TITLE: Harmonic Unit Disk Problem with Boundary Function - Dirichlet Problem
QUESTION [0 upvotes]: Solve $u_{xx}+u_{yy}=0$ in the disk $r<1$ having the boundary function: $$u=4\cos(2\theta)+\sin^2(3\theta)$$
I am pretty sure this is a Dirichlet problem... but I can't seem to get a sensible answer.

REPLY [0 votes]: The functions $r^{|n|}e^{in\theta}$ are harmonic for $n=0,\pm 1,\pm 2,\cdots$ because these are the polar forms of the functions $z^{n}$ and $\overline{z^n}$ for $n=0,1,2,3,\cdots$. Rewrite your function in terms of these:
$$
     4\cos(2\theta)+\sin^2(3\theta)=4\cos(2\theta)+\frac{1}{2}\{1-\cos(6\theta)\}.
$$
The following function is harmonic
$$
          u(r,\theta)=\frac{1}{2}+4r^2\cos(2\theta)-\frac{1}{2}r^6\cos(6\theta),
$$
and
$$
           u(1,\theta)=4\cos(2\theta)+\sin^2(3\theta).
$$
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