Let $u(x)$ be harmonic for $x \in \Bbb R^3$. Suppose that $\nabla u \in L^2 (\Bbb R^3)$. Prove that $u$ is a constant.

Question

Let $u(x)$ be harmonic for $x \in \Bbb R^3$. Suppose that $\nabla u \in L^2 (\Bbb R^3)$. Prove that $u$ is a constant.

I tried with Green's equation but it didn't work. Could anyone tell me which method I should use?

Answer 1

Note that $\int_{\mathbb{R}} |\nabla u(x)|^2 dx=-\int_{\mathbb{R}}u(x)\Delta u(x)dx=0$. So $\nabla u(x)=0$ and $u$ is identically a constant.