Suppose I know that $$ \int_{\Omega} \varphi |Dv|^2 = \int_{\Omega} v\; Dv\cdot D\varphi $$ holds for every $\varphi \in C^1_c(\Omega)$ (this is an easy identity for harmonic functions but that is more or less beside the point of my question).
Can I let $\varphi$ approximate the characteristic function of a ball i.e. $$ \varphi \to \mathbf{1}_{B_{\rho}(x_0)} $$ in order to deduce directly something like $$ \int_{B_{\rho}(x_0)}|Dv|^2 = \int_{\partial B_{\rho}(x_0)} v\; Dv\cdot \frac{(x-x_0)}{\rho} $$ without doing another argument in between?
I feel like I've seen people do this, but you can't actually $W^{1,2}$ converge to $\mathbf{1}_{B_{\rho}(x_0)}$ or anything like that so in what sense does the second identity hold?