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\centerline{\bf Oral Examination Syllabus}
\centerline{\bf Andrei Zherebtsov} %Topics for the Oral Exam.
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\centerline{\bf Major Topic: Partial Differential Equations}
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\begin{enumerate}
  \item Laplace Equation
    \begin{itemize}
     \item  [a)] The fundamental solution 
     \item  [b)] $\Delta u = f$ in $\Omega$, $\Omega$ bounded domain, $f$ H\"{o}lder continuous 
     \item  [c)] Mean value formula and maximum principles 
     \item  [d)] Harnack's inequality 
     \item  [e)] Green's function for a ball and for half space 
     \item  [f)] Perron's method 
    \end{itemize}
  \item Sobolev Spaces 
    \begin{itemize}
     \item  [a)] Density of smooth functions in Sobolev spaces 
     \item  [b)] Global extensions 
     \item  [c)] Trace theorem 
     \item  [d)] Sobolev inequalities (Gagliardo-Nirenberg-Sobolev, Morrey) 
     \item  [e)] Compact imbeddings 
     \item  [f)] Poincar\'{e}'s inequality 
    \end{itemize}

  \item Second Order Elliptic Equations
    \begin{itemize}
     \item  [a)] Strong and weak maximum principles
     \item  [b)] Uniqueness of the Dirichlet boundary value problems
     \item  [c)] Definition of weak solutions
     \item  [d)] Existence: Lax-Milgram, energy estimates, Fredholm 
alternative
     \item  [e)] Interior and boundary regularity of solutions
    \end{itemize} 
 
  \item Classical solutions.  Schauder approach.
\end{enumerate}

\centerline{\bf Minor Topic: Functional Analysis}
\begin{enumerate}
     \item  Hilbert Spaces
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       \begin{itemize}
         \item [a)] Riesz lemma
         \item [b)] Orthonormal bases
         \item [c)] Dual of a Hilbert space
         \item [d)] Lax-Milgram Theorem
       \end{itemize} 
     \item  Hahn-Banach theorems(analytic and geometric forms)
     \item  Baire category theorem,principle of uniform boundedness, open mapping theorem, closed graph theorem
     \item  Weak topologies, weak * topology, Banach-Alaoglu theorem
     \item  Compact operators. Riesz-Fredholm theory. Spectrum of a compact operator
     \item  Spectral decomposition of compact autoadjoint operators. 
    \end{enumerate}


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