Quiz 6 T5102

[Victor Ivrii]

Solve
\begin{align*}
& \Delta u:=u_{xx}+u_{yy}=0&& \text{in }  r> a\\[3pt]
& u|_{r=a}=f(\theta),\\[3pt]
& \max |u| <\infty.
\end{align*}
where we use polar coordinates $(r,\theta)$ and $f(\theta)=\left\{\begin{aligned}
&1  &&0<\theta<\pi\\
-&1 &&\pi<\theta<2\pi.
\end{aligned}\right.$

The expected answer: solution as a series.

[Jingxuan Zhang]

The general bounded solution of the DE is
\begin{equation}u=\frac{a_0}{2}+\sum_n r^{-n}(a_n \cos n\theta + b_0\sin n\theta)\label{a}\end{equation}
Now $f$ is odd so $a_n\equiv0$ and
\begin{equation}b_n=\frac{2a^n}{\pi}\int_0^\pi \sin n\theta\,d\theta=\left\{\begin{array} &\frac{4a^n}{n\pi}&\text{n odd}\\0&\text{n even}\end{array}\right.\label{b}\end{equation}

Combining $(1),(2)$ we have the final solutoin
\begin{equation}\frac{4}{\pi}\sum_k (\frac{a}{r})^{2k+1} \frac{\sin(2k+1)\theta}{2k+1}\label{c}\end{equation}