Cascade of minimizers for a nonlocal isoperimetric problem in thin domains

TL;DR

This paper analyzes a nonlocal isoperimetric problem in thin domains, showing that striped patterns minimize the energy for small thickness, with the number of stripes depending on a parameter, and extends results to higher dimensions.

Contribution

It demonstrates that striped patterns are the energy minimizers in thin domains for large nonlocal interaction strength, and generalizes the findings to higher dimensions.

Findings

Striped patterns minimize the energy in thin domains as thickness approaches zero.

Number of stripes grows like rd power of the nonlocal parameter b3.

Results are extended to higher-dimensional settings.

Abstract

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.