Pavel Exner and Vladimir Lotoreichik
Spectral optimization for Robin Laplacian in domains without cut loci
(383K, pdf)

ABSTRACT.  In this paper we deal with spectral optimization for the Robin Laplacian on a family of planar domains without cut loci, namely a fixed-width strip built over a smooth closed curve and the exterior of a convex set with a smooth boundary. We show that if the curve length is kept fixed, the first eigenvalue referring to the fixed-width strip is for any value of the Robin parameter maximized by a circular annulus. Furthermore, we prove that the second eigenvalue in the exterior of a convex domain $\Omega$ 
corresponding to a negative Robin parameter does not exceed the analogous quantity for a disk whose boundary has a curvature larger or equal to the maximum of that for $\partial\Omega$.