Let $ \displaystyle{ U \subset \mathbb R ^n }$ open, bounded and connented.
If $ u \in C(U) \cap C(\bar U)$ such that:
$$ \Delta u =0 \quad \text { in U}$$
$$u=g \geq 0 \quad \text {in} \quad \partial U $$
and $ g >0 $ on a point in $ \partial U$ then $ u>0 $ in whole $U$.
I know that I have to use the maximum principle someway but for some reason I cant't solve it.
Any help ?
Thank's in advance!