Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Optimization of Neumann Eigenvalues under convexity and geometric constraints

Beniamin Bogosel, Antoine Henrot, Marco Michetti

Abstract

In this paper we study optimization problems for Neumann eigenvalues $μ_k$ among convex domains with a constraint on the diameter or the perimeter. We work mainly in the plane, though some results are stated in higher dimension. We study the existence of an optimal domain in all considered cases. We also consider the case of the unit disk, giving values of the index $k$ for which it can be or cannot be extremal. We give some numerical examples for small values of $k$ that lead us to state some conjectures.

Optimization of Neumann Eigenvalues under convexity and geometric constraints

Abstract

In this paper we study optimization problems for Neumann eigenvalues among convex domains with a constraint on the diameter or the perimeter. We work mainly in the plane, though some results are stated in higher dimension. We study the existence of an optimal domain in all considered cases. We also consider the case of the unit disk, giving values of the index for which it can be or cannot be extremal. We give some numerical examples for small values of that lead us to state some conjectures.
Paper Structure (4 sections, 12 theorems, 108 equations, 3 figures)

This paper contains 4 sections, 12 theorems, 108 equations, 3 figures.

Table of Contents

  1. Introduction
  2. existence of optimal shapes
  3. analysis of the disk
  4. Some numerical results

Key Result

Lemma 2.2

Let $h_\epsilon$ be a sequence of functions in $\mathcal{L}$ that converges in $L^2(0,1)$ to a function $h\in \mathcal{L}$, then for any decomposition $h_{\epsilon}=h_{\epsilon}^++h_{\epsilon}^-$ as a sum of two nonnegative concave functions, and we set we have

Figures (3)

  • Figure 1: Convex shapes minimizing the $k$-th Neumann eigenvalue for shapes with unit diameter, $2 \leq k \le 9$.
  • Figure 2: Minimizers for the $k$-th Neumann eigenvalue among shapes with unit perimeter.
  • Figure 3: Maximizers for the $k$-th Neumann eigenvalue among shapes with unit perimeter.

Theorems & Definitions (26)

  • Definition 2.1: Sturm-Liouville eigenvalues
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • proof
  • Theorem 2.5
  • proof
  • Theorem 2.6
  • ...and 16 more