Erik Lindgren

ArticleApproximation of the least Rayleigh quotient for degree p homogeneous functionals

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Journal of Functional Analysis 272 (12), 2017

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We present two novel methods for approximating minimizers of the abstract Rayleigh quotient $Phi(u)/ u ^p$. Here $Phi$ is a strictly convex functional on a Banach space with norm $ cdot $, and $Phi$ is assumed to be positively homogeneous of degree $pin (1,infty)$. Minimizers are shown to satisfy $partial Phi(u)- lambdamathcal{J}_p(u)ni 0$ for a certain $lambdain mathbb{R}$, where $mathcal{J}_p$ is the subdifferential of $frac{1}{p} cdot ^p$. The first approximation scheme is based on inverse iteration for square matrices and involves sequences that satisfy $$ partial Phi(u_k)- mathcal{J}_p(u_{k-1})ni 0 quad (kin mathbb{N}). $$ The second method is based on the large time behavior of solutions of the doubly nonlinear evolution $$ mathcal{J}_p(dot v(t))+partialPhi(v(t))ni 0 quad(a.e.;t>0) $$ and more generally $p$-curves of maximal slope for $Phi$. 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 $Phi(u)/ u ^p$. 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.