Differences Between Robin and Neumann Eigenvalues

Let $$\Omega {\subset } {\mathbb {R}}^2$$ Ω ⊂ R 2 be a bounded planar domain, with piecewise smooth boundary $$\partial \Omega $$ ∂ Ω . For $$\sigma >0$$ σ > 0 , we consider the Robin boundary value problem $$\begin{aligned} -\Delta f =\lambda f, \qquad \frac{\partial f}{\partial n} + \sigma f = 0 \text{ on } \partial \Omega \end{aligned}$$ - Δ f = λ f , ∂ f ∂ n + σ f = 0 on ∂ Ω where $$ \frac{\partial f}{\partial n} $$ ∂ f ∂ n is the derivative in the direction of the outward pointing normal to $$\partial \Omega $$ ∂ Ω . Let $$0<\lambda ^\sigma _0\le \lambda ^\sigma _1\le \ldots $$ 0 < λ 0 σ ≤ λ 1 σ ≤ … be the corresponding eigenvalues. The purpose of this paper is to study the Robin–Neumann gaps $$\begin{aligned} d_n(\sigma ):=\lambda _n^\sigma -\lambda _n^0 . \end{aligned}$$ 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 $$2{\text {length}}(\partial \Omega )/{\text {area}}(\Omega )\cdot \sigma $$ 2 length ( ∂ Ω ) / area ( Ω ) · σ and in the smooth case, give an upper bound of $$d_n(\sigma )\le C(\Omega ) n^{1/3}\sigma $$ 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.