Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Global Jacobian and $\Gamma$-convergence in a two-dimensional
  Ginzburg-Landau model for boundary vortices   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 $\Gamma$-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.

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The categories of this paper are: physics
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The categories of this paper are: chem-ph
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The categories of this paper are: q-alg
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The categories of this paper are: cs
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The categories of this paper are: chao-dyn
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The categories of this paper are: cond-mat
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The categories of this paper are: hep-th
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The categories of this paper are: adap-org
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The categories of this paper are: mtrl-th
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The categories of this paper are: hep-ex
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The categories of this paper are: nucl-th
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The categories of this paper are: bayes-an
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The categories of this paper are: nlin
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The categories of this paper are: hep-lat
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The categories of this paper are: q-fin
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The categories of this paper are: math-ph