Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Branched Microstructures in the Ginzburg-Landau Model of Type-I
  Superconductors   We consider the Ginzburg-Landau energy for a type-I superconductor in the
shape of an infinite three-dimensional slab, with two-dimensional periodicity,
with an applied magnetic field which is uniform and perpendicular to the slab.
We determine the optimal scaling law of the minimal energy in terms of the
parameters of the problem, when the applied magnetic field is sufficiently
small and the sample sufficiently thick. This optimal scaling law is proven via
ansatz-free lower bounds and an explicit branching construction which refines
further and further as one approaches the surface of the sample. Two different
regimes appear, with different scaling exponents. In the first regime, the
branching leads to an almost uniform magnetic field pattern on the boundary; in
the second one the inhomogeneity survives up to the boundary.

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The categories of this paper are: supr-con
8
The categories of this paper are: hep-ex
9
The categories of this paper are: nlin
6
The categories of this paper are: astro-ph
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The categories of this paper are: alg-geom
0
The categories of this paper are: hep-ph
4
The categories of this paper are: cond-mat
15
The categories of this paper are: nucl-ex
16
The categories of this paper are: stat
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The categories of this paper are: dg-ga
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The categories of this paper are: atom-ph
3
The categories of this paper are: physics
5
The categories of this paper are: gr-qc
21
The categories of this paper are: adap-org
30
The categories of this paper are: q-alg
18
The categories of this paper are: econ