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Journal of Convex Analysis 25 (2018), No. 1, [final page numbers not yet available]
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
david.krejcirik@fjfi.cvut.cz

Vladimir Lotoreichik
Department of Theoretical Physics, Nuclear Physics Institute, Czech Academy of Sciences, 25068 Rez, Czech Republic
lotoreichik@ujf.cas.cz



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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