Quantitative equipartition of the Ginzburg-Landau energy with applications (Q2879957)

The authors investigate the local structure of asymptotical minimizers of the Ginzburg-Landau functional \(E_{\epsilon}(u)=\frac{1}{2}\int_{B}\left\{|\nabla u|^2+\frac{1}{2\epsilon^2}(1-|u|^2)^2\right\}\,dx\) where \(B\) is the unit disk in the complex plane, and \(u\in H^1(B;\mathbb C)\) is required to have index \(\pm 1\) about some point of \(B\). For small values of \(\epsilon\) the energy of local minimizers is concentrated at vortices, with each vortex carrying \(\pi|\log\epsilon|+O(1)\) energy [\textit{F. Bethuel, H. Brézis} and \textit{F. Hélein}, Ginzburg-Landau vortices. Progress in Nonlinear Differential Equations and their Applications. 13. Boston, MA: Birkhäuser; (1994; Zbl 0802.35142)]. Moreover, the energy of the vortex is asymptotically equidistributed between the horizontal and vertical partial derivatives, see [\textit{F. H. Lin}, Commun. Pure Appl. Math. 52, No. 6, 737--761 (1999; Zbl 0929.35076)].NEWLINENEWLINEThe present article gives a further insight into the structure of a vortex. Informally, it shows that the fluid tensor \(\nabla u\otimes \nabla u\) is essentially diagonal in the sense of \(L^1\) estimates. To make this precise, one first needs to make sure that \(u\) is close to being minimal for \(E_\epsilon\), and that \(B\) contains a single vortex. These requirements are naturally expressed by the inequalities \(\|\det \nabla u-\pi \delta_0\|_{\dot{W}^{-1,1}(B)}\leq 1/4\) and \(E_{\epsilon}(u)\leq \pi |\log \epsilon|+K_0\) where \(K_0\) is a constant. The conclusion of the theorem is that the matrix \(\frac{1}{2}(\nabla u\otimes \nabla u)\) is within \(L^1\) distance of \(\sqrt{K_1 |\log\epsilon|}\) from the constant diagonal matrix \(\frac{\pi}{2}|\log\epsilon|I\). Here \(K_1=C(C+K_0)e^{K_0/\pi}\) and \(C\) is a universal constant.NEWLINENEWLINEThe order of the error bound \(\sqrt{|\log\epsilon|}\) is shown to be optimal by means of the following interesting example, which is inspired by [\textit{R. L. Jerrard}, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 1, No. 4, 733--768 (2002; Zbl 1170.35318)]: \(u_\epsilon (re^{i\theta}) = \min(\epsilon^{-1}r, 1 ) \exp(i\theta+i\alpha_\epsilon \cos 2\theta )\). With the choice \(\alpha_\epsilon = |\log\epsilon|^{-1/2}\) this function satisfies the assumptions of the main theorem, but the off-diagonal entries of \(\nabla u\otimes \nabla u\) are of order \(|\log\epsilon|^{1/2}\).