I'm studying PDE and at the moment I'm reading L. Evans' book.
The strong maximum principle states that; if $u\in C^2(U)\cup C(\bar U)$ is harmonic in $U$, where $U$ is connected and if there exists $x_0$ such that $u(x_0)=\max _\bar U u$, then $u$ is constant within $U$.
A little later Evans states that if $U$ is connected and $u$ is a solution of the pde $\Delta u=0$ in $U$ and $u=g$ on $\partial U$, where $g\geq 0$. Then $u$ is positive in $U$ everywhere if $g$ is positive somewhere on $\partial U$.
Why is this true? I can't see how this follows immediately from the strong maximum principle. I'm guessing it is trivial and I'm just over thinking it. Can somebody help me?