# Finding the maximum of a harmonic function

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.

• What do you mean the maximum is not defined? If you believe the maximum occurs at $(1,0)$ shouldn't you say the maximum is $0$? Regardless, be careful - by using $\theta$ as a coordinate you're modelling the circle as an interval, so the maximum isn't necessarily a critical point - it could occur at an endpoint. Commented Jul 12, 2017 at 1:21
• "The maximum value is not defined" what did you mean by this?
– user223391
Commented Jul 12, 2017 at 1:22

For (a), the maximum value of $\theta^2$ on $[-\pi, \pi]$ is $\pi^2$, occurring at the endpoints. Note that $\theta = \pi$ and $\theta =-\pi$ both correspond to the same point $(-1,0)$.
• @Wolfgang-1 Can you just picture the graph of the function $x^2$ for $-\pi \le x \le \pi$? Where does it take its maximum value? Commented Jul 12, 2017 at 1:57
• @ErickWong $\pi$? Commented Jul 12, 2017 at 1:59