Consider $f(x_1,x_2)=\chi_{B_1(0,0)}(x_1,x_2)$.
1) Is $\nabla f\in (H^1_0(\mathbb{R}^2;\mathbb{R}^2))^*$?
2) $\langle \nabla f , u \rangle= ?$ with $u\in H_0^1(\mathbb{R^2})$?
For the first question I had no problem, but for the second one I have some troubles, it would be correct to use an integration by parts?