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Journal of Convex Analysis 25 (2018), No. 1, 319--337
Copyright Heldermann Verlag 2018

Optimisation of the Lowest Robin Eigenvalue in the Exterior of a Compact Set

David Krejcirík
Department of Mathematics, Faculty of Nuclear Sciences and Physical Engineering, Technical University, Trojanova 13, 12000 Prague 2, Czech Republic

Vladimir Lotoreichik
Department of Theoretical Physics, Nuclear Physics Institute, Czech Academy of Sciences, 25068 Rez, Czech Republic

We consider the problem of geometric optimisation of the lowest eigenvalue of the Laplacian in the exterior of a compact planar set, subject to attractive Robin boundary conditions. Under either a constraint of fixed perimeter or area, we show that the maximiser within the class of exteriors of convex sets is always the exterior of a disk. We also argue why the results fail without the convexity constraint and in higher dimensions.

Keywords: Robin Laplacian, negative boundary parameter, exterior of a convex set, lowest eigenvalue, spectral isoperimetric inequality, spectral isochoric inequality, parallel coordinates.

MSC: 35P15; 58J50

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