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Estimates for the first and second Steklov-Dirichlet eigenvalues

Rossano Sannipoli

TL;DR

This work analyzes the Steklov-Dirichlet eigenvalue problem for the Laplacian in annular domains $Ω_r = Ω_0 minus B_r$, establishing that the first eigenvalue vanishes as the hole shrinks and that the corresponding eigenfunctions converge to a fixed constant on $Ω_0$. It then shows the second Steklov-Dirichlet eigenvalue converges to the first non-trivial Steklov eigenvalue of the unperforated domain, with associated eigenfunctions converging in $H^1$, and uses this to derive isoperimetric inequalities for small holes via Brock and Weinstock results. A radial-case analysis and a corrector function $ω^ε$ underpin the proofs, providing precise convergence and comparison tools that inform shape optimization for perforated domains. The findings illuminate how small perforations influence Steklov-Dirichlet spectra and yield quantitative bounds under measure and perimeter constraints.

Abstract

In this paper, we deal with the Steklov-Dirichlet eigenvalue problem for the Laplacian in annular domains. More precisely, we consider $Ω_r = Ω_0 \setminus \overline{B}_r$, where $Ω_0 \subset \mathbb{R}^n$, $n \geq 2$, is an open, bounded set with a Lipschitz boundary, and $B_r$ is the ball centered at the origin with radius $r > 0$, such that $\overline{B}_r \subset Ω_0$. In the first part of the paper, we focus on the first Steklov-Dirichlet eigenvalue $σ_1(Ω_r)$ and prove that the sequence of corresponding normalized eigenfunctions converges to a particular constant as $r \to 0^+$. This will allow us to prove an isoperimetric inequality for $ σ_1(Ω_r)$ when $r$ is small enough, under a measure constraint. The second part is focused on the second Steklov-Dirichlet eigenvalue $σ_2(Ω_r)$. We prove that it converges to the first non-trivial Steklov eigenvalue $\overlineσ_1(Ω_0)$ of the non-perforated domain $Ω_0$. This result, together with the Brock and Weinstock inequalities, respectively, allows us to prove two isoperimetric inequalities for small holes.

Estimates for the first and second Steklov-Dirichlet eigenvalues

TL;DR

This work analyzes the Steklov-Dirichlet eigenvalue problem for the Laplacian in annular domains , establishing that the first eigenvalue vanishes as the hole shrinks and that the corresponding eigenfunctions converge to a fixed constant on . It then shows the second Steklov-Dirichlet eigenvalue converges to the first non-trivial Steklov eigenvalue of the unperforated domain, with associated eigenfunctions converging in , and uses this to derive isoperimetric inequalities for small holes via Brock and Weinstock results. A radial-case analysis and a corrector function underpin the proofs, providing precise convergence and comparison tools that inform shape optimization for perforated domains. The findings illuminate how small perforations influence Steklov-Dirichlet spectra and yield quantitative bounds under measure and perimeter constraints.

Abstract

In this paper, we deal with the Steklov-Dirichlet eigenvalue problem for the Laplacian in annular domains. More precisely, we consider , where , , is an open, bounded set with a Lipschitz boundary, and is the ball centered at the origin with radius , such that . In the first part of the paper, we focus on the first Steklov-Dirichlet eigenvalue and prove that the sequence of corresponding normalized eigenfunctions converges to a particular constant as . This will allow us to prove an isoperimetric inequality for when is small enough, under a measure constraint. The second part is focused on the second Steklov-Dirichlet eigenvalue . We prove that it converges to the first non-trivial Steklov eigenvalue of the non-perforated domain . This result, together with the Brock and Weinstock inequalities, respectively, allows us to prove two isoperimetric inequalities for small holes.
Paper Structure (12 sections, 13 theorems, 89 equations)

This paper contains 12 sections, 13 theorems, 89 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations
  4. Friedrich inequalities
  5. On the Steklov eigenvalue problem
  6. On the Steklov-Dirichlet eigenvalue problem
  7. The radial case
  8. About the nodal domains
  9. A corrector function
  10. The main results
  11. Results related to $\sigma_1$ and the corresponding eigenfunctions
  12. Results related to $\sigma_2$ and the corresponding eigenfunctions

Key Result

Theorem 1.1

Let $\Omega_r=\Omega_0\setminus\overline{B}_r$, where $\Omega_0$ is an open, bounded and connected set in $\mathbb R^n$ with Lipschitz boundary and $B_r$ a ball of radius $r>0$ centered at the origin, such that $\overline{B}_r \subset \Omega_0$. There exists a $r_1^M=r_1^M(\Omega_0)$ depending only where $R_M>0$ is the radius of the ball with the same measure as $\Omega_0$. The equality case hold

Theorems & Definitions (25)

  • Theorem 1.1
  • Theorem 1.2: S. Basak, A. Chorwadwala, S. Verma
  • Theorem 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Remark 1.6
  • Remark 2.1
  • Proposition 2.2
  • Proposition 2.3
  • proof
  • ...and 15 more