Differences Between Robin and Neumann Eigenvalues.

Let be a bounded planar domain, with piecewise smooth boundary . For , we consider the Robin boundary value problem where is the derivative in the direction of the outward pointing normal to . Let be the corresponding eigenvalues. The purpose of this paper is to study the Robin-Neumann gaps For a wide class of planar domains we show that there is a limiting mean value, equal to and in the smooth case, give an upper bound of 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.