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On the isoperimetric and isodiametric inequalities and the minimisation of eigenvalues of the Laplacian

Sam Farrington

Abstract

We consider the problem of minimising the $k$-th eigenvalue of the Laplacian with some prescribed boundary condition over collections of convex domains of prescribed perimeter or diameter. It is known that these minimisation problems are well-posed for Dirichlet eigenvalues in any dimension $d\geq 2$ and any sequence of minimisers converges to the ball of unit perimeter or diameter respectively as $k\to +\infty$. In this paper, we show that the same is true in the case of Neumann eigenvalues under diameter constraint in any dimension and under perimeter constraint in dimension $d=2$. We also consider these problems for mixed Dirichlet-Neumann eigenvalues, under an additional geometric constraint, and discuss some applications of our proof techniques.

On the isoperimetric and isodiametric inequalities and the minimisation of eigenvalues of the Laplacian

Abstract

We consider the problem of minimising the -th eigenvalue of the Laplacian with some prescribed boundary condition over collections of convex domains of prescribed perimeter or diameter. It is known that these minimisation problems are well-posed for Dirichlet eigenvalues in any dimension and any sequence of minimisers converges to the ball of unit perimeter or diameter respectively as . In this paper, we show that the same is true in the case of Neumann eigenvalues under diameter constraint in any dimension and under perimeter constraint in dimension . We also consider these problems for mixed Dirichlet-Neumann eigenvalues, under an additional geometric constraint, and discuss some applications of our proof techniques.
Paper Structure (9 sections, 23 theorems, 105 equations, 3 figures)

This paper contains 9 sections, 23 theorems, 105 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Bounding Dirichlet and Neumann counting functions
  3. Proofs of Theorems \ref{['thm:neumann_perim']}, \ref{['thm:neumann_diam']} and \ref{['thm:neumann_per_diam']}
  4. Geometric stability of Weyl's law
  5. Mixed boundary conditions
  6. Properties of $\mathcal{O}_{d,L}$
  7. Continuity of the $\zeta_{k}$
  8. Proof of Theorems \ref{['thm:Zaremba_surface_measure']} and \ref{['thm:Zaremba_diamater']}
  9. An open question

Key Result

Theorem 1.1

For each $k\geq 1$, there exists a minimiser to prob:dir_perim. Moreover, any sequence of minimisers Hausdorff converges to the ball of unit perimeter as $k\to +\infty$.

Figures (3)

  • Figure 1: An example of symmetric Zaremba boundary conditions on a kite about its axis of symmetry, with Dirchlet boundary conditions denoted in blue and Neumann boundary conditions denoted in red.
  • Figure 2: Numerically computed optimal solutions to the isoperimetric problem with unit perimeter over $\mathcal{O}_{2,L}$ with $L=0.5$ (top left), $L=1$ (top right), $L=2$ (bottom left) and $L=4$ (bottom right).
  • Figure 3: Illustation of the two cases arising from Question \ref{['ques:DN-Polya']} with $\partial Q_{\delta}$ shown in red and $\Gamma$ shown in blue.

Theorems & Definitions (43)

  • Theorem 1.1: van-den-Berg-2015
  • Theorem 1.2: van-den-Berg-2015
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Remark 1.6
  • Theorem 2.1: Minkowski-Steiner, see Gruber-2007
  • Proposition 2.2
  • proof
  • Remark 2.3
  • ...and 33 more