Let $u$ be a harmonic function on $U$ connected open set of $\mathbb{R}^n$ and suppose there is a open set $V\subset U$, such that $u(x)=0$ for every $x\in V.$ Show that $u=0$ in $U$.
So, I tried to use of Maximum Principle but I not found a contradiction. Let be $x_0\in\partial V$ and $B=B(x_0,r)\subset U$. If I show that $u(x_0)$ is maximum or minimum is done. So, suppose there is $y\in B$ such that $u(y)\in\partial B$ is maximum in $B$.
How can I show this is not gonna happen? Some hint?
Thanks!