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

Result code in IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F61389005%3A_____%2F20%3A00524214" target="_blank" >RIV/61389005:_____/20:00524214 - isvavai.cz</a>

Alternative codes found

RIV/68407700:21340/20:00328025

Result on the web

<a href="https://doi.org/10.1007/s11118-018-9752-0" target="_blank" >https://doi.org/10.1007/s11118-018-9752-0</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1007/s11118-018-9752-0" target="_blank" >10.1007/s11118-018-9752-0</a>

Result language

angličtina

Original language name

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

Original language description

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.

Czech name

—

Czech description

—

Type

J_imp - Article in a specialist periodical, which is included in the Web of Science database

CEP classification

—

OECD FORD branch

10101 - Pure mathematics

Project

Result was created during the realization of more than one project. More information in the Projects tab.

Continuities

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Publication year

2020

Confidentiality

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Name of the periodical

Potential Analysis

ISSN

0926-2601

e-ISSN

—

Volume of the periodical

52

Issue of the periodical within the volume

4

Country of publishing house

NL - THE KINGDOM OF THE NETHERLANDS

Number of pages

14

Pages from-to

601-614

UT code for WoS article

000528380600003

EID of the result in the Scopus database

2-s2.0-85059834304