Calculus of Variations and Geometric Measure Theory

M. Morini - P. Sternberg

Cascade of minimizers for a nonlocal isoperimetric problem in thin domains

created by morini on 12 Aug 2013
modified on 01 Apr 2014


Accepted Paper

Inserted: 12 aug 2013
Last Updated: 1 apr 2014

Journal: SIAM Journal on Mathematical Analysis
Year: 2014


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 \[ E^{\gamma}_{\Omega_\varepsilon}(u):= P_{\Omega_\varepsilon}(\{u(x)=1\})+\gamma\int_{\Omega_\varepsilon}(\nabla v)^2\,dx \] and the minimization is taken over competitors $u\in BV(\Omega_\varepsilon;\{\pm 1\})$ satisfying a mass constraint $\int_{\Omega_\varepsilon}u=m\, \mathrm{meas}(\Omega_\varepsilon)$, for some $m\in (-1,1)$. Here $P_{\Omega_\varepsilon}(\{u(x)=1\})$ denotes the perimeter of the set $\{u(x)=1\}$ in $\Omega_\varepsilon$ 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 show that a striped pattern is the minimizer for $\varepsilon\ll 1$ with the number of stripes growing like $\gamma^{1/3}$ as $\gamma\to\infty.$ We then present generalizations of this result to higher dimensions.