The Oseen-Frank theory of liquid crystals - Liu, L (2019)
This thesis concerns the minimization of the Oseen-Frank bulk free energy. The structure is as following: in Chapter 1, we will give a brief introduction to the Oseen-Frank theory and the Landau-de Gennes theory. Also we will introduce some established results related to the two theories. In Chapter 2, we define first in Section 2.1 the degree for a vector field n ∈ H1/2(𝕊1; 𝕊1); and then in Section 2.2 the degree for an H1/2 vector field from a Lipschitz boundary ∂w to 𝕊1: In Section 2.3, we prove the fact that vector fields subject to some given boundary conditions and degree constraints in a given exterior N-connected domain can be written explicitly, and the result is stated in Proposition 2.3.6. We began in Chapter 3 by focusing on the existence and uniqueness of minimizers for a modified one-constant Oseen-Frank energy subject to some prescribed boundary conditions in a 2D circular domain Ω = {x ∈ ℝ2 | 0 < a < |x| < ∞} and derive some ‘nice’ asymptotic behaviour at ∞ of the minimizer. In Chapter 4, we make the problem studied in Chapter 3 more complicated by adding more ‘holes’ in the domain. Then by introducing the homotopy classes for vector fields subject to prescribed boundary conditions, we prove that there exists a unique minimizer for a modified oneconstant Oseen-Frank energy in each homotopy class and we still have the ‘nice’ asymptotic behaviour of the minimizer in each homotopy class. Also, there exists a minimizer in the union of these homotopy classes, although this minimizer may not be unique. Then in Chapter 5, we work with line fields on the boundary in the given exterior N-connected Lipschitz domain. By introducing an auxiliary vector field and modifying the definition of homotopy classes in Chapter 4, we prove the uniqueness of the minimizing line field in each homotopy class. Also the result proved in Chapter 5 applies to both orientable and non-orientable line fields on the boundaries, and when the given boundary line field is orientable, it is equivalent to our result proved in Chapter 4. Finally, in Chapter 6, we study the 2D non-equal constant case (i.e. without assuming k1 = k2 = k3, and k4 = 0) in a one-circular domain. In particular, in Section 6.1 we assume the vector fields are radius-independent and derive the minimizers to the Oseen-Frank bulk free energy subject to prescribed degree constraint on the circular boundary. Then in Section 6.2 we show that the result proved in Section 6.1 will also hold when one of the Frank elastic constant is zero (degenerate case), but will be in a different function space W1,1(Ω;𝕊1). At last in Section 6.3 we will show that the minimizers derived in Section 6.1 and Section 6.2 are in general not minimizers for the radius-dependent case.