The sol'n of Poisson’s equation with Neumann boundary conditions in unique up to a constant

Question

Show that if $u(x),v(x)$ are solutions of $f(x)=Δw$ in $Ω$, with boundary condition $\frac{∂w}{∂n}=g$ on $∂Ω$ then $u(x)-v(x)=$ constant

Answer 1

Supposing that $u,v$ are two solution, we see that $w = u-v$ satisfies the equation $$\Delta w = 0 \text{ in } \Omega, \,\,\,\,\,\, \frac{\partial w}{\partial n} = 0 \text{ on } \partial\Omega.$$ Multiplying by $w$ and integrating (then integrating by parts) gives $$\int_{\Omega} w\Delta w = 0 \implies -\int_{\Omega} \lvert\nabla w \rvert^2 + \int_{\partial\Omega} w \underbrace{\frac{\partial w}{\partial n}}_{=0} = 0 \implies \int_{\Omega} \lvert \nabla w \rvert^2 = 0.$$ But this implies that $\nabla w = 0$ so $w$ is constant.