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\title{Math 124B: Take home final}
\date{}


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\begin{document}
\maketitle

Any theorems covered during lecture or in Strauss' PDE textbook, including its appendix can be referenced.  Any other theorems or nontrivial claims must be provided with proof. A claim is nontrivial if you do not know why its true., i.e. no ``From Theorem 2 of Book X''.

\textbf{Due in class on Thursday or online Friday}

\begin{enumerate}
\item Derive the Laplacian in spherical coordinates.

\item Compute the sum $\displaystyle \sum_{n=1}^\infty \frac{1}{n^8}$.  You may use previously established sums.  Then explain how one could obtain the sum for $\displaystyle \sum_{n=1}^\infty \frac{1}{n^{2k}}$ for any $k \in \mathbb{N}$.
\item Derive Poisson's formula for $D= \{(r,\theta) \in \mathbb{R}^2 \ | \ r>a > 0\}$, given by
\begin{equation*}
u(r,\theta) = (r^2-a^2)\int_0^{2\pi} \frac{h(\phi)}{a^2-2ar\cos(\theta - \phi)+r^2}\frac{d\phi}{2\phi}
\end{equation*}
where $u$ is a solution of
\begin{equation*}
\begin{cases}
u_{xx}+u_{yy} = 0 & \text{ for } r>a \\
u = h(\theta) & \text{ for } r=a \\
u \text{ is bounded as } r\to \infty.
\end{cases}
\end{equation*}

\item A function $u(x,y)$ is subharmonic in $D$ if $\Delta u\geq 0$ in $D$.  Prove that its maximum value is attained on $\dd D$.  Then show that it also satisfies the mean value inequality
\begin{equation*}
u(0) \leq \frac{1}{2\pi r}\int_0^{2\pi}u(r,\theta)d\theta
\end{equation*}
for all $r>0$ such that $B(0,r) \subset D$.

\item Show that the second smallest eigenvalue for the Neumann function is $\lambda_2 > 0$.

\item Let $g(x)$ be a function on $\dd D$.  Consider the minimum of the functional
\begin{equation*}
\frac{1}{2}\iiint_D|\nabla w|^2dV - \iiint_D fw dV
\end{equation*}
among all $C^2$ functions $w$ for which $w = g$ on $\dd D$.  Show that a solution of this minimum problem leads to a solution of the Dirichlet problem
\begin{equation*}
\begin{cases}
-\Delta u =f & \text{ in }D\\
u= g &\text{ on }\dd D.
\end{cases}
\end{equation*}

\item The Neumann function $N(x,y)$ for a domain $D$ is defined much like the Green's function, except for the boundary condition is replaced by
\begin{equation*}
\frac{\dd N}{\dd n} = c, \quad \text{ on }  \dd D,
\end{equation*}
for some constant $c$.  Show that $c= \frac{1}{A}$ where $A$ is the area of $\dd D$, ($c=0$ if $A=\infty$).  Then state and prove a theorem expressing the solution of the Neumann problem in terms of the Neumann function, (Theorem 7.3.1 in Strauss).

\item Compute the eigenvalues of the 2 dimensional rectangle,
\begin{equation*}
\begin{cases}
\Delta u +\lambda u =0 &\text{ on } [0,a]\times[0,b]\\
u= 0 & \text{ on }\dd D.
\end{cases}
\end{equation*}
(Use separation of variables)
\end{enumerate}

\end{document}