Thesis (Ph. D.)--University of Texas at Austin, 2000. / Vita. Includes bibliographical references (leaves 74-76). Available also in a digital version from Dissertation Abstracts.

Ip Tsz Ho. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2002. / Includes bibliographical references (leaves 86-89). / Chapter 0 --- Introduction --- p.5 / Chapter 1 --- Preliminary --- p.11 / Chapter 1.1 --- Background in analysis --- p.11 / Chapter 1.1.1 --- Holder Continuity --- p.11 / Chapter 1.1.2 --- Hausdorff Measure --- p.12 / Chapter 1.1.3 --- Weak Derivatives --- p.13 / Chapter 1.2 --- Basic Facts of Harmonic Functions --- p.14 / Chapter 1.2.1 --- Harmonic Approximation --- p.14 / Chapter 1.2.2 --- Elliptic Regularity --- p.15 / Chapter 1.3 --- Background in geometry --- p.16 / Chapter 1.3.1 --- Notations and Symbols --- p.16 / Chapter 1.3.2 --- Nearest Point Projection --- p.16 / Chapter 2 --- Monotonicity formula and Regularity of Harmonic maps --- p.17 / Chapter 2.1 --- Energy Minimizing Maps --- p.17 / Chapter 2.2 --- Variational Equations --- p.18 / Chapter 2.3 --- Monotonicity Formula --- p.21 / Chapter 2.4 --- A Technical Lemma --- p.22 / Chapter 2.5 --- Luckhau's Lemma --- p.28 / Chapter 2.6 --- Reverse Poincare Inequality --- p.40 / Chapter 2.7 --- ε-Regularity of Energy Minimizing Maps --- p.45 / Chapter 3 --- Monotonicity Formulae and Vanishing Theorems --- p.52 / Chapter 3.1 --- Stress energy tensor and basic formulae for harmonic p´ؤforms --- p.52 / Chapter 3.2 --- Monotonicity formula --- p.59 / Chapter 3.2.1 --- Monotonicity Formula for Harmonic Maps --- p.64 / Chapter 3.2.2 --- Bochner-Weitzenbock Formula --- p.65 / Chapter 3.3 --- Conservation Law and Vanishing Theorem --- p.68 / Chapter 4 --- On conformally compact Einstein Manifolds --- p.71 / Chapter 4.1 --- Energy Decay of Harmonic Maps with Finite Total Energy --- p.73 / Chapter 4.2 --- Vanishing Theorem of Harmonic Maps --- p.81 / Bibliography --- p.86

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Thesis (Ph. D.)--University of Texas at Austin, 1998. / Vita. Includes bibliographical references (leaves 77-80). Available also in a digital version from Dissertation Abstracts.

Let f_1, f_2 be univalent harmonic mappings of some planar domain D into the complex plane C. This thesis contains results concerning conditions under which the convolution f_1 ∗ f_2 or the convex combination tf_1 + (1 − t)f_2 is univalent. This is a long-standing problem, and I provide several partial solutions. I also include applications to minimal surfaces.

This manuscript demonstrates the well-posedness (existence, uniqueness, and regularity of solutions) of the Cauchy problem for simplified equations of nematic liquid crystal hydrodynamic flow in three dimensions for initial data that is uniformly locally L3(R3) integrable (L3U(R3)). The equations examined are a simplified version of the equations derived by Ericksen and Leslie. Background on the continuum theory of nematic liquid crystals and their flow is provided as are explanations of the related mathematical literature for nematic liquid crystals and the Navier–Stokes equations.

This manuscript demonstrates the regularity and uniqueness of some geometric heat flows with critical nonlinearity. First, under the assumption of smallness of renormalized energy, several issues of the regularity and uniqueness of heat flow of harmonic maps into a unit sphere or a compact Riemannian homogeneous manifold without boundary are established. For a class of heat flow of harmonic maps to any compact Riemannian manifold without boundary, satisfying the Serrin's condition, the regularity and uniqueness is also established. As an application, the hydrodynamic flow of nematic liquid crystals in Serrin's class is proved to be regular and unique. The natural extension of all the results to the heat flow of biharmonic maps is also presented in this manuscript.

This thesis concerns the regularity of holonomic minimisers of variational integrals in the context of direct methods in the calculus of variations. Specifically, we consider Sobolev mappings from a bounded domain into a connected compact Riemannian manifold without boundary, to which such mappings are said to be holonomically constrained. For a general class of strictly quasiconvex integral functionals, we give a direct proof of local C^1,α-Hölder continuity, for some 0 < α < 1, of holonomic minimisers off a relatively closed 'singular set' of Lebesgue measure zero. Crucially, the proof constructs comparison maps using the universal covering of the target manifold, the lifting of Sobolev mappings to the covering space and the connectedness of the covering space. A certain tangential A-harmonic approximation lemma obtained directly using a Lipschitz approximation argument is also given. In the context of holonomic minimisers of regular variational integrals, we also provide bounds on the Hausdorff dimension of the singular set by generalising a variational difference quotient method to the holonomically constrained case with critical growth. The results are analogous to energy-minimising harmonic maps into compact manifolds, however in this case the proof does not use a monotonicity formula. We discuss several applications to variational problems in condensed matter physics, in particular those concerning the superfluidity of liquid helium-3 and nematic liquid crystals. In these problems, the class of mappings are constrained to an orbit of 'broken symmetries' or 'manifold of internal states', which correspond to a sub-group of residual symmetries.

In this paper we prove structure results for the singular sets of stationary harmonic maps and mean curvature flows local to particular singularities. The original work is contained in Chapter 5 and Chapter 8. Chapters 1-5 are concerned with energy minimising maps and stationary harmonic maps. Chapters 6-8 are concerned with mean curvature flows and Brakke flows. In the case of stationary harmonic maps we consider a singularity at which the spine dimension is maximal, and such that the weak tangent map is homotopically non-trivial, and has minimal density amongst singularities of maximal spine dimen- sion. Local to such a singularity we show the singular set is a bi-Hölder continuous homeomorphism of the unit disk of dimension equal to the maximal spine dimension. A weak tangent map is translation invariant along a subspace, and invariant under dilations, so it completely defined by its values on a sphere. Such a map is said to be homotopically non-trivial if the mapping of a sphere into some target manifold cannot be deformed by a homotopy to a constant map. For an n-dimensional mean curvature flow we consider a singularity at which we can find a shrinking cylinder as a tangent flow, that collapses on an (n−1)-dimensional plane. Local to such a singularity we show that all singularities have such a cylindrical tangent, or else have lower Gaussian density than that of the shrinking cylinder. The subset of cylindrical singularities can be shown to be contained in a finite union of parabolic (n − 1)-dimensional Lipschitz submanifolds. In the case that the mean curvature flow arises from elliptic regularisation we can show that all singularities local to a cylindrical singularity with (n − 1)-dimensional spine are either cylindrical singularities with (n − 1)-dimensional spine, or contained in a parabolic Hausdorff (n − 2)-dimensional set.