Return to search

A Refined Saddle Point Theorem and Applications

Under adequate conditions on $g$, we show the density in $L^2((0,\pi),(0,2\pi))$ of the set of functions $p$ for which \begin{equation*} u_{tt}(x,t)-u_{xx}(x,t)= g(u(x,t)) + p(x,t) \end{equation*} has a weak solution subject to \begin{equation*} \begin{aligned} u(x,t)&=u(x,t+2\pi)\\ u(0,t)&=u(\pi,t)=0. \end{aligned} \end{equation*}
To achieve this, we prove a Saddle Point Principle by means of a refined variant of the deformation lemma of Rabinowitz.
Generally, inf-sup techniques allow the characterization of critical values by taking the minimum of the maximae on some particular class of sets. In this version of the Saddle Point Principle, we introduce sufficient conditions for the existence of a saddle-structure which is not restricted to finite-dimensional subspaces.

Identiferoai:union.ndltd.org:CLAREMONT/oai:scholarship.claremont.edu:hmc_theses-1032
Date31 May 2012
CreatorsEnniss, Harris
PublisherScholarship @ Claremont
Source SetsClaremont Colleges
Detected LanguageEnglish
Typetext
Formatapplication/pdf
SourceHMC Senior Theses
Rights© Harris Enniss, http://creativecommons.org/licenses/by-nc-sa/3.0/

Page generated in 0.0022 seconds