Isoperimetric relations between Dirichlet and Neumann eigenvalues
Analysis of PDEs
2019-06-25 v1 Mathematical Physics
Differential Geometry
math.MP
Spectral Theory
Abstract
Inequalities between the Dirichlet and Neumann eigenvalues of the Laplacian have received much attention in the literature, but open problems abound. Here, we study the number of Neumann eigenvalues no greater than the first Dirichlet eigenvalue. Based on a combination of analytical and numerical results, we conjecture that this number is controlled by the isoperimetric ratio of the domain. This has applications to the nodal deficiency of eigenfunctions and is closely related to a long-standing conjecture of Yau on the Hausdorff measure of nodal sets.
Cite
@article{arxiv.1906.10061,
title = {Isoperimetric relations between Dirichlet and Neumann eigenvalues},
author = {Graham Cox and Scott Scott MacLachlan and Luke Steeves},
journal= {arXiv preprint arXiv:1906.10061},
year = {2019}
}
Comments
15 pages, 8 figures