Global minimizers for the Ginzburg-Landau functional below the first critical magnetic field
Let \(\Omega\) be a smooth, bounded and simply connected domain in \({\mathbb{R}}^2\). The paper deals with the study of global minimizers of the Ginzburg-Landau functional \[ J(u,A)=\frac{1}{2}\int_\Omega |\nabla_Au|^2+|h-h_{ex}|^2-h^2_{ex}+ \frac{1}{2\varepsilon^2}(1-|u|^2)^2 \] that corresponds to the free energy of a superconductor in a prescribed, constant magnetic field \(h_{ex}\). The minimization of \(J(u,A)\) is performed with respect to the class \[ D=\left\{(u,A);\frac{1}{2}\int_\Omega |\nabla_Au|^2+\frac{1}{2\varepsilon^2} (1-|u|^2)^2<{\mathcal M}|\log\varepsilon |\right\}, \] where \({\mathcal M}\) is a prescribed (large) positive constant. Minimizers of \(J(u,A)\) solve the associated Euler equations (1) \(-\nabla_A^2u= \frac{1}{\varepsilon^2}u(1-|u|^2)\) and (2) \(-\star dh=(iu,d_Au)\), where the unknowns are the complex-valued order parameter \(u\in H^1(\Omega, {\mathbb{C}})\) and the \(U(1)\) connection \(A\in H^1(\Omega , {\mathbb{R}}^2)\). Let \(H_{c_1}\) denote the first critical field. The main result of the paper asserts that there exists a value \(H'_{c_1}\simeq H_{c_1}\) (more precisely, \(H'_{c_1}=H_{c_1}+O(|\log |\log\varepsilon ||)\)) such that, if \(\varepsilon >0\) is small enough and \(h_{ex}\leq H'_{c_1}\), then a globally minimizing solution of the Ginzburg-Landau system (1)-(2) satisfies \(|u|\geq 3/4\) on \(\Omega\), and is a vortex-free solution. The proof is based on variational methods and uses refined elliptic estimates.
scientific article; zbMATH DE number 1441850