Minimising convex combinations of low eigenvalues∗. (7th April 2014)

• Record Type: Journal Article
• Authors: Iversen, Mette; Mazzoleni, Dario
• Is Part Of: ESAIM. Volume 20:Number 2(2014:Apr.)
• Page Start: 442
• Page End: 459
• Publication Date: 2014-04-07
• Subjects: System analysis -- Periodicals; Calculus of variations -- Periodicals; Mathematical analysis -- Periodicals; Mathematical optimization -- Periodicals; Control theory -- Periodicals; 515.64
• DOI: 10.1051/cocv/2013070
• Languages: English
• ISSNs: 1292-8119

Abstract

<abstract abstract-type="normal" xml:lang="en"> <title> <x content-type="archive" xml:space="preserve">Abstract</x> </title> <p>We consider the variational problem </p> <p> inf{<italic>αλ</italic>₁(<italic>Ω</italic>) + <italic>βλ</italic>₂(<italic>Ω</italic>) + (1 − <italic>α</italic> − <italic>β</italic>)<italic>λ</italic>₃(<italic>Ω</italic>) | Ω open in ℝ^{<italic>n</italic>}, |<italic>Ω</italic>| ≤ 1}, </p> <p> for <italic>α, β</italic> ∈ [0, 1], <italic>α</italic> + <italic>β</italic> ≤ 1, where <italic>λ</italic>_{<italic>k</italic>}(<italic>Ω</italic>) is the <italic>k</italic>th eigenvalue of the Dirichlet Laplacian acting in <italic>L</italic>²(<italic>Ω</italic>) and |<italic>Ω</italic>| is the Lebesgue measure of <italic>Ω</italic>. We investigate for which values of <italic>α, β</italic> every minimiser is connected.</p> </abstract>