Local and global minimality of the lamella for the anisotropic Ohta-Kawasaki energy

TL;DR

This paper investigates the local and global minimality of lamellar structures in an anisotropic Ohta-Kawasaki energy model, extending previous isotropic results and analyzing the effects of anisotropy on minimal configurations.

Contribution

It adapts second variation methods to anisotropic energies, proving local minimality of lamellae under ellipticity and establishing rigidity and global minimality results.

Findings

Lamellae are locally minimal under uniform ellipticity of anisotropy.

If the Wulff shape has horizontal facets, lamellae are isolated local minimizers.

Global minimality results are shown for planar cases.

Abstract

In this paper we consider the volume-constrained minimization of a variant of the Ohta-Kawasaki functional with an anisotropic surface energy replacing the standard perimeter. Following and suitably adapting the second variation approach devised in arXiv:1211.0164, we prove local minimality results for the horizontal lamellar configuration, in analogy with the isotropic case, under the assumption that the anisotropy is uniformly elliptic. If instead the Wulff shape of the anisotropy has upper and lower horizontal facets, we prove that the lamella exhibits a rigid behavior and is an isolated local minimizer for all parameter values. We conclude by showing some global minimality results, mostly focusing on the planar case.