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On Pleijel's nodal domain theorem for the Robin problem

Asma Hassannezhad, David Sher

Abstract

We prove an improved Pleijel nodal domain theorem for the Robin eigenvalue problem. In particular we remove the restriction, imposed in previous work, that the Robin parameter be non-negative. We also improve the upper bound in the statement of the Pleijel theorem. In the particular example of a Euclidean ball, we calculate the explicit value of the Pleijel constant for a generic constant Robin parameter and we show that it is equal to the Pleijel constant for the Dirichlet Laplacian on a Euclidean ball.

On Pleijel's nodal domain theorem for the Robin problem

Abstract

We prove an improved Pleijel nodal domain theorem for the Robin eigenvalue problem. In particular we remove the restriction, imposed in previous work, that the Robin parameter be non-negative. We also improve the upper bound in the statement of the Pleijel theorem. In the particular example of a Euclidean ball, we calculate the explicit value of the Pleijel constant for a generic constant Robin parameter and we show that it is equal to the Pleijel constant for the Dirichlet Laplacian on a Euclidean ball.
Paper Structure (4 sections, 8 theorems, 86 equations)

This paper contains 4 sections, 8 theorems, 86 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['Rpleijel']}
  3. Pleijel constant for the Robin problem on a Euclidean ball
  4. Proof of Lemma \ref{['lem:technical']}

Key Result

Theorem 1.1

There exists a positive constant $\epsilon(d)$ depending only on $d$ such that for any open, bounded, connected domain $\Omega\subset \mathbb{R}^d$ with $C^{1,1}$ boundary we have where $\gamma(d)$ is as in eq:BDS.

Theorems & Definitions (19)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.4
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Lemma 2.3
  • proof : Proof of Lemma \ref{['distance function']}
  • Remark 2.4
  • ...and 9 more