Differences Between Robin and Neumann Eigenvalues.

Let Ω âŠ‚ 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.