Calculus of Variations and Geometric Measure Theory

M. Goldman - E. Runa

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

created by goldman on 22 Nov 2016
modified on 01 Apr 2019


Accepted Paper

Inserted: 22 nov 2016
Last Updated: 1 apr 2019

Journal: Calc. Var. and PDE
Year: 2016


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.