This should be an easy application, since it is given as a remark without proof in Evan's pde book.
Let U be connected and u harmonic in U. Then u is positive everywhere in U if u is positive somewhere on the boundary of U.
My approach: Use the strong minimum principle. If u is equal to or less than 0 somewhere in U then u will attain its minimum in U. That means u has a local minimum, so u is constant and equals to the minumum (which is equal or less than 0). But it is given that u has a positive value on the boundary.
Somehow my proof doesn't feel right.. Can somebody tell me what I am missing here?
The assumption in the book is that $u\ge 0$ everywhere on the boundary and $u>0$ somewhere on the boundary. Also, $u\in C^2(U)\cap C(\overline{U})$.
Your argument is correct. Here's a restatement in different words:
Suppose that $u=0$ at some interior point. By the strong maximum principle applied to $-u$, it follows that $u\equiv 0$. This contradicts the assumption that $u$ is positive somewhere.