Suppose that $u$ is a harmonic function in the disk $D = \{ r < 1 \}$ such that $u|_{r=1} = \theta^2$ for $-\pi \leq \theta \leq \pi$. Without finding the solution $u$:
a.) Find the maximum value of $u$ in $\overline{D}$.
b.) Find $u(0,0)$, the value of $u$ at the origin.
Attempted solution a.) - By the maximum principle, the maximum is achieved on $\partial D$. On $\partial D$, the function $u(1,0)$ can be handled as a single variable function. The critical points of the function satisfy $$\frac{\partial u}{\partial \theta} = 2\theta = 0$$ That happens at $\theta = 0$. As $u(1,0) = 0$ the maximum value is not defined.
Attempted solution b.) - The value of $u$ at the origin can be found by the mean value property: \begin{align*} u(0,0) &= \frac{1}{2\pi}\int_{0}^{2\pi}\theta^2 d\theta\\ &= \frac{1}{2\pi}\left[\frac{\theta^2}{3} \Big|_0^{2\pi} \right]\\ &= \frac{4}{3}\pi^2 \end{align*}
I want to know if these solutions are correct. Any suggestions are appreciated.
