Skyrmions in two dimensional cholesteric liquid crystals

In this paper we consider a two dimensional cholesteric liquid crystal with an applied external field, whose field configurations u:R2→S2 are constant at infinity, and can be classified by their topological degree. We look for non trivial, topologically stable configurations (chiral skyrmions) by minimizing the Oseen-Frank energy functional over the functions u with deg(u)=-1. The energy functional depends on the splay (K1), twist (K2) and bend (K3) elastic constants, and, assuming that K1=K2, we show the existence of skyrmions provided the cholesteric pitch q0 is small enough; moreover we study the compactness of such skyrmions as q0→0, and their limit. Our results generalize some previous results in the literature obtained under the assumption that K1=K2=K3 (the well known “one-constant approximation” assumption). Moreover, in order to overcome the lack of compactness of the energy functional, we use (instead of the concentration-compactness principle) the fact that the field configurations with energy below a suitable threshold do not cover twice S2 minus a neighborhood of the North Pole, together with a truncation argument at infinity.