On the optimality of stripes in a variational model with non-local interactions

TL;DR

This paper investigates the formation of stripe patterns in a variational model with competing local and non-local interactions, demonstrating convergence to periodic stripes near a critical parameter value through advanced mathematical analysis.

Contribution

It extends previous discrete system analyses to a continuous setting, overcoming key challenges with slicing and rigidity techniques to prove pattern optimality.

Findings

Minimizers converge to periodic stripes as parameter approaches critical value

The continuous model differs from discrete cases in energy contributions

Mathematical techniques developed to handle non-local interactions

Abstract

We study pattern formation for a variational model displaying competition between a local term penalizing interfaces and a non-local term favoring oscillations. By means of a $\Gamma$--convergence analysis, we show that as the parameter J converges to a critical value J c, the minimizers converge to periodic one-dimensional stripes. A similar analysis has been previously performed by other authors for related discrete systems. In that context, a central point is that each " angle " comes with a strictly positive contribution to the energy. Since this is not anymore the case in the continuous setting, we need to overcome this difficulty by slicing arguments and a rigidity result.