Is there $f\in H^1(\mathbb T^n)$ such that $ \textrm{div}(f)=\sum_{j=1}^n \partial_j f=1$?

Question

Is there any $f\in H^1(\mathbb T^n)$ such that: $$\textrm{div}(f):=\sum_{j=1}^n \partial_j f=1,$$ where $1$ stands for the constant function $x\longmapsto 1$.

Thanks.

Answer 1

Well, the torus being compact, the function must have a maximum somewhere. At this point the divergence is 0.