# Smooth solutions of the Laplace equation with Neumann data

Let $D$ a connected domain, $\Delta u = 0$ on $D$ and $\partial_n u = 0$ on $\partial D$. Using energy methods I can show, that two solutions of this problem are unique up to a constant. Now, how can I show (again, using energy methods) that the only smooth solutions are constants as well?

Since you already have uniqueness up to constant, all that remains to do is to say that $u\equiv 0$ is a solution. By uniqueness, all solutions are of the form $0+\mathrm{const}$.
For completeness: the energy method here amounts to showing that $\int_D |\nabla u|^2=0$. The point is that $$\int_D |\nabla u|^2=\frac12 \int_{\partial D} \partial_n(u^2)= \int_{\partial D} u\, \partial_n u =0 \tag 1$$ where the first identity is the divergence theorem, and $$\frac{1}{2}\operatorname{div} \nabla (u^2) = \operatorname{div}(u\nabla u) = \nabla u\cdot \nabla u + u\Delta u = |\nabla u|^2.$$ (Alternatively, one uses Green's formula that cuts out the middle part with $u^2$.)