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

Michael Goldman (1) , Eris Runa (2)

1 LJLL - Laboratoire Jacques-Louis Lions
2 MPI-MiS - Max Planck Institute for Mathematics in the Sciences

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 Γ−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.

Domaines: Equations aux dérivées partielles [math.AP]; Physique mathématique [math-ph]

Michael Goldman, Eris Runa. On the optimality of stripes in a variational model with non-local interactions. Calculus of Variations and Partial Differential Equations, 2019, 3. ⟨hal-01400481v2⟩