I have a question on a proof we did on a lecture in PDE in which we proofed the following:
Let X $\subset \mathbb R^n$ be a open, connected and bounded space and let $f:X \to \mathbb R$ be harmonic and $f \in C^2 (X) \cap C^1(\overline X)$. Assume that $f$ is equal to $0$ on $\partial X \cap B(x_o,r)$ for some $x_0 \in X$ and $r>0$. Then $f=0$.
In the proof we show that the function
\begin{equation} F(x) = \begin{cases} f(x) & x \in B(x_o,r)\cap \overline X \\ 0 & x \in B(x_o,r) \setminus X \end{cases} \end{equation} is harmonic itself and thus $f=0$ but I am not entirely sure why this is sufficient. I thought one would assume that there is some $x \in B(x_o,r)\cap \overline X$ such that $F(x)\neq 0$ then it is not constant and therefore it must have its maximum on the Boundary but i don't see why that would be a false statement.