Title:

Calculus of variations and its application to liquid crystals

The thesis concerns the mathematical study of the calculus of variations and its application to liquid crystals. In the first chapter we examine vectorial problems in the calculus of variations with an additional pointwise constraint so that any admissible function n ε W^{1,1}(ΩM), and M is a manifold of suitable regularity. We formulate necessary and sufficient conditions for any given state n to be a strong or weak local minimiser of I. This is achieved using a nearest point projection mapping in order to use the more classical results which apply in the absence of a constraint. In the subsequent chapters we study various static continuum theories of liquid crystals. More specifically we look to explain a particular cholesteric fingerprint pattern observed by HP Labs. We begin in Chapter 2 by focusing on a specific cholesteric liquid crystal problem using the theory originally derived by Oseen and Frank. We find the global minimisers for general elastic constants amongst admissible functions which only depend on a single variable. Using the oneconstant approximation for the OseenFrank free energy, we then show that these states are global minimisers of the threedimensional problem if the pitch of the cholesteric liquid crystal is sufficiently long. Chapter 3 concerns the application of the results from the first chapter to the situations investigated in the second. The local stability of the onedimensional states are quantified, analytically and numerically, and in doing so we unearth potential shortcomings of the classical OseenFrank theory. In Chapter 4, we ascertain some equivalence results between the continuum theories of Oseen and Frank, Ericksen, and Landau and de Gennes. We do so by proving lifting results, building on the work of Ball and Zarnescu, which relate the regularity of line and vector fields. The results prove to be interesting as they show that for a director theory to respect the head to tail symmetry of the liquid crystal molecules, the appropriate function space for the director field is S BV^{2} (Ω,S^{2,/sup>). We take this idea and in the final chapter we propose a mathematical model of liquid crystals based upon the OseenFrank free energy but using special functions of bounded variation. We establish the existence of a minimiser, forms of the EulerLagrange equation, and find solutions of the EulerLagrange equation in some simple cases. Finally we use our proposed model to reexamine the same problems from Chapter 2. By doing so we extend the analysis we were able to achieve using Sobolev spaces and predict the existence of multidimensional minimisers consistent with the known experimental properties of highchirality cholesteric liquid crystals.
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