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 ?
2026-04-06 06:14:21.1775456061
A harmonic function $\varphi$ with $D\varphi \in L^q(\mathbb R^n)$ is constant.
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