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 ‖⋅‖, and Φ is assumed to be positively homogeneous of degree p∈(1,∞). Minimizers are shown to satisfy ∂Φ(u)−λJp(u)∋0 for a certain λ∈R, where Jp is the subdifferential of 1p‖⋅‖p. The first approximation scheme is based on inverse iteration for square matrices and involves sequences that satisfy∂Φ(uk)−Jp(uk−1)∋0(k∈N). The second method is based on the large time behavior of solutions of the doubly nonlinear evolutionJp(v˙(t))+∂Φ(v(t))∋0(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.