Abstract
Variational problems for the liquid crystal energy of mappings from three-dimensional domains into the real projective plane are studied. More generally, we study the dipole problem, the relaxed energy, and density properties concerning the conformal $p$-energy of mappings from $n$-dimensional domains that are constrained to take values into the $p$-dimensional real projective space, for any positive integer $p$. Furthermore, a notion of optimally connecting measure for the singular set of such class of maps is given.
Highlights
A liquid crystal is a state of a matter, called mesomorphic, intermediate between a crystalline solid and a normal isotropic liquid, in which long rod-shaped molecules display orientational order
The form of the energy was derived by Oseen [32] on the basis of a molecular theory, and by Frank [15] as a consequence of Galilean invariance. This means that the energy density satisfies the invariance properties
Notice that a formula similar to (0.7), and to (0.8) for n = 3, holds true if we replace Bn with any bounded domain Ω ⊂ Rn, or with e.g. Ω = Sn, the n-sphere in Rn+1. This clearly yields that the relaxed energy is a non-local functional, for n ≥ 3
Summary
We collect some well-known facts about maps taking values into real projective spaces, focusing on the case of maps into the real projective plane. 34], if A is connected, for any continuous map U : A → RPp there are exactly two continuous functions vi : A → Sp such that [vi]p = U , for i = 1, 2, with v2(x) = −v1(x) for every x ∈ A. The manifold RPp is orientable if and only if p is odd This yields that the degree of a continuous map v : Σp → Sp, where Σp is a copy of Sp, satisfies degSp (−v) = degSp (v) if p is odd, whereas degSp (−v) = − degSp (v) if p is even. Embedding of RPp. In a similar way, for every p ≥ 2 integer we can find a smooth map gp : Sp → RN(p) ,. Proposition 1.4 The following properties hold true for every p ≥ 2 : i) RPp is a smooth, compact, connected submanifold without boundary, orientable if and only if p is odd; ii).
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