20-2 Pavel Exner and Vladimir Lotoreichik

Spectral optimization for Robin Laplacian in domains without cut loci (383K, pdf) Jan 8, 20

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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 $\partial\Omega$.

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