In the theory of 2D Ginzburg-Landau vortices, the Jacobian plays a crucial role for the detection of topological singularities. We introduce a related distributional quantity, called the global Jacobian that can detect both interior and boundary vortices for a 2D map u. We point out several features of the global Jacobian, in particular, we prove an important stability property. This property allows us to study boundary vortices in a 2D Ginzburg-Landau model arising in thin ferromagnetic films, where a weak anchoring boundary energy penalising the normal component of u at the boundary competes with the usual bulk potential energy. We prove an asymptotic expansion by Γ-convergence at the second order for this mixed boundary/interior energy in a regime where boundary vortices are preferred. More precisely, at the first order of the limiting expansion, the energy is quantised and determined by the number of boundary vortices detected by the global Jacobian, while the second order term in the limiting energy expansion accounts for the interaction between the boundary vortices.

在二维Ginzburg-Landau涡旋理论中，雅可比矩阵对拓扑奇异性的检测起着至关重要的作用。我们介绍了一个相关的分布量，称为全局雅可比分布，它可以检测2D映射u的内部和边界涡。我们指出了全局雅可比行列式的几个特征，特别是，我们证明了重要的稳定性。该特性使我们能够研究薄铁磁薄膜中产生的二维Ginzburg-Landau模型中的边界涡，其中弱锚固边界能会破坏u的正态分量在边界处与通常的体势能竞争。我们证明了在首选边界涡旋的情况下，对于这种混合的边界/内部能量，通过二阶Γ收敛的渐近展开。更精确地，在极限膨胀的第一阶，通过整体雅可比矩阵检测到的边界涡旋的数量来量化和确定能量，而在极限能量膨胀中的第二阶项解释了边界涡旋之间的相互作用。