ArticleApproximation of the least Rayleigh quotient for degree p homogeneous functionals
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Journal of Functional Analysis 272 (12), 2017
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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 $pin (1,infty)$. Minimizers are shown to satisfy $partial Phi(u)- lambdamathcal{J}_p(u)ni 0$ for a certain $lambdain mathbb{R}$, 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 $$ partial Phi(u_k)- mathcal{J}_p(u_{k-1})ni 0 quad (kin mathbb{N}). $$ The second method is based on the large time behavior of solutions of the doubly nonlinear evolution $$ mathcal{J}_p(dot v(t))+partialPhi(v(t))ni 0 quad(a.e.;t>0) $$ 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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The Bespoke Consort will present invigorating concerts of crossover chamber music sans boundaries including neo-Baroque escapades, Impressionistic tone poems, original tangos, quirky musical novelties, Alberto Ginastera’sEstancia ballet suite, plus rollicking arrangements of Erik Satie, Leroy Anderson, Rising Storm and Birdsongs of the Mesozoic chestnuts. A pleasant and uplifting time is guaranteed for all!
Saturday, May 4 @ The Lilypad
1353 Cambridge Street (Inman Square), Cambridge MA ❖www.lilypadinman.com5:00–6:30 $10 Please join the musicians afterwards for a dinner across the street at the S&S Restaurant. This program is supported in part by a grant from Associated Chamber Music Players.Sunday, May 5 Private Party
Boston MA TBA *Saturday, June 1 @ First Unitarian Universalist Society of Middleboro
25 South Main Street, Middleborough, MA 02346 7:30 PM Free–donations accepted. ❖uumiddleboro.orgThis program is supported in part by a grant from the Middleborough Cultural Council, a local agency which is supported by the Mass Cultural Council, a state agency.Sunday, June 2 Private Concert
Milton, MA 2 PM *
* Please inquire about availability if you are interested in attending either of the private events.
On Saturday, May 18 at 2PM, Erik and friends will also perform a 10-minute Leroy Anderson set (“Blue Tango,” “The Syncopated Clock,” and “Belle of The Ball”) during the Brockton Symphony Orchestra Chamber Pops Gala! Christ Congregational Church, 1350 Pleasant Street, Brockton MA. Tickets $25. www.brocktonsymphony.org
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“Erik Lindgren is a model of musical schizophrenia—the type who’ll play Stravinsky and Louie Louie, back-to-back, with one foot in the conservatory and the other in the garage.” — Brett Milano, The Boston Globe
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