Differences Between Robin and Neumann Eigenvalues

Let Omega subset of R-2 be a bounded planar domain, with piecewise smooth boundary partial derivative Omega. For sigma > 0, we consider the Robin boundary value problem where partial derivative f/partial derivative n is the derivative in the direction of the outward pointing normal to partial derivative Omega. Let 0 < lambda(sigma)(0) <= lambda(sigma)(1) <=... be the corresponding eigenvalues. The purpose of this paper is to study the Robin-Neumann gaps d(n)(sigma) : = lambda(sigma)(n) - lambda(0)(n). For a wide class of planar domains we show that there is a limiting mean value, equal to 2 length(partial derivative Omega)/ area(Omega) center dot sigma and in the smooth case, give an upper bound of d(n)(sigma) <= C(Omega)n(1/3)s 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.

Automated summary

The paper examines the properties of Robin-Neumann gaps on various planar domains, providing a limiting mean value and specific upper and lower bounds in different scenarios. Further research is conducted on specific properties of different geometric structures.