness and regularity for classes of open sets under geometric constraints

Mohamed Barkatou

Abstract

This article introduces innovative classes of open sets in \(\mathbb{R}^{N}\), where \(N=2, 3\), characterized by a geometric property associated with the inward normal. The focus lies on proving compactness results for the Hausdorff topology within these classes. Furthermore, the paper establishes the equivalence of convergences, encompassing Hausdorff, compact, and characteristic functions, for select classes. We also investigate the regularity of the thickness function associated with these domains and analyze how the regularity of the fixed convex set \(C\) influences the boundary regularity of the admissible shapes.

Hausdorff compactness and regularity for classes of open sets under geometric constraints

Abstract

This article introduces innovative classes of open sets in , where , characterized by a geometric property associated with the inward normal. The focus lies on proving compactness results for the Hausdorff topology within these classes. Furthermore, the paper establishes the equivalence of convergences, encompassing Hausdorff, compact, and characteristic functions, for select classes. We also investigate the regularity of the thickness function associated with these domains and analyze how the regularity of the fixed convex set influences the boundary regularity of the admissible shapes.

Paper Structure

This paper contains 23 sections, 25 theorems, 88 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and definitions
  3. Minimizing the perimeter on $\mathcal{O}_C$
  4. Analysis of Extremal Solutions
  5. Limit of sequence of open sets satisfying $C$-GNP with volumes tending to $\infty$
  6. Regularity of the thickness function and its implications for boundary regularity under C-GNP
  7. Regularity of the thickness function
  8. Existence of a Bilipschitz Map between $\partial\Omega \setminus C$ and $\partial C$ under C-GNP
  9. Regularity of $\Phi$
  10. Regularity of the Inverse Map
  11. Counterexample without the Condition
  12. Implications for shape regularity
  13. Regularity under Shape Optimization
  14. Example: Constant Thickness
  15. Bilipschitz Applications and Preservation of the C-GNP Property
  16. ...and 8 more sections

Key Result

Proposition 2.6

Let $B_{C}$ be the smallest ball of center $O$ containing $C$. Let $\Omega_{n}$ be a sequence of open connected subsets which satisfy the $B_{C}$-GNP. If $Vol(\Omega_{n})$ tends to $\infty$ then $\Omega_{n}$ converges to the ball $B_{\infty}^{O}$ according to Definition 2.5. $\blacktriangleleft$$\bl

Theorems & Definitions (67)

  • Definition 2.1
  • Example 2.2
  • Example 2.3
  • Example 2.4
  • Definition 2.5
  • Proposition 2.6
  • Lemma 2.7
  • proof
  • proof : Proof of Proposition 2.5
  • Lemma 2.8
  • ...and 57 more