Could someone explain what the mean value property means, and how I can apply it to a question?
Mean value property: Let $u$ be a harmonic function in a disk $D$, and continuous in its closure $D$, then the value of $u$ at the center of the disk is equal to the average of $u$ on its circumference.
It is 10.1 in this link. http://www.math.ucsb.edu/~grigoryan/124B/lecs/lec10.pdf
How would I apply it to the following?
If $u$ is harmonic function in disk $D=\{r<2\}$ and $u(\theta)=3\sin(2\theta) +1$ for $r = 2$, without finding the solution calculate $u$ at the origin.
Thanks!