Optimisation of the Lowest Robin Eigenvalue in the Exterior of a Compact Set, II: Non-Convex Domains and Higher Dimensions

0524214 - ÚJF 2021 RIV NL eng J - Journal Article
Krejčiřík, D. - Lotoreichik, Vladimir
Optimisation of the Lowest Robin Eigenvalue in the Exterior of a Compact Set, II: Non-Convex Domains and Higher Dimensions.
Potential Analysis. Roč. 52, č. 4 (2020), s. 601-614. ISSN 0926-2601. E-ISSN 1572-929X
R&D Projects: GA ČR GA17-01706S
Institutional support: RVO:61389005
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
OECD category: Pure mathematics
Impact factor: 1.416, year: 2020
Method of publishing: Limited access
https://doi.org/10.1007/s11118-018-9752-0

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.
Permanent Link: http://hdl.handle.net/11104/0308592