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

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

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