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\begin{document}


\centerline{\textbf{Problem Set 4: Math 201B}}
\centerline{Due: Friday, January 28}

\bigskip\noindent
\textbf{1.} Let $D \subset \Rl^2$ be the unit disc and $f\in C(\partial D)$
a continuous function defined on the unit circle $\partial D$. Suppose
that $u : \overline{D} \to \Rl$ is a function $u\in C^2(D) \cap C(\overline{D})$
such that
\begin{align}
\begin{split}
&\Delta u = 0\qquad \mbox{in $D$},
\\
&u = f\qquad \mbox{on $\partial D$}.
\end{split}
\label{dirichlet}
\end{align}
(a) Show that
\[
\max_{\overline{D}} u = \max_{\partial D} f.
\]
\textsc{Hint.} Let $u^\eps(x,y) = u(x,y) + \eps(x^2+y^2)$
and show that $u^\eps$ cannot have an interior maximum for any
$\eps > 0$.

\smallskip\noindent
(b) Deduce that a solution of \eq{dirichlet} is unique and is therefore
given by
\[
u(r,\theta) = (P_r\ast f)(\theta)
\]
in $0\le r < 1$ where $P_r$ is the Poisson kernel.

\bigskip\noindent
\textbf{2.} Define $f\in L^2(\Ts)$ by
\[
f(x) = |x|\qquad \mbox{for $|x| < \pi$}.
\]
Show that $f \in H^1(\Ts)$ and compute its weak derivative
$f^\prime\in L^2(\Ts)$.
Is $f^\prime\in H^1(\Ts)$? For what values of $s >0$ is it true that $f \in H^s(\Ts)$?

\bigskip
\noindent
\textbf{3.} Suppose that $f : [0,L] \to \Rl$ is a smooth function \textit{e.g.} $f\in C^1([0,L])$
such that $f(0) = f(L) = 0$. Prove that
\[
\int_0^L \left[f(x)\right]^2\, dx \le \left(\frac{L}{\pi}\right)^2 \int_0^L \left[f^\prime(x)\right]^2 \, dx. 
\]
Show that the constant in this inequality is sharp. Why do you need to assume that $f(0) = f(L) =0$? Show that
you cannot estimate the $L^2$-norm of a smooth, square-integrable function $f : [0,\infty)\to \Rl$ such that
$f(0) = 0$ in terms of the $L^2$ norm of its derivative.

\newpage\noindent
\textbf{4.} Suppose that $u(x,t)$ is a solution of the following initial value problem
for the heat equation
\begin{align*}
&u_t = u_{xx}\qquad\qquad \mbox{$x\in \Ts$, $t >0$}
\\
&u(x,0) = f(x)\qquad \mbox{$x\in \Ts$}
\end{align*}
where $f\in C(\Ts)$ and
\[
u\in C^2\left(\Ts \times (0,\infty)\right) \cap C\left(\Ts \times [0,\infty)\right).
\]
%You can assume that the solution is unique. (There is a maximum-principle proof of uniqueness similar to the one in problem 1.) 
(a) Show that
\[
u(x,t) = \left(\theta_t\ast f\right)(x)\qquad \mbox{for $t >0$}
\]
where
\[
\theta_t(x) = \frac{1}{2\pi} \sum_{n\in \Ir} e^{-n^2 t} e^{inx}.
\]
(b) Show that $u \in C^\infty\left(\Ts \times (0,\infty)\right)$.
\end{document}

Let $\Ts^d = \Ts \times \Ts \times \dots \times \Ts$ be the
$d$-dimensional torus and write $\mathbf{x} = (x_1,x_2,\dots,x_d)\in \Ts^d$.

\smallskip\noindent
(a) Show that
\[
\mathcal{B} = \left\{e^{i\vec{n}\cdot\vec{x}} : \vec{n} \in \Ir^d\right\}
\]
is an orthogonal set in  $L^2(\Ts)$.

\smallskip\noindent
(b) If $\{\phi_n : n\in \Nl\}$
is an approximate identity in $L^2(\Ts)$ and
\[
\Phi_n(\vec{x}) = \phi_n(x_1) \phi_n(x_2)\dots \phi_n(x_d),
\]
show that $\{\Phi_n : n\in \Nl\}$ is an approximate identity in $L^2(\Ts^d)$.
Deduce that $\mathcal{B}$ is an orthogonal basis of $L^2(\Ts)$.

\smallskip\noindent
(c) If $f\in L^2(\Ts^d)$ and
\[
f(\vec{x}) = \sum_{\vec{n}\in \Ir^d} c_\vec{n} e^{i\vec{n}\cdot\vec{x}}
\]
give an expression for the Fourier coefficient $c_\vec{n}$.