% LaTeX file -- AM/PM 331, Fall 2011, Assignment 7

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\begin{center}
\textbf{\large AMath/PMath 331 \\ Assignment 7\\
Due Friday December 2}
\end{center}
\vskip 2em 

\begin{Ex}

\item  
\begin{parts}
\item Let
 $u(r,\theta) = (3 - 4r^2 +r^4) + (8r^2 - 8r^4) \sin^2 \theta + 8r^4 \sin^4 \theta$.
 Compute $\Delta u (r, \theta)$.
\item Let $u(r,\theta) = \log r$.  Compute $\Delta u$ and $u(1,\theta)$.
Explain why $u$ is not a solution of the heat equation for the
boundary function $f(\theta)=0$.
\end{parts}



\item \begin{parts}
\item Prove that $\dfrac{1-r}{1+r} \le 2\pi P(r,\theta) \le \dfrac{1+r}{1-r}$.
\item Let $f$ be a \textit{positive} continuous
 $2\pi$-periodic function with harmonic extension $u(r,\theta)$.
 Prove that $u(r,\theta) \ge 0$.
\item Again $f \ge 0$. Prove \textit{Harnack's inequality}: 
 $\Big(\dfrac{1-r}{1+r}\Big) u(0,0) \le u(r,\theta) \le \Big(\dfrac{1+r}{1-r}\Big) u(0,0) .$
\end{parts}


\item Which of the following are the Fourier series of a continuous function?
Explain---but do not try to calculate the sum.
\begin{parts}
 \item $2 + \sum_{n=1}^\infty \tfrac1{n^3} \cos n\theta + \tfrac1{3^n} \sin n\theta$.
 \item $1 + \sum_{n=1}^\infty \tfrac{(-1)^n}{n} \cos n\theta + \tfrac1{2^n} \sin n\theta$.
 \hint look at $\dlim_{r\to 1}u(r,\pi)$.
\end{parts}

\item 
\begin{parts}
\item Compute the Fourier series of the function $f(\theta) = |\sin \theta|$.\\
\hint use symmetry and the addition formula for $\sin(n\theta \pm \theta)$.
\item Show that the Fourier series converges uniformly. 
\item Hence evaluate $\dsum_{n=1}^\infty \dfrac{(-1)^n}{4n^2-1}$. 
\hint what value of $\theta$ is needed here?
\end{parts}


\item 
\begin{parts}
\item Suppose $u(x,y)$ is a solution of the heat problem on
$\ol{\bD}$ (given in rectangular coordinates).
Let $D_R(x_0,y_0)$ be a small disk of radius $R$ contained inside $\bD$.
Establish the \textit{mean value property}:
\[
  u(x_0,y_0) = \frac1{2\pi} \int_{-\pi}^{\pi}
  u(x_0 + R\cos\theta, y_0 + R\sin\theta) \, d\theta .
\]
\hint The restriction of $u$ to $\ol{D_R(x_0,y_0)}$ is the
solution to the heat problem on this disk.  Use the Poisson formula
for the value at the centre of this disk.

\item  Suppose that $u(x,y)$ is a continuous function on $\ol{\bD}$ 
that satisfies the mean value property.
Prove that if $u(x,y)$ attains its maximum value at an
interior point, then it must be constant.
\hint consider the value of $u$ on concentric circles around points attaining the maximum.
\end{parts}

\item \textbf{Bonus.} Prove that a continuous function on $\bD$ satisfing
the mean value property is harmonic. \quad
\hint Fix a point $(x_0,y_0)$ in $\bD$ and let
$D_R(x_0,y_0)$ be a small disk contained inside $\bD$.
Let $v(x,y)$ be the solution of the steady-state heat problem on
$D_R(x_0,y_0)$ that agrees with $u$ on the boundary circle.
Then $u-v$ satisfies the mean value property.
Show that $u=v$, and hence deduce that $\Delta u(x_0,y_0)=0$.




\end{Ex}
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