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Approximation of the least Rayleigh quotient for degree p homogeneous functionals
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).ORCID iD: 0000-0003-4309-9242
2017 (English)In: Journal of Functional Analysis, ISSN 0022-1236, E-ISSN 1096-0783, Vol. 272, no 12, 4873-4918 p.Article in journal (Refereed) Published
Abstract [en]

We present two novel methods for approximating minimizers of the abstract Rayleigh quotient Φ(u)/‖u‖p. Here Φ is a strictly convex functional on a Banach space with norm ‖⋅‖, and Φ is assumed to be positively homogeneous of degree p∈(1,∞). Minimizers are shown to satisfy ∂Φ(u)−λJp(u)∋0 for a certain λ∈R, where Jp is the subdifferential of 1p‖⋅‖p. The first approximation scheme is based on inverse iteration for square matrices and involves sequences that satisfy ∂Φ(uk)−Jp(uk−1)∋0(k∈N). The second method is based on the large time behavior of solutions of the doubly nonlinear evolution Jp(v˙(t))+∂Φ(v(t))∋0(a.e.t>0) and more generally p-curves of maximal slope for Φ. We show that both schemes have the remarkable property that the Rayleigh quotient is nonincreasing along solutions and that properly scaled solutions converge to a minimizer of Φ(u)/‖u‖p. These results are new even for Hilbert spaces and their primary application is in the approximation of optimal constants and extremal functions for inequalities in Sobolev spaces.

Place, publisher, year, edition, pages
Academic Press Inc. , 2017. Vol. 272, no 12, 4873-4918 p.
Keyword [en]
Doubly nonlinear evolution, Inverse iteration, Large time behavior, Nonlinear eigenvalue problem
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-207480DOI: 10.1016/j.jfa.2017.02.024ISI: 000400539700001ScopusID: 2-s2.0-85015702520OAI: oai:DiVA.org:kth-207480DiVA: diva2:1108053
Note

Export Date: 22 May 2017; Article; CODEN: JFUAA; Correspondence Address: Lindgren, E.; Department of Mathematics, KTHSweden; email: eriklin@kth.se; Funding details: KVA, Royal Swedish Academy of Sciences; Funding details: 2012-3124, VR, Vetenskapsrådet; Funding text: Supported by the Swedish Research Council, grant no. 2012-3124. Partially supported by the Royal Swedish Academy of Sciences. QC 20170612

Available from: 2017-06-12 Created: 2017-06-12 Last updated: 2017-06-12Bibliographically approved

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