Minimising convex combinations of low eigenvalues^∗

¹ School of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, United Kingdom
Mette.Iversen@bris.ac.uk
² Dipartimento di Matematica, Università degli Studi di Pavia, Via Ferrata 1, 27100 Pavia, Italy
dario.mazzoleni@unipv.it
³ Department Mathematik, Friedrich−Alexander Universität Erlangen-Nürnberg, Cauerstrasse, 11, 91058 Erlangen, Germany
mazzoleni@math.fau.de

Received: 9 January 2013
Revised: 28 June 2013

Abstract

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, where λ_k(Ω) is the kth eigenvalue of the Dirichlet Laplacian acting in L²(Ω) and |Ω| is the Lebesgue measure of Ω. We investigate for which values of α, β every minimiser is connected.