I am interested in the following statement:

Let $F$ be a vector field in $\mathbb{R}^n$ that is $C^1$-smooth in a domain $U$, continuous up to the boundary $\partial U$, and vanishing on $\partial U$. Then $\int_U div(F)=0$.

Naively, if $U_n$ is a smooth domain approximating $U$ from inside and $|F|\leq 1/n$ on $\partial U_n$, then the regular divergence theorem gives $|\int_{\partial U_n} F\cdot N| \leq Area(\partial U_n)/n$ which might very well diverge. So it looks like one needs a finite perimeter condition. However, is there a known example of a construction in a case of non-finite perimeter where the integral of the divergence is not zero?