Let $\varphi$ be an harmonic function such that $D\varphi \in L^q(\mathbb R^n)$ for $q \in (1, +\infty)$. I read in *Partial Differential Equations* of *Quin Han* and *Fanghua Lin* that for $q = 2$, $\varphi$ has to be constant. My professor told me that it is possible to generalize this result to any $q \in (1,+\infty)$ but I couldn't find a reference of this anywhere. Would someone please tell me where I could find a proof ?



A harmonic function $\varphi$ with $D\varphi \in L^q(\mathbb R^n)$ is constant
If $\phi$ is harmonic over $\mathbb{R}^n$, then all its partial derivatives $\partial\_i \phi$ are harmonic. As a consequence, all we are left to prove is that any harmonic function that belongs to $L^p$ is zero. $\phi$ will have all its partial derivatives vanish so it is constant.


Let $\psi$ be an harmonic function that belongs to $L^p$ for some $p \in [1, \infty)$. By the mean value property, we have, for any $x \in \mathbb{R}^n$, that
$$
\psi(x) = \frac{1}{\mathrm{Vol}(B\_r(x))} \int\_{B\_r(x)} \psi(y) dy^n.
$$
We now use Hölder's inequality to get
$$
|\psi(x)| \leq \frac{1}{\mathrm{Vol}(B\_r(x))} \|\psi\|\_{L^p(B\_r(x))} \mathrm{Vol}(B\_r(x))^{1-\frac{1}{p}} \leq \mathrm{Vol}(B\_r(x))^{-\frac{1}{p}} \|\psi\|\_{L^p(\mathbb{R}^n)}.
$$
Letting $r$ tend to $+\infty$, we conclude that $\psi(x) = 0$.


