% ShiTing Bao's Orals February 24, 2005
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\title{\bf Syllabus for Oral Examination}
\author{ ShiTing Bao}
\date{}
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\section*{Major topic: Partial Differential Equations}

\noindent $\bullet$ {\bf Laplacian Equation}\\

(1) Fundamental solution

(2) Mean-value formulas

(3) Properties of harmonic functions:

    \hspace{0.55cm} maximum principle, smoothness, local estimates, Liouville's theorem,

    \hspace{0.55cm} analyticity, Harnack's inequality,

(4) Green's function for a ball and for a half plane

(5) Energy methods\\

\noindent$\bullet$ {\bf Sobolev Spaces}\\

(1) Definition of Sobolev spaces

(2) Approximation by smooth functions

(3) Extensions

(4) Sobolev inequalities:

    \hspace{0.53cm} Gagliardo-Nirenberg-Sobolev inequality,
    Morrey's inequality

(5) Compact embedding theorem\\

\noindent$\bullet$ {\bf Second Order Elliptic Equations}\\

(1) Definition of weak solutions

(2) Existence of weak solutions:

    \hspace{0.53cm} Lax-Milgram theorem, energy estimate and Fredholm alternative

(3) Regularity

(4) Maximum principles
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\section*{Minor topic: Functional Analysis}

 \noindent$\bullet$ {\bf Banach Spaces}\\

(1) Linear normed spaces

(2) Barie category theorem

(3) Dual and reflexive spaces

(4) Alaoglu theorem\\

 \noindent$\bullet$ {\bf Hilbert Spaces}\\

(1) Riezs Lemma

(2) Orthonormal sets and Bases

(3) Bessel's inequality\\

\noindent$\bullet$ {\bf Linear Operators}\\

(1) Uniform Bounded Principle

(2) Open Mapping Theorem and Closed Graph Theorem

(3) Fixed Points Theorem

(4) Strong, Weak and Weak* Convergence\\

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