Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
5772207 | Journal of Functional Analysis | 2017 | 46 Pages |
Abstract
We present two novel methods for approximating minimizers of the abstract Rayleigh quotient Φ(u)/âuâp. Here Φ is a strictly convex functional on a Banach space with norm ââ
â, and Φ is assumed to be positively homogeneous of degree pâ(1,â). Minimizers are shown to satisfy âΦ(u)âλJp(u)â0 for a certain λâR, where Jp is the subdifferential of 1pââ
âp. The first approximation scheme is based on inverse iteration for square matrices and involves sequences that satisfyâΦ(uk)âJp(ukâ1)â0(kâN). The second method is based on the large time behavior of solutions of the doubly nonlinear evolutionJp(vË(t))+âΦ(v(t))â0(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)/âuâp. 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.
Related Topics
Physical Sciences and Engineering
Mathematics
Algebra and Number Theory
Authors
Ryan Hynd, Erik Lindgren,