On the Isoperimetric and Isodiametric Inequalities and the Minimisation of Eigenvalues of the Laplacian

Farrington, Sam

Abstract

We consider the problem of minimising the k-th eigenvalue of the Laplacian with some prescribed boundary condition over collections of convex domains of prescribed perimeter or diameter. It is known that these minimisation problems are well-posed for Dirichlet eigenvalues in any dimension d≥2 and any sequence of minimisers converges to the ball of unit perimeter or diameter respectively as k→+∞. In this paper, we show that the same is true in the case of Neumann eigenvalues under diameter constraint in any dimension and under perimeter constraint in dimension d=2. We also consider these problems for Robin eigenvalues and mixed Dirichlet–Neumann eigenvalues, under an additional geometric constraint.

Citation

Farrington, S. (2025). On the Isoperimetric and Isodiametric Inequalities and the Minimisation of Eigenvalues of the Laplacian. Journal of Geometric Analysis, 35(2), Article 62. https://doi.org/10.1007/s12220-024-01887-0

• Journal Article Type: Article
• Acceptance Date: Dec 17, 2024
• Online Publication Date: Jan 4, 2025
• Publication Date: Feb 1, 2025
• Deposit Date: Jan 5, 2025
• Publicly Available Date: Jan 6, 2025
• Journal: The Journal of Geometric Analysis
• Print ISSN: 1050-6926
• Electronic ISSN: 1559-002X
• Publisher: Springer
• Peer Reviewed: Peer Reviewed
• Volume: 35
• Issue: 2
• Article Number: 62
• DOI: https://doi.org/10.1007/s12220-024-01887-0
• Keywords: 49R05, Mixed boundary conditions, Spectral shape optimisation, 35J25, Isoperimetric inequality, 49Q10, 35P15, Weyl’s law, Isodiametric inequality
• Public URL: https://durham-repository.worktribe.com/output/3324558