Potential Analysis. An International Journal Devoted to the Interactions between Potential Theory, Probability Theory, Geometry and Functional Analysis 46 (4), 705-737, 2017
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.
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