The Robin Laplacian - spectral conjectures, rectangular theorems

Abstract

The first two eigenvalues of the Robin Laplacian are investigated along with their gap and ratio. Conjectures by various authors for arbitrary domains are supported here by new results for rectangular boxes. Results for rectangular domains include that: the square minimizes the first eigenvalue among rectangles under area normalization, when the Robin parameter $\alpha \in \mathbb{R}$ is scaled by perimeter; that the square maximizes the second eigenvalue for a sharp range of $\alpha$-values; that the line segment minimizes the Robin spectral gap under diameter normalization for each $\alpha \in \mathbb{R}$; and the square maximizes the spectral ratio among rectangles when $\alpha >0$. Further, the spectral gap of each rectangle is shown to be an increasing function of the Robin parameter, and the second eigenvalue is concave with respect to $\alpha$. Lastly, the shape of a Robin rectangle can be heard from just its first two frequencies, except in the Neumann case.

44 pages, 7 figures

References in corpus (2)

• On the lowest eigenvalue of Laplace operators with mixed boundary conditions
• Parallel coordinates in three dimensions and sharp spectral isoperimetric inequalities

Cited by in corpus (8)

• Spectral optimisation of Dirac rectangles
• A numerical study of the Dirichlet-to-Neumann operator in planar domains
• The Robin problem on rectangles
• Spectral inequality for Dirac right triangles
• Faber-Krahn inequalities for Schrödinger operators with point and with Coulomb interactions
• The $\overline{\partial }$-Robin Laplacian
• Spectral ratios and gaps for Steklov eigenvalues of balls with revolution-type metrics
• Reverse Faber-Krahn and Szego-Weinberger type inequalities for annular domains under Robin-Neumann boundary conditions