Timeline for Can gradient zero implies that a function is constant with Hörmander vector fields
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| Feb 21 at 8:39 | comment | added | Houa | Thanks for your answers. But what I worry about is whether $v$ can be a weak solution of $-\triangle_Xu=0$ in the distribution sense, i.e., $\langle v,-\triangle_X\phi\rangle$ for $\phi\in C_0^\infty(\mathbb{R}^n)$. I think this should be correct since the condition $Xv=0$ a.e. is too strong. By the definition of weak derivative we obtain $-\triangle_Xu=0$ in the distribution sense. Then $v$ smooth and we can prove the conclusion. | |
| Feb 20 at 16:43 | history | answered | Bazin | CC BY-SA 4.0 |