Cascade of Minimizers for a Nonlocal Isoperimetric Problem in Thin Domains

Abstract

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