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 $$u:\mathbb {R}^2\rightarrow S^2$$ u : R 2 → S 2 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 $$\deg (u)=-1$$ deg ( u ) = - 1 . The energy functional depends on the splay ( $$K_1$$ K 1 ), twist ( $$K_2$$ K 2 ) and bend ( $$K_3$$ K 3 ) elastic constants, and, assuming that $$K_1=K_2$$ K 1 = K 2 , we show the existence of skyrmions provided the cholesteric pitch $$q_0$$ q 0 is small enough; moreover we study the compactness of such skyrmions as $$q_0\rightarrow 0$$ q 0 → 0 , and their limit. Our results generalize some previous results in the literature obtained under the assumption that $$K_1=K_2=K_3$$ K 1 = K 2 = K 3 (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 $$S^2$$ S 2 minus a neighborhood of the North Pole, together with a truncation argument at infinity.