Let Ω ⊂ R 2 be a bounded planar domain, with piecewise smooth boundary ∂ Ω . For σ > 0 , we consider the Robin boundary value problem - Δ f = λ f , ∂ f ∂ n + σ f = 0 on ∂ Ω where ∂ f ∂ n is the derivative in the direction of the outward pointing normal to ∂ Ω . Let 0 < λ 0 σ ≤ λ 1 σ ≤ … 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 length ( ∂ Ω ) / area ( Ω ) · σ and in the smooth case, give an upper bound of d n ( σ ) ≤ C ( Ω ) n 1 / 3 σ 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.
Bibliographical noteFunding Information:
We thank Michael Levitin and Iosif Polterovich for the their comments. This research was supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 786758) and by the ISRAEL SCIENCE FOUNDATION (Grant No. 1881/20).
© 2021, The Author(s).
ASJC Scopus subject areas
- Statistical and Nonlinear Physics
- Mathematical Physics