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\centerline{{\sc Problem set 1}}
\centerline{Math 218A, Winter 2001}

\bigskip\noindent
{\bf 1.} Derive the Green's function representation for the solution
of Laplace's equation in the half-plane,
\beann
&&u_{xx} + u_{yy} = 0\qquad y>0,\\
&&u(x,0) = f(x),
\eeann
where $u(x,y)$ is assumed to decay suuficiently quickly as $y\to \infty$.


\bigskip\noindent
{\bf 2.} Find the Green's function for Laplace's equation in the unit
ball in $\Rl^3$, and write out the Green's function representation of the
solution of
\beann
&&-\Delta u = f\qquad |x| < 1,\\
&&u = 0 \qquad |x|=1.
\eeann
{\sc Hint.} Use the method of images.



\bigskip\noindent
{\bf 3.} A function $u\in C^2(\Omega)$ is \emph{subharmonic} in a domain $\Omega$ if
$\Delta u \ge 0$, and \emph{superharmonic} if $\Delta u \le 0$. Prove that
if a subharmonic function attains its maximum at any point of $\Omega$,
then it is a constant function. Suppose that
\[
u, v, w \in C^2(\Omega) \cap C(\overline{\Omega}),
\]
where $u$ is subharmonic, $v$ is harmonic, and $w$ is superharmonic. If
$u \le v \le w$ on $\partial \Omega$, prove that
$u \le v \le w$ on $\overline{\Omega}$.


\bigskip\noindent
{\bf 4.} Suppose that $\{u_n\}$ is a decreasing sequence of
harmonic functions defined in a domain $\Omega$ such that
$\{u_n(x)\}$ converges for some popint $x\in \Omega$. Prove that
the sequence converges uniformly on every closed, bounded
subset of $\Omega$.
{\sc Hint.} Harnack's inequality.


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