I am reading a paper of Comte and Mironescu [CM96], where they discuss properties of solutions $v = v_{\epsilon}: G \to \mathbf{C}$ of critical points of the (non-magnetic) Ginzburg–Landau functional $E_\epsilon(v) = \frac{1}{2} \int_G \lvert \nabla v \rvert^2 + \frac{1}{4 \epsilon^2} \int_G (1 - \lvert v \rvert^2)^2$, defined on a starshaped domain $G \subset \mathbf{R}^2$.

How is the identity $(1)$ proved?

In the course of a derivation of an estimate for such a critical point $v$ (on page 203), which is decomposed as $v = \rho \mathrm{e}^{\mathrm{i} \varphi}$, they use an integral identity stating that \begin{equation} \tag{1} \int_{\partial B_{C\epsilon}(x_j^\epsilon)} \rho^2 \frac{\partial \varphi}{\partial \nu} = 0. \end{equation} The integral is over the boundary of a disk $B_{C\epsilon}(x_j^\epsilon)$ of radius $C\epsilon$ around a 'bad point' $x_j^\epsilon$. (These disks around points $x_1^\epsilon,\dots,x_J^\epsilon$ are constructed so that $\lvert v \rvert \geq 1/2$ in their complement.)

The authors refer to a paper of Bethuel, Brezis and Helein [BBH93], and specifically to the proof of Proposition 1 therein, but I don't see the connection: no identity of the sort is used there.

[BBH93] F. Bethuel, H. Brezis, F. Helein. Asymptotics for minimizers of a Ginzburg–Landau functional, Calculus of Variations and PDE 1 (1993), pp. 123-148.

[CM96] M. Comte and P. Mironescu. Remarks on nonminimizing solutions of Ginzburg-Landau type equation, Asymptotic Analysis 13 (1996) pp. 199-215.