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Optimisation of the Lowest Robin Eigenvalue in the Exterior of a Compact Set, II: Non-Convex Domains and Higher Dimensions
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SYSNO ASEP 0524214 Document Type J - Journal Article R&D Document Type Journal Article Subsidiary J Článek ve WOS Title Optimisation of the Lowest Robin Eigenvalue in the Exterior of a Compact Set, II: Non-Convex Domains and Higher Dimensions Author(s) Krejčiřík, D. (CZ)
Lotoreichik, Vladimir (UJF-V) ORCID, SAINumber of authors 2 Source Title Potential Analysis. - : Springer - ISSN 0926-2601
Roč. 52, č. 4 (2020), s. 601-614Number of pages 14 s. Publication form Print - P Language eng - English Country NL - Netherlands Keywords Robin Laplacian ; Negative boundary parameter ; Exterior of a compact set ; Lowest eigenvalue ; Spectral isoperimetric inequality ; Spectral isochoric inequality ; Parallel coordinates ; Critical coupling ; Willmore energy Subject RIV BE - Theoretical Physics OECD category Pure mathematics R&D Projects GA17-01706S GA ČR - Czech Science Foundation (CSF) Method of publishing Limited access Institutional support UJF-V - RVO:61389005 UT WOS 000528380600003 EID SCOPUS 85059834304 DOI 10.1007/s11118-018-9752-0 Annotation We consider the problem of geometric optimisation of the lowest eigenvalue of the Laplacian in the exterior of a compact set in any dimension, subject to attractive Robin boundary conditions. As an improvement upon our previous work (Krejcirik and Lotoreichik J. Convex Anal. 25, 319-337, 2018), we show that under either a constraint of fixed perimeter or area, the maximiser within the class of exteriors of simply connected planar sets is always the exterior of a disk, without the need of convexity assumption. Moreover, we generalise the result to disconnected compact planar sets. Namely, we prove that under a constraint of fixed average value of the perimeter over all the connected components, the maximiser within the class of disconnected compact planar sets, consisting of finitely many simply connected components, is again a disk. In higher dimensions, we prove a completely new result that the lowest point in the spectrum is maximised by the exterior of a ball among all sets exterior to bounded convex sets satisfying a constraint on the integral of a dimensional power of the mean curvature of their boundaries. Furthermore, it follows that the critical coupling at which the lowest point in the spectrum becomes a discrete eigenvalue emerging from the essential spectrum is minimised under the same constraint by the critical coupling for the exterior of a ball. Workplace Nuclear Physics Institute Contact Markéta Sommerová, sommerova@ujf.cas.cz, Tel.: 266 173 228 Year of Publishing 2021 Electronic address https://doi.org/10.1007/s11118-018-9752-0
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