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Optimal bounds for the first two Steklov eigenvalues of Euclidean domains

Denis Vinokurov

Abstract

We establish upper bounds for the first two nonzero Steklov eigenvalues of bounded domains in Euclidean spaces of dimension $d \geq 3$, under a natural normalization involving volume and boundary measure, and show that these bounds are sharp for $d \geq 7$.

Optimal bounds for the first two Steklov eigenvalues of Euclidean domains

Abstract

We establish upper bounds for the first two nonzero Steklov eigenvalues of bounded domains in Euclidean spaces of dimension , under a natural normalization involving volume and boundary measure, and show that these bounds are sharp for .

Paper Structure

This paper contains 8 sections, 7 theorems, 36 equations.

Table of Contents

  1. Introduction and main results
  2. Acknowledgements
  3. Preliminaries
  4. Proofs of the main results
  5. The ball maximizes Lg
  6. Proof of Theorem \ref{['thm:max-lambda-1']}
  7. The two balls maximize Lg
  8. Proof of Theorem \ref{['thm:max-lambda-2']}

Key Result

Proposition 1.4

Let $\Omega \subset \mathbb{R}^d$ be a bounded $C^1$-domain. Then there exists a family of $C^1$-domains $\Omega^\varepsilon \subset \Omega$ such that $|\Omega^\varepsilon|\to |\Omega|$ and $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (14)

  • Definition 1.1
  • Remark 1.2
  • Remark 1.3
  • Proposition 1.4: Girouard-Karpukhin-Lagace:2021:continuity-of-eigenval
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Remark 1.8
  • Proposition 2.1
  • proof
  • ...and 4 more