A free discontinuity model for smectic thin films

ABSTRACT

We attempt to describe surface defects in smectic A thin films by formulating a free discontinuity problem – that is, a variational problem in which the order parameter is allowed to have jump discontinuities on some (unknown) set. The free energy functional contains an interfacial energy, which penalises dislocations of the smectic layers at the jump. We discuss mathematical issues related to the existence of minimisers and provide examples of minimisers in some simplified settings.

Graphical abstract

KEYWORDS:

• Smectic liquid crystals
• surface defects
• special functions of bounded variation
• BV ellipticity

Acknowledgments

We would like to thank Emmanuelle Lacaze for many very helpful discussions. We also thank Giacomo Albi and Marco Caliari from the University of Verona for their help with the numerical simulations, and the referee for their careful reading of the manuscript and comments.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Data availability statement

Data sharing is not applicable to this article as no new data were created or analysed in this study.

Notes

1. In [23], we will show that ${\phi }_{\alpha }$ is not only BV-elliptic, but also jointly convex, in the sense of [18], Definition 5.17. Joint convexity is a sufficient condition for BV-ellipticity, but it is not known whether it is a necessary condition as well.