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On the Effectless Cut Method for Laplacian Eigenvalues in any dimensions

Vincenzo Amato, Nunzia Gavitone, Francesca de Giovanni

Abstract

In this paper, we study the optimization of the first Laplacian eigenvalue on axisymmetric doubly connected domains under positive Robin boundary conditions. Under additional geometric constraints, we prove that spherical shells maximize this eigenvalue. Our approach combines known isoperimetric inequalities for mixed Laplacian eigenvalues with a higher-dimensional extension of the effectless cut technique introduced by Hersch to study multiply connected membranes of given area fixed along their boundaries.

On the Effectless Cut Method for Laplacian Eigenvalues in any dimensions

Abstract

In this paper, we study the optimization of the first Laplacian eigenvalue on axisymmetric doubly connected domains under positive Robin boundary conditions. Under additional geometric constraints, we prove that spherical shells maximize this eigenvalue. Our approach combines known isoperimetric inequalities for mixed Laplacian eigenvalues with a higher-dimensional extension of the effectless cut technique introduced by Hersch to study multiply connected membranes of given area fixed along their boundaries.

Paper Structure

This paper contains 11 sections, 19 theorems, 128 equations.

Table of Contents

  1. Introduction
  2. Effectless cut in higher dimension
  3. Critical points
  4. Proof of the main results
  5. Applications
  6. Appendix
  7. Boundary and essential boundary
  8. Quermassintegrals
  9. Some useful Lemmas
  10. The radial cases
  11. Reverse Faber-Krahn for mixed eigenvalue involving a NBC

Key Result

Theorem 1.1

Let $\Omega = \Omega_{out} \setminus \overline{\Omega}_{in} \in \mathcal{A}$. Let $u \in C^{1}(\overline{\Omega})$ be a positive eigenfunction associated with the first eigenvalue of problem probprinc. Then there exists an open set $G$ such that and where $\partial_\ast G$ denotes the essential boundary of $G$, i.e., the subset of $\partial G$ on which the outward unit normal is well defined (se

Theorems & Definitions (40)

  • Definition 1
  • Theorem 1.1
  • Definition 2
  • Theorem 1.2
  • Proposition 1
  • Remark 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • ...and 30 more