Let $\Delta=\{z\in\mathbb{C}: |z|<1\}$ Assume $u\in C(\overline{\Delta}\setminus \{1\})$ such that it is harmonic in $\Delta$ and $u(\xi)=0$ for $\xi\in S^1\setminus \{1\}$.
(a) Find an example $u$ as above which is not identically 0
(b) Find some nontrivial condition which guarantees $u$ to be $0$. Make your condition as sharp as possible.
(c) Find all such nonnegative $u$'s
How would you do all of these? Using Poisson kernel?