Spectral optimization for Robin Laplacian on domains admitting parallel coordinates

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AbstractIn this paper we deal with spectral optimization for the Robin Laplacian on a family of planar domains admitting parallel coordinates, 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 Ω corresponding to a negative Robin parameter does not exceed the analogous quantity for the exterior of a disk whose boundary has a curvature larger than or equal to the maximum of that for .