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 $$\mbox{ \LARGE Oral Exam Syllabus \ \ \ \ \ \ \ \ \ \  } $$ 

$$ \large \begin{array}{ll}  \mbox{Student:}              & \mbox{German Enciso} \\
                       \mbox{Committee:}            & \mbox{Prof. Yanyan Li} \\
                                                    & \mbox{Prof. Eduardo Sontag} \\
                                                    & \mbox{Prof. Michael Vogelius} \\
								    & \mbox{Prof. Richard Wheeden}  \\
                       \mbox{Date:}                 & \mbox{Monday, December $16^{th}$} \\
                       \mbox{Time \& Place:}       & \mbox{10 am, Hill 425} 
				  
     \end{array}  \ \ \ \ \ \ \  \ \ \ \ \  \ \ \ \ \       $$

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Subject I:  Harmonic Analysis

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1) The classic construction of the Fourier Transform.

2) The Lebesgue set, and the maximal function.

3) Marcinkievitch interpolation theorem.

4) Calder\'{o}n-Zygmund decomposition.

5) Singular integral theorems.

6) Hilbert and Riesz Transforms.

7) Poisson Kernel.

8) Spherical Harmonics



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Sources: Stein, Elias, {\em Singular Integrals and Differentiability Properties}, Princeton \linebreak .  \ \ 
University Press 1970.

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Subject II: Numerical solutions to PDEs, with background in theoretical PDEs

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{\em Background: }

1) Maximum principle.

2) Harnack Inequality.

3) Mean Value Property.

4) Green's Function on the unit ball 

5) Poisson Formula




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{\em Numerical Methods: }

6) Discrete maximum principle.

7) The discretisation of the problem

$$  \left\{ \begin{array}{ll}   -\triangle u = f & \mbox{in} \ \Omega \\
                                   \ \ \    u=\phi & \mbox{in} \ \partial \Omega  
     \end{array}   \right.  \ \ \ \ \ \ \ \ \ \ \ \ \    $$

and its solution using finite difference methods.

8)  Variational principles.

9) Finite elements solution of the above problem.


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Sources:  Evans, L.C., {\em Partial Differential Equations}, American Mathematical Society, 1998; Braess, Dietrich, {\em Finite Elements}, Cambridge University Press 2001.










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