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Regularity in shape optimization under convexity constraint

Jimmy Lamboley, Raphaël Prunier

Abstract

This paper is concerned with the regularity of shape optimizers of a class of isoperimetric problems under convexity constraint. We prove that minimizers of the sum of the perimeter and a perturbative term, among convex shapes, are C 1,1-regular. To that end, we define a notion of quasi-minimizer fitted to the convexity context and show that any such quasi-minimizer is C 1,1-regular. The proof relies on a cutting procedure which was introduced to prove similar regularity results in the calculus of variations context. Using a penalization method we are able to treat a volume constraint, showing the same regularity in this case. We go through some examples taken from PDE theory, that is when the perturbative term is of PDE type, and prove that a large class of such examples fit into our C 1,1-regularity result. Finally we provide a counterexample showing that we cannot expect higher regularity in general.

Regularity in shape optimization under convexity constraint

Abstract

This paper is concerned with the regularity of shape optimizers of a class of isoperimetric problems under convexity constraint. We prove that minimizers of the sum of the perimeter and a perturbative term, among convex shapes, are C 1,1-regular. To that end, we define a notion of quasi-minimizer fitted to the convexity context and show that any such quasi-minimizer is C 1,1-regular. The proof relies on a cutting procedure which was introduced to prove similar regularity results in the calculus of variations context. Using a penalization method we are able to treat a volume constraint, showing the same regularity in this case. We go through some examples taken from PDE theory, that is when the perturbative term is of PDE type, and prove that a large class of such examples fit into our C 1,1-regularity result. Finally we provide a counterexample showing that we cannot expect higher regularity in general.
Paper Structure (20 sections, 23 theorems, 181 equations, 3 figures)

This paper contains 20 sections, 23 theorems, 181 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Motivations:
  3. State of the art about regularity theory with convexity constraint:
  4. Strategy of proof and plan of the paper:
  5. Regularity in shape optimization
  6. Regularity for quasi-minimizers of the perimeter under convexity constraint
  7. Regularity with volume constraint
  8. Preliminaries
  9. Main result
  10. Examples and applications
  11. First examples
  12. PDE and Spectral examples
  13. A general existence result
  14. Selected examples
  15. Torsional rigidity and Dirichlet eigenvalues
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $n\in {\mathbb N}^*$, $N\geq2$, $F:(0,+\infty) \times (0,+\infty)\times (0,+\infty)^n\times{\mathbb R}_{+}^n \rightarrow{\mathbb R}$ be locally Lipschitz. Let $R:{\mathcal{K}}^N\rightarrow{\mathbb R}$ be defined by the formula where $\tau(K)$ is the torsional rigidity of $K$, $\lambda_{1}(K),\ldots,\lambda_{n}(K)$ are the $n$ first Dirichlet eigenvalues of $K$, and $\mu_{1}(K),\ldots,\mu_{n}(

Figures (3)

  • Figure 1: Cutting procedure
  • Figure 2: Localization of $\omega_r$
  • Figure 3: Convex body in cartesian graph

Theorems & Definitions (46)

  • Theorem 1.1
  • Definition 2.1
  • Remark 2.2
  • Theorem 2.3
  • Corollary 2.4
  • Remark 2.5
  • Remark 2.6
  • Remark 2.7
  • Proposition 2.8
  • Proposition 2.9
  • ...and 36 more