Differences Between Robin and Neumann Eigenvalues

Let Ω ⊂ R² be a bounded planar domain, with piecewise smooth boundary ∂Ω. For σ> 0 , we consider the Robin boundary value problem -Δf=λf,∂f∂n+σf=0on∂Ωwhere ∂f∂n is the derivative in the direction of the outward pointing normal to ∂Ω. Let 0<λ0σ≤λ1σ≤… be the corresponding eigenvalues. The purpose of this paper is to study the Robin–Neumann gaps dn(σ):=λnσ-λn0.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.