On the optimality of periodic stripes in a continuous variational model with non-local interaction

In this talk, we will discuss the continuous version of a variational problem proposed by Giuliani, Lebowitz and Lieb. The model has a perimeter term and a non-local term, which are in competition. We compute the $\mathrm{\Gamma }$-limit as the parameter $J$ converges to its critical value $J_c$. The $\mathrm{\Gamma }$-limit will turn out to be finite only on stripes. By using the convergence of the minimizers, we show that minimizers converges to periodic stripes. This work is in collaboration with Michael Goldman.