Harmonic function vanishes with its normal derivative on a part of boundary; can Green's formula be applied to broken boundary?

Question

Let $\Omega$ be an open domain, and let $\Sigma$ be a smooth and nonempty portion of the boundary. Let $u$ be a harmonic function in $\Omega$ and $u=D_\nu u=0$ on $\Sigma$. ($D_\nu$ is the derivative along the outward vector). How to prove $u=0$ in $\Omega$?

Can I use Green's formula to this "portion" of boundary?

Answer 1

A couple solutions are given here. What do you mean by using Green's formula to a "portion"?