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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?

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  • $\begingroup$ "somewhere on the boundary" - do you mean everywhere on the boundary? Otherwise it is clearly wrong, take $u=\text{Re}\, z$ in the unit disc. $\endgroup$ – A.Γ. Jul 19 '15 at 15:20
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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.

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