Approximation of the least Rayleigh quotient for degree p homogeneous funetionals

作者：Hynd Ryan; Lindgren Erik^*

来源：Journal of Functional Analysis, 2017, 272(12): 4873-4918.

DOI：10.1016/j.jfa.2017.02.024

• Nonlinear eigenvalue problem
• Doubly nonlinear evolution
• Inverse iteration
• Large time behavior

We present two novel methods for approximating minimizers of the abstract Rayleigh quotient Phi(u)/parallel to u parallel to(p). Here Phi is a strictly convex functional on a Banach space with norm parallel to center dot parallel to, and Phi is assumed to be positively homogeneous of degree p is an element of (1,infinity). Minimizers are shown to satisfy partial derivative Phi(u) - lambda j(p)(u) there exists 0 for a certain lambda is an element of R, where J(p) is the subdifferential of 1/p parallel to center dot parallel to(p.) The first approximation scheme is based on inverse iteration for square matrices and involves sequences that satisfy
partial derivative Phi(u(k)) - j(p()u(k-1)) there exists 0 (k is an element of N)
The second method is based on the large time behavior of solutions of the doubly nonlinear evolution
j(p)((v) over circle (t)) + partial derivative Phi(v(t)) there exists 0 (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)/parallel to u parallel to(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.

• 出版日期2017-6-15