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Minimising convex combinations of low eigenvalues∗

Published online by Cambridge University Press:  07 March 2014

Mette Iversen and
Dario Mazzoleni
Mette Iversen
Affiliation:
School of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, United Kingdom. Mette.Iversen@bris.ac.uk
Dario Mazzoleni
Affiliation:
Dipartimento di Matematica, Università degli Studi di Pavia, Via Ferrata 1, 27100 Pavia, Italy; dario.mazzoleni@unipv.itDepartment Mathematik, Friedrich−Alexander Universität Erlangen-Nürnberg, Cauerstrasse, 11, 91058 Erlangen, Germany; mazzoleni@math.fau.de

Abstract

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We consider the variational problem

        inf{αλ1(Ω) + βλ2(Ω) + (1 − α − β)λ3(Ω) | Ω open in ℝn, |Ω| ≤ 1},

for α, β ∈ [0, 1], α + β ≤ 1, whereλk(Ω)is the ktheigenvalue of the Dirichlet Laplacian acting in L2(Ω) and|Ω| isthe Lebesgue measure of Ω. We investigate for which values of α, β every minimiser isconnected.

Keywords

EigenvaluesDirichlet–LaplacianShape Optimization

Information

Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 20 , Issue 2 , April 2014, pp. 442 - 459
Copyright
© EDP Sciences, SMAI, 2014

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