Cascade of minimizers for a nonlocal isoperimetric problem in thin domains

For \Omega_\e=(0,\e)\times (0,1) a thin rectangle, we consider minimization of the two-dimensional nonlocal isoperimetric problem given by \inf_u E^{\gamma}_{\Omega_\e}(u) where E^{\gamma}_{\Omega_\e}(u):= P_{\Omega_\e}(\{u(x)=1\})+\gamma\int_{\Omega_\e}\abs{\nabla{v}}^2\,dx and the minimization is taken over competitors u\in BV(\Omega_\e;\{\pm 1\}) satisfying a mass constraint \fint_{\Omega_\e}u=m for some $m\in (-1,1)$. Here P_{\Omega_\e}(\{u(x)=1\}) denotes the perimeter of the set $\{u(x)=1\}$ in \Omega_\e, \fint denotes the integral average and $v$ denotes the solution to the Poisson problem -\Delta v=u-m\;\mbox{in}\;\Omega_\e,\quad\nabla v\cdot n_{\partial\Omega_\e}=0\;\mbox{on}\;\partial\Omega_\e,\quad\int_{\Omega_\e}v=0. We show that a striped pattern is the minimizer for \e\ll 1 with the number of stripes growing like $\gamma^{1/3}$ as $\gamma\to\infty.$ We then present generalizations of this result to higher dimensions.Comment: 20 pages, 2 figure

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

Sharp interface limit of an energy modelling nanoparticle-polymer blends

We identify the $\Gamma$-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