Smooth solutions of the Laplace equation with Neumann data

Question

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?

Answer 1

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 $u^2$.)