TITLE: Uniqueness theorem of harmonic function
QUESTION [1 upvotes]: Consider $\Omega=\{(x,y)\in R^{2} |y>0\}$. If $u(x,y)=xy$, then $u$ is harmonic in $\Omega$ and $u=0$ on $\partial \Omega$.  But the constant function $0$ is also harmonic with the same boundary condition.  How do we show that this does not contradict the uniqueness theorem of harmonic function.

REPLY [0 votes]: Proof of uniqueness of solution depends on Maximum principle.
The maximum principle has assumption bounded domain.
As in your problem domain is Upper half-plane , so uniqueness theorem will not hold.
{"set_name": "stack_exchange", "score": 1, "question_id": 3710476}