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

Kód výsledku v 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>

Nalezeny alternativní kódy

RIV/68407700:21340/20:00328025

Výsledek na webu

<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>

Jazyk výsledku

angličtina

Název v původním jazyce

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

Popis výsledku v původním jazyce

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.

Název v anglickém jazyce

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

Popis výsledku anglicky

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.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

Výsledek vznikl pri realizaci vícero projektů. Více informací v záložce Projekty.

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2020

Kód důvěrnosti údajů

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

Název periodika

Potential Analysis

ISSN

0926-2601

e-ISSN

—

Svazek periodika

52

Číslo periodika v rámci svazku

4

Stát vydavatele periodika

NL - Nizozemsko

Počet stran výsledku

14

Strana od-do

601-614

Kód UT WoS článku

000528380600003

EID výsledku v databázi Scopus

2-s2.0-85059834304