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.Conjectures with fixed Robin parameter include: a strengthened Rayleigh-Bossel inequality for the first eigenvalue of a convex domain under area normalization; a Szegő-type upper bound on the second eigenvalue of a convex domain; the gap conjecture saying the line segment minimizes the spectral gap under diameter normalization; and the Robin-PPW conjecture on maximality of the spectral ratio for the ball. Questions for a varying Robin parameter include monotonicity of the spectral gap and the spectral ratio, as well as concavity of the second eigenvalue.Results for rectangular domains include that: the square minimizes the first eigenvalue among rectangles under area normalization, when the Robin parameter α ∈ R is scaled by perimeter; that the square maximizes the second eigenvalue for a sharp range of α-values; that the line segment minimizes the Robin spectral gap under diameter normalization for each α ∈ R; and the square maximizes the spectral ratio among rectangles when α > 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 α.Lastly, the shape of a Robin rectangle can be heard from just its first two frequencies, except in the Neumann case.

The Robin problem on rectangles

“…Our goal is to study arithmetic properties and statistics of the Robin eigenvalues on a rectangle. For results related to shape optimization for the first two eigenvalues of the Robin Laplacian on a rectangle, see [7].…”

Courant-sharp Robin eigenvalues for the square: The case of negative Robin parameter

“…(See also Lemma IV.4 of [10]). The corresponding eigenvalues −β 0 (h) 2 /π 2 , −β 1 (h) 2 /π 2 behave like −h 2 as h → −∞.…”

“…We derive the Robin eigenfunctions of an interval for h < 0 (see also Section V of [10] and Subsection 4.3.1 of [3], for example). We wish to solve the following problem:…”

Existence and regularity of optimal shapes for spectral functionals with Robin boundary conditions

“…Similar problems of spectral optimization with Neumann boundary conditions or Robin conditions with negative parameter have been shown to be different in nature, in the former case the first eigenvalue is maximal on the disk, and this is shown with radically different method, mainly building appropriate test functions since the eigenvalues are defined as an infimum through the Courant-Fisher min-max formula. Let us also mention several maximization result for Robin boundary condition with parameter that scales with the perimeter, obtained in [22], [16] with similar methods.…”