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\begin{center}
{\Large Oral Qualifying Examination for \underline{Augusto C\'esar Ponce}}
\end{center}

\vspace{3mm}

\noindent
{\large {\bf Topic 1} : Linear Theory for Second Order Elliptic PDE's}\\

	The linear theory for second order elliptic PDE's comprises many approaches regarding the study of the problem
\begin{equation}\label{eq}
\begin{cases}
Lu = f				& \text{in } \Omega\\
\hspace{1.6ex} u = \varphi		& \text{on } \partial\Omega
\end{cases},
\end{equation}
where $\Omega$ is a domain in $\bb R^n$ and the operator $L$ may be in either the nondivergence form
$$
Lu := a^{ij} \partial_{ij}u + b^i \partial_i u + c
$$
or in the divergence form
$$
Lu := \partial_i \big(a^{ij} \partial_{j}u \big) + b^i \partial_i u + c.
$$
In both cases, the functions $a^{ij}$, defined in $\Omega$, must satisfy the ellipticity condition
$$
a^{ij} \xi_i\xi_j \geq \lambda |\xi|^2, \: \forall \xi \in \bb R^n,
$$
for some $\lambda>0$.

	Some questions immediately arise from the study of \eqref{eq}, specially about its well-posedness:\\
a) what one means by a solution of \eqref{eq};\\
b) if there is such a solution;\\
c) if this solution is unique;\\
d) if there is a continuous dependence of the solution on the data given in the problem.

	In order to answer these questions, a number of results are available to us. The first topic for my Oral Qualifying Examination should include some of them:
\begin{itemize}
\item Schauder Theory (classical solutions);
\item Sobolev spaces (weak solutions);
\item $L^p$ Theory (strong solutions);
\item Maximum principles;
\item Functional Analysis (Weak topology, Riesz Representation Theorem, Lax-Milgram Theorem and Fredholm Alternative).\\
\end{itemize}

\noindent
{\bf References}\\
$[1]$ {\sc Br\'ezis}, Ha\"\i m. {\em Analyse fonctionnelle} :\@ Th\'eorie et applications. 2nd ed. Paris :\@ Dunod, 1999.\\
$[2]$ {\sc Evans}, Lawrence C. {\em Partial differential equations}. Providence, RI : American Mathematical Society, 1998. (Graduate Studies in Mathematics, 19)\\
$[3]$ {\sc Gilbarg}, David; {\sc Trudinger}, Neil S. {\em Elliptic Partial Differential Equation of Second Order}. 2nd ed. Berlin :\@ Springer, 1998.  (Grundlehren der mathematischen Wissenschaften, 224)

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\newpage

\begin{center}
{\Large Oral Qualifying Examination for \underline{Augusto C\'esar Ponce}}
\end{center}

\vspace{3mm}

\noindent
{\large {\bf Topic 2} : The Semilinear Dirichlet Problem}\\

	Despite the fact that the linear theory has independent interest, its development is essential if one wants to study more general nonlinear PDE's. As a second topic for my Oral Qualifying Examination, I propose to analyze the semilinear Dirichlet problem
\begin{equation}\label{eq}
\begin{cases}
- \triangle u = f(u)			& \text{in } \Omega\\
\hspace{3.9ex} u = 0 			& \text{on } \partial\Omega
\end{cases},
\end{equation}
where $\Omega$ is a domain in $\bb R^n$ and $f : \bb R \rightarrow \bb R$ is smooth.

	I intend to focus on how the linear theory provides us the right abstract space setting in order to guarantee the existence of classical solutions of \eqref{eq}. Some nonlinear methods are readily available in this case:
\begin{itemize}
\item Weak convergence techniques;
\item Degree theory and fixed point methods;
\item Critical Point Theory (minimax characterisation of critical values, in particular the Mountain Pass Theorem);
\item Implicit Function Theorem;
\item Method of sub and supersolutions.
\end{itemize}

Under some additional hypotheses on $f$, we may easily prove the existence of classical solutions of \eqref{eq} in the following cases (here $0 < \lambda_1 < \lambda_2 \leq \cdots$ denotes the sequence of eigenvalues of $-\triangle$ on $\Omega$, with Dirichlet condition):
\begin{itemize}
\item $\displaystyle \limsup_{s\rightarrow \pm \infty}{\frac{f(s)}{s}} < \lambda_1$;
\item $\lambda_j < \displaystyle \lim_{s\rightarrow -\infty}{\frac{f(s)}{s}}, \: \displaystyle \lim_{s\rightarrow +\infty}{\frac{f(s)}{s}}< \lambda_{j+1}$;
\item $\displaystyle \lim_{s\rightarrow \pm \infty}{\frac{f(s)}{s}} = + \infty$ and $|f(s)| \leq C(|s|^{p-1}+1)$, for some $2 < p < 2^\ast$;
\item $\displaystyle \lim_{s\rightarrow + \infty}{\frac{f(s)}{s}} = + \infty$, $f>0$, $f$ is increasing and convex (no bounds on $f$ are imposed); in this case one should replace $f$ by $\lambda f$, where $\lambda > 0$, and study the set of $\lambda$'s for which the problem \eqref{eq} has a nonnegative solution.\\
\end{itemize}


\noindent
{\bf References}\\
$[1]$ {\sc Br\'ezis}, Ha\"\i m. {\em Analyse fonctionnelle} :\@ Th\'eorie et applications. 2nd ed. Paris :\@ Dunod, 1999.\\
$[2]$ {\sc Br\'ezis}, Ha\"\i m; {\sc Nirenberg}, Louis. {\em Nonlinear Functional Analysis and Applications}. (in preparation)\\
$[3]$ {\sc Deimling}, Klaus. {\em Nonlinear functional analysis}. 1st ed., Berlin :\@  Springer-Verlag, 1985.\\
$[4]$ {\sc Kavian}, Otared. {\em Introduction \`a la th\'eorie des points critiques} \/ et applications aux probl\`emes elliptiques. Paris :\@ Springer-Verlag, 1993.\\
$[5]$ {\sc Willem}, Michel. {\em Minimax theorems}. Boston :\@ Birkh\"auser, 1996.


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