I'm trying to solve the following question (this is just for practice):
If $u$ is harmonic within $\mathbb{R}^n$ with $\int_{\mathbb{R}^n}|Du|^2 dx \leq C$ for some $C > 0$, then show that u is a constant in $\mathbb{R}^n$.
I guess the idea is to somehow show that $Du = 0$ which implies $u$ is constant, or otherwise show that $u$ is bounded and thus constant by Liouville's theorem. I can't quite see how to do this though. Of course if it were on a bounded domain $U$ I know I could use the integration by parts formula $$0 = – \int_U u \Delta u dx = \int_U |Du|^2 dx – \int_{\partial U} u^2 dS$$ which would imply that $u$ is bounded … but then Liouville's theorem doesn't apply because it's not defined on all of $\mathbb{R}^n$ (I think).
Can anyone point me in the right direction? (Also, out of curiosity, is there some kind of analogue of integration by parts for unbounded domains?)
Best Answer
If I understand your problem, you are trying to prove that an harmonic function on $\mathbb R^N$ whose gradient is $L^2(\mathbb R^N)$ is necessarily constant.
First of all, you should point out what definition of harmonic function you use; I will use the fact that a continuous function $u$ which satisfies the mean property is harmonic (it is equivalent to $u \in C^2$ and $\Delta u =0$). I remark also the fact that an harmonic function is necessarily $C^{\infty}$ (indeed, analytic) - you can easily see this by using mollifiers.
If $u$ is harmonic, then $\partial_{x_i}u$ is harmonic for every $i=1,\ldots , N$. It follows that $\nabla u$ has the mean property: using Cauchy-Schwarz, $$ \vert \nabla u(x) \vert =\left\vert \frac{1}{\omega_N R^N}\int_{B(x,R)}\nabla u(y)dy \right\vert \leq \frac{1}{\omega_N R^N}\int_{B(x,R)} \vert \nabla u(y) \vert dy\leq \frac{\sqrt{\omega_N R^N}}{\omega_N R^N}\left(\int_{B(x,R)}|\nabla u(y)|^2 dy\right)^{1/2}\leq\frac{||\nabla u||_2}{\sqrt{\omega_N R^N}} $$
Let $R \to + \infty$ and you get $\nabla u(x)=0$, which is the claim. Hope this helps (but I am afraid of misunderstanding the question).
Remark: if this is correct, we can substitute the hypothesis $\nabla u \in L^2(\mathbb R^N)$ with the more general one $\nabla u \in L^p(\mathbb R^N)$, for some $p \in [1,+\infty]$.