approximated by smooth S 2 -valued maps. More recently, the authors in proved, as a special case of more general results, that if u 2 W 1;1 \ L 1 ðR 2 ; S 1 Þ and the distributional Jacobian of u is a Radon measure, then this measure must be atomic. Similar results are found in the work of Giaquinta, Modica, and Soou$ c cek on Cartesian currents and analog of this result in the space H 1=2 is proven by Bourgain et al. [7].
In this paper, we continue this analysis with different scalings. Suppose that for a sequence of functions fu e g, E e ðu e Þ4Kg e ;
jln ej4g e (e À2 : ð1:1Þ
Since e 2 g e tends to zero by assumption, the potential term in E e forces ju e j to be close to one in most of the domain. However, ju e j is still close to zero around vortices. In view of the results of , this contributes to the energy at least by an amount of jln ej jjJ ðu e Þjj M . Moreover, mentioned before, this ''vortex energy'' concentrates near the vortices: on the union of balls with small radii. We call this ''vortex set'' V e . Then, on this set, the energy E e ðu e Þ is approximately bounded from below by Z
V e e e ðu e Þ dx5jln ej jjJ ðu e Þjj M :
A precise mathematical statement of this fact is demonstrated in Section 5 as a sharp lower bound of the energy in terms of the Jacobian. Away from the vortices only the gradient term is active. Since for
and since away from the vortices ju e j is near one, in this region jru e j 2 % jjðu e Þj 2 ¼ ju e  ru e j 2 : Hence, approximately Z U=V e e e ðu e Þ dx5 Z U=V e jru e j 2 dx5 Z U=V e jjðu e Þj 2 dx: This reasoning indicates that the functional E e ðu e Þ is approximately bounded from below by 1 2 jjjðu e Þw U=V e jj 2 2 þ jln ej jjJ ðu e Þjj M :
The excess energy between E e and the above expression is due to the extra winding around the vortices.
JERRARD AND SONER