Differences between Robin and Neumann eigenvalues

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 \[ -Δf =λf, \qquad \frac{\partial f}{\partial n} + σf = 0 \mbox{ on } \partial Ω\] where $\frac{\partial f}{\partial n}$ is the derivative in the direction of the outward pointing normal to $\partial \Omega$. Let $0<{\lambda }_0^{\sigma }\le {\lambda }_1^{\sigma }\le \dots$ be the corresponding eigenvalues. The purpose of this paper is to study the Robin-Neumann gaps \[ d_n(σ):=λ_n^σ-λ_n^0 . \] For a wide class of planar domains we show that there is a limiting mean value, equal to $2\mathrm{length}(\partial \Omega )/\mathrm{area}(\Omega )\cdot \sigma$ and in the smooth case, give an upper bound of $d_n(\sigma )\le C(\Omega )n^{1/3}\sigma$ 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.

Several changes. Added references and comments about higher dimensions and variable Robin function