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Boundary Perturbations of Steklov Eigenvalues

Lihan Wang

TL;DR

The paper addresses whether non-zero Steklov eigenvalues are generically simple under smooth boundary perturbations of a domain, reframing the problem through the Dirichlet-to-Neumann operator $\Lambda$. A two-step variational framework tracks how the pulled-back operator $L_h=h^{*}\Lambda h^{*-1}$ changes with boundary deformations, and two evaluation maps convert eigenvalue degeneracy into a transversality problem. By applying Henry's generalized transversality theorem and a weak unique continuation principle for Steklov eigenfunctions, the authors prove there exists a residual set of embeddings $h$ for which all non-zero Steklov eigenvalues on $\Omega_h$ are simple. This result extends prior work on metric perturbations, providing a robust geometric basis for the generic simplicity of the Steklov spectrum with implications for shape optimization and inverse problems in spectral geometry.

Abstract

We consider the dependence of non-zero Steklov eigenvalues on smooth perturbations of the domain boundary. We prove that these eigenvalues are generically simple under such boundary perturbations. This result complements our previous work on metric perturbations, thereby establishing generic simplicity Steklov eigenvalues under both fundamental geometric variations.

Boundary Perturbations of Steklov Eigenvalues

TL;DR

The paper addresses whether non-zero Steklov eigenvalues are generically simple under smooth boundary perturbations of a domain, reframing the problem through the Dirichlet-to-Neumann operator . A two-step variational framework tracks how the pulled-back operator changes with boundary deformations, and two evaluation maps convert eigenvalue degeneracy into a transversality problem. By applying Henry's generalized transversality theorem and a weak unique continuation principle for Steklov eigenfunctions, the authors prove there exists a residual set of embeddings for which all non-zero Steklov eigenvalues on are simple. This result extends prior work on metric perturbations, providing a robust geometric basis for the generic simplicity of the Steklov spectrum with implications for shape optimization and inverse problems in spectral geometry.

Abstract

We consider the dependence of non-zero Steklov eigenvalues on smooth perturbations of the domain boundary. We prove that these eigenvalues are generically simple under such boundary perturbations. This result complements our previous work on metric perturbations, thereby establishing generic simplicity Steklov eigenvalues under both fundamental geometric variations.
Paper Structure (7 sections, 16 theorems, 109 equations)

This paper contains 7 sections, 16 theorems, 109 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Variation formulas and the weak unique continuity principle
  4. Main results
  5. First evaluation map
  6. Second evaluation map
  7. Proof of Main theorem

Key Result

Theorem 1.1

Let $\Omega\in \mathbb{R}^n$ be a bounded domain with $C^m$ boundary for $m\geq 2$. Then there exists a residual set of $C^m$ embedding $h$ of $\Omega$ into $\mathbb{R}^n$ such that all non-zero Steklov eigenvalues on $h(\Omega)$ are simple.

Theorems & Definitions (35)

  • Theorem 1.1: Main Theorem
  • Remark 1.2
  • Proposition 2.1
  • Proposition 2.2
  • Lemma 2.3
  • Remark 2.4
  • Lemma 3.1: Variation of $u^h$
  • proof
  • Lemma 3.2: Variation of $L_h$
  • proof
  • ...and 25 more