HMC Senior Theses

5-31-2012

Document Type

Open Access Senior Thesis

Degree Name

Bachelor of Science

Mathematics

Alfonso Castro

Jon T. Jacobsen

© Harris Enniss

Terms of Use & License Information

This work is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 License.

Abstract

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.

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