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Simplicity of eigenvalues for elliptic problems with mixed Steklov-Robin boundary condition

Marco Ghimenti, Anna Maria Micheletti, Angela Pistoia

Abstract

This paper investigates the spectral properties of two classes of elliptic problems characterized by mixed Steklov-Robin boundary conditions. Our main objective is to prove that, for a generic domain, all the eigenvalues are simple. This result is established by employing domain perturbation techniques and analyzing the transversality of the associated operators.

Simplicity of eigenvalues for elliptic problems with mixed Steklov-Robin boundary condition

Abstract

This paper investigates the spectral properties of two classes of elliptic problems characterized by mixed Steklov-Robin boundary conditions. Our main objective is to prove that, for a generic domain, all the eigenvalues are simple. This result is established by employing domain perturbation techniques and analyzing the transversality of the associated operators.
Paper Structure (14 sections, 12 theorems, 89 equations)

This paper contains 14 sections, 12 theorems, 89 equations.

Table of Contents

  1. Introduction
  2. Notations
  3. Variational setting and proof of Theorem \ref{['thm:main-simp']}
  4. The no splitting condition
  5. Perturbations supported in $S$
  6. Perturbations supported in $W$
  7. Proof of Theorem \ref{['thm:main-simp']}
  8. Proof of Theorem \ref{['thm:2-simp']}
  9. Problem with coefficients
  10. No splitting condition for problem (\ref{['pot-down']})
  11. No splitting condition for problem (\ref{['pot-up']})
  12. Proof of Theorem \ref{['thm:pot']}
  13. Proof of Theorem \ref{['thm:div']}
  14. Computation of derivatives

Key Result

Theorem 1

For any $\varepsilon >0$, there exists $\psi \in \mathcal{D}$, with $\|\psi\|_{C^2}<\varepsilon$, for which all the eigenvalues of the problem are simple. Also, $\psi$ can be chosen to leave one of the two components of the boundary, $S$ or $W$, unchanged.

Theorems & Definitions (19)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Corollary 5
  • Theorem 6
  • Corollary 7
  • Remark 8
  • Proposition 9
  • proof
  • ...and 9 more