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?