Differences between Robin and Neumann eigenvalues

TL;DR

This paper investigates the differences between Robin and Neumann eigenvalues for planar domains, establishing their asymptotic behavior, bounds, and convergence properties, with specific results for rectangles and disks.

Contribution

It provides new bounds and asymptotic analysis of Robin-Neumann eigenvalue gaps, including mean value limits and convergence results for various domain classes.

Findings

Mean value of gaps equals boundary length over area times sigma

Upper bounds on gaps for smooth domains and rectangles

Convergence of gaps to mean value in ergodic billiards

Abstract

Let $\Omega\subset \mathbb R^2$ be a bounded planar domain, with piecewise smooth boundary $\partial \Omega$. For $\sigma>0$, we consider the Robin boundary value problem \[ -\Delta f =\lambda f, \qquad \frac{\partial f}{\partial n} + \sigma f = 0 \mbox{ on } \partial \Omega \] where $ \frac{\partial f}{\partial n} $ is the derivative in the direction of the outward pointing normal to $\partial \Omega$. Let $0<\lambda^\sigma_0\leq \lambda^\sigma_1\leq \dots $ be the corresponding eigenvalues. The purpose of this paper is to study the Robin-Neumann gaps \[ d_n(\sigma):=\lambda_n^\sigma-\lambda_n^0 . \] For a wide class of planar domains we show that there is a limiting mean value, equal to $2{\rm length}(\partial\Omega)/{\rm area}(\Omega)\cdot \sigma$ and in the smooth case, give an upper bound of $d_n(\sigma)\leq C(\Omega ) n^{1/3}\sigma $ and a uniform lower bound. For ergodic billiards we show that along a density-one subsequence, the gaps converge to the mean value. We obtain further properties for rectangles, where we have a uniform upper bound, and for disks, where we improve the general upper bound.