On Some Spectral Problems and Asymptotic Limits Occuring in the Analysis of Liquid Crystals

Abstract

On the basis of de Gennes‘ theory of analogy between liquid crystals and superconductivity, the second author introduced the critical wave number Q_c3 of liquid crystals, which is an analog of the upper critical field H_c3 for superconductors, and he predicted the existence of a surface smectic state, which was supposed to be an analog of the surface superconducting state. In this article we study this problem and our study relies on the Landau-de Gennes functional of liquid crystals in connection with a simpler functional called the reduced Ginzburg-Landau functional which appears to be relevant when some of the elastic constants are large. We discuss the behavior of the minimizers of these functionals. We describe briefly some results obtained by Bauman-Carme Calderer-Liu-Phillips, and present more recent results on the reduced Ginzburg-Landau functional obtained by the authors. This paper is partially extracted of lectures given by the first author in Recife and Serrambi in August 2008.