While i am revising for exam : I am facing some problems to understand things clearly . Here are my doubts :
a) If $u$ solves $\Delta u =0 , x\in \Omega; u=g , x\in \partial \Omega$ for non constant boundary data $g$ with $g\ge0$ and $g(x_0)\> 0$ for some $x_0 \in \partial \Omega $ then why is it true that $u(x)>0$ for all $x\in \Omega$ ?
b) It's about Harnack inequality . Let $V$ be open and connected and $V\subset\subset \Omega$ ie $V$ is compactly contained in $\Omega$ then there existz a constant $C<\infty$ such that $sup_V u \le C inf_V u$ . My question is why only upto $V$ and not upto $\bar V $ ie why not upto closure ?