Approximation of the least Rayleigh quotient for degree $p$ homogeneous functionals

TL;DR

This paper introduces two innovative methods for approximating minimizers of the Rayleigh quotient for degree p homogeneous functionals, with applications in Sobolev inequalities and extremal functions.

Contribution

The paper presents novel inverse iteration and evolution-based schemes for approximating Rayleigh quotient minimizers in Banach spaces, extending known results even in Hilbert spaces.

Findings

Both schemes ensure the Rayleigh quotient decreases along solutions.

Properly scaled solutions converge to minimizers of the quotient.

Methods are applicable to Sobolev space inequalities.

Abstract

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 $p\in (1,\infty)$. Minimizers are shown to satisfy $\partial \Phi(u)- \lambda\mathcal{J}_p(u)\ni 0$ for a certain $\lambda\in \mathbb{R}$, 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 $$ \partial \Phi(u_k)- \mathcal{J}_p(u_{k-1})\ni 0 \quad (k\in \mathbb{N}). $$ The second method is based on the large time behavior of solutions of the doubly nonlinear evolution $$ \mathcal{J}_p(\dot v(t))+\partial\Phi(v(t))\ni 0 \quad(a.e.\;t>0) $$ 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.