On the existence of skyrmions in planar liquid crystals

Abstract

The study of topologically nontrivial field configurations is an important topic in many branches of physics and applied sciences. In this paper we are interested to the existence of such structures, the so-called skyrmions, in the context of liquid crystals. More precisely, we consider a two-dimensional nematic or cholesteric liquid crystal. In the nematic case we use a Bogomol'nyi type decomposition in order to get a topological lower bound on the configurations with a given degree for the full Oseen-Frank energy functional, and so we can find a global minimum of degree $\pm 1$ for the energy. Then we consider the cholesteric case in presence of an electric field under the one constant approximation assumption, and, by using the concentration-compactness method, we prove the existence of a minimum again on the configurations of degree $\pm 1$, for sufficiently large electric fields.