Droplet phase in a nonlocal isoperimetric problem under confinement
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abstract

We address small volume-fraction asymptotic properties of a nonlocal
isoperimetric functional with a confinement term, derived as the sharp
interface limit of a variational model for self-assembly of diblock copolymers
under confinement by nanoparticle inclusion. We introduce a small parameter
$\eta$ to represent the size of the domains of the minority phase, and study
the resulting droplet regime as $\eta\to 0$. By considering confinement
densities which are spatially variable and attain a nondegenerate maximum, we
present a two-stage asymptotic analysis wherein a separation of length scales
is captured due to competition between the nonlocal repulsive and confining
attractive effects in the energy. A key role is played by a parameter $M$ which
gives the total volume of the droplets at order $\eta^3$ and its relation to
existence and non-existence of Gamow's Liquid Drop model on $\mathbb{R}^3$. For
large values of $M$, the minority phase splits into several droplets at an
intermediate scale $\eta^{1/3}$, while for small $M$ minimizers form a single
droplet converging to the maximum of the confinement density.