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 \eqref{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}
\label{1}\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.