Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The $\mathbf \Gamma$-limit of the two-dimensional Ohta-Kawasaki energy.
  I. Droplet density   This is the first in a series of two papers in which we derive a
$\Gamma$-expansion for a two-dimensional non-local Ginzburg-Landau energy with
Coulomb repulsion, also known as the Ohta-Kawasaki model in connection with
diblock copolymer systems. In that model, two phases appear, which interact via
a nonlocal Coulomb type energy. We focus on the regime where one of the phases
has very small volume fraction, thus creating small "droplets" of the minority
phase in a "sea" of the majority phase. In this paper we show that an
appropriate setting for $\Gamma$-convergence in the considered parameter regime
is via weak convergence of the suitably normalized charge density in the sense
of measures. We prove that, after a suitable rescaling, the Ohta-Kawasaki
energy functional $\Gamma$-converges to a quadratic energy functional of the
limit charge density generated by the screened Coulomb kernel. A consequence of
our results is that minimizers (or almost minimizers) of the energy have
droplets which are almost all asymptotically round, have the same radius and
are uniformly distributed in the domain. The proof relies mainly on the
analysis of the sharp interface version of the energy, with the connection to
the original diffuse interface model obtained via matching upper and lower
bounds for the energy. We thus also obtain a characterization of the limit
charge density for the energy minimizers in the diffuse interface model.

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The categories of this paper are: nucl-th
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The categories of this paper are: funct-an
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The categories of this paper are: comp-gas
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The categories of this paper are: patt-sol
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The categories of this paper are: econ
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The categories of this paper are: nucl-ex
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The categories of this paper are: hep-th
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The categories of this paper are: eess
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The categories of this paper are: ao-sci
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The categories of this paper are: bayes-an
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The categories of this paper are: atom-ph
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The categories of this paper are: supr-con
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The categories of this paper are: adap-org
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The categories of this paper are: acc-phys
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The categories of this paper are: hep-ex
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The categories of this paper are: chao-dyn