38 research outputs found
Sharp interface limit of an energy modelling nanoparticle-polymer blends
We identify the -limit of a nanoparticle-polymer model as the number
of particles goes to infinity and as the size of the particles and the phase
transition thickness of the polymer phases approach zero. The limiting energy
consists of two terms: the perimeter of the interface separating the phases and
a penalization term related to the density distribution of the infinitely many
small nanoparticles. We prove that local minimizers of the limiting energy
admit regular phase boundaries and derive necessary conditions of local
minimality via the first variation. Finally we discuss possible critical and
minimizing patterns in two dimensions and how these patterns vary from global
minimizers of the purely local isoperimetric problem.Comment: Minor changes. Rephrased introduction. This version is to appear in
Interfaces and Free Boundarie
Axisymmetric critical points of a nonlocal isoperimetric problem on the two-sphere
On the two dimensional sphere, we consider axisymmetric critical points of an
isoperimetric problem perturbed by a long-range interaction term. When the
parameter controlling the nonlocal term is sufficiently large, we prove the
existence of a local minimizer with arbitrary many interfaces in the
axisymmetric class of admissible functions. These local minimizers in this
restricted class are shown to be critical points in the broader sense (i.e.,
with respect to all perturbations). We then explore the rigidity, due to
curvature effects, in the criticality condition via several quantitative
results regarding the axisymmetric critical points.Comment: 26 pages, 6 figures. This version is to appear in ESAIM: Control,
Optimisation and Calculus of Variation
Existence of Ground States of Nonlocal-Interaction Energies
We investigate which nonlocal-interaction energies have a ground state
(global minimizer). We consider this question over the space of probability
measures and establish a sharp condition for the existence of ground states. We
show that this condition is closely related to the notion of stability (i.e.
-stability) of pairwise interaction potentials. Our approach uses the direct
method of the calculus of variations.Comment: This version is to appear in the J Stat Phy
On the existence of minimizing sets for a weakly-repulsive non-local energy
We consider a non-local interaction energy over bounded densities of fixed
mass . We prove that under certain regularity assumptions on the interaction
kernel these energies admit minimizers given by characteristic functions of
sets when is sufficiently small (or even for every , in particular
cases). We show that these assumptions are satisfied by particular interaction
kernels in power-law form, and give a certain characterization of minimizing
sets. Finally, following a recent result of Davies, Lim and McCann, we give
sufficient conditions on the interaction kernel so that the minimizer of the
energy over probability measures is given by Dirac masses concentrated on the
vertices of a regular -gon of side length 1 in