Harris Enniss
Thesis
A Refined Saddle Point Theorem and Applications
- Advisor
- Alfonso Castro
- Second Reader(s)
- Jon T. Jacobsen
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.
Proposal
Existence and Regularity Properties of the p-Laplacian Operator
