Differences Between Robin and Neumann Eigenvalues

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