Notice that the time is 12:30, not 12:00!
\newline
\vskip\baselineskip
The following is joint work with Sylvia Serfaty and Cyrill Muratov.
We study the asymptotic behavior of the screened sharp interface
Ohta-Kawasaki energy in dimension 2 using the framework of Γ-convergence.
In that model, two phases appear, and they interact via a nonlocal Coulomb
type energy. We focus on the regime where one of the phases has very small
volume fraction, thus creating ``droplets" of that phase in a sea of the
other phase. We consider perturbations to the critical volume fraction
where droplets first appear, show the number of droplets increases
monotonically with respect to the perturbation factor, and describe their
arrangement in all regimes, whether their number is bounded or unbounded.
When their number is unbounded, the most interesting case we compute the
Γ limit of the `zeroth' order energy and yield averaged information for
almost minimizers, namely that the density of droplets should be uniform.
We then go to the next order, and derive a next order Γ-limit energy,
which is exactly the ``Coulombian renormalized energy W" introduced in the
work of Sandier/Serfaty, and obtained there as a limiting interaction
energy for vortices in Ginzburg-Landau. The derivation is based on their
abstract scheme, that serves to obtain lower bounds for 2-scale energies
and express them through some probabilities on patterns via the
multiparameter ergodic theorem. Without thus appealing at all to the
Euler-Lagrange equation, we establish here for all configurations which
have ``almost minimal energy," the asymptotic roundness and radius of the
droplets as done by Muratov, and the fact that they asymptotically shrink
to points whose arrangement should minimize the renormalized energy W, in
some averaged sense. This leads to expecting to see hexagonal lattices of
droplets.
