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Minimisation of Laplacian eigenvalue with indefinite weight under inhomogeneous Robin boundary condition

Baruch Schneider, Diana Schneiderová, Yifan Zhang

Abstract

This paper explores a certain Laplacian eigenvalue optimisation problem with indefinite weight under inhomogeneous Robin boundary condition. The minimum principal eigenvalue is fully determined in one dimension by formulating the problem as a shape optimisation one. The result is verified numerically using a shooting method.

Minimisation of Laplacian eigenvalue with indefinite weight under inhomogeneous Robin boundary condition

Abstract

This paper explores a certain Laplacian eigenvalue optimisation problem with indefinite weight under inhomogeneous Robin boundary condition. The minimum principal eigenvalue is fully determined in one dimension by formulating the problem as a shape optimisation one. The result is verified numerically using a shooting method.
Paper Structure (6 sections, 11 theorems, 68 equations, 1 figure, 1 table)

This paper contains 6 sections, 11 theorems, 68 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. The 1-dimensional case
  3. Remark on homogeneous Robin boundary
  4. Transfer–matrix shooting
  5. Choice of parameters
  6. Outputs

Key Result

Lemma 1

For $\phi\in H^1(\Omega)$, let and Then $\lambda$ is a principal eigenvalue of eq:pb, if and only if $\mu(\lambda)=0$.

Figures (1)

  • Figure 1: Canonical examples for the six non-degenerate cases of Theorem \ref{['thm:onedim']}.

Theorems & Definitions (17)

  • Lemma 1: Smoller_1994, Theorem 11.4 and 11.10
  • Lemma 2: Afrouzi1999, Lemma 2
  • Theorem 3
  • Theorem 4
  • proof
  • Theorem 5
  • Proposition 6
  • Lemma 7
  • proof
  • Lemma 8
  • ...and 7 more