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.

score 0 · Answer 1 · answered Dec 04 '15 at 17:22

0

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

answered Dec 04 '15 at 17:22

Thomas

7,470

But for this approach you need continuity of $f$ and strong differentiability at the point where the maximum is attained. – gerw Dec 04 '15 at 18:38

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

1 Answers1