This thesis is about regularity and uniqueness of minimizers of integral functionals of the form F(u) := ∫Ω F(∇u(x)) dx; where F∈C2(RNn) is a strongly quasiconvex integrand with p-growth, Ω⊆RnRn is an open bounded domain and u∈W1,pg(Ω,RN) for some boundary datum g∈C1,α(‾Ω, RN). The first contribution of this work is a full regularity result, up to the boundary, for global minimizers of F provided that the boundary condition g satisfies that ΙΙ∇gΙΙLP < ε for some ε > 0 depending only on n;N, the parameters given by the strong quasiconvexity and p-growth conditions and, most importantly, on an arbitrary but fixed constant M > 0 for which we require that ΙΙ∇gΙΙO,α < M. Furthermore, when the domain Ω is star-shaped, we extend the regularity result to the case of W1,p-local minimizers. On the other hand, for the case of global minimizers we exploit the compactness provided by the aforementioned regularity result to establish the main contribution of this thesis: we prove that, under essentially the same smallness assumptions over the boundary condition g that we mentioned above, the minimizer of F in W1,pg is unique. This result appears in contrast to the non-uniqueness examples previously given by Spadaro [Spa09], for which the boundary conditions are required to be suitably large.

published_or_final_version / abstract / Chemistry / Doctoral / Doctor of Philosophy

published_or_final_version / Chemistry / Doctoral / Doctor of Philosophy

published_or_final_version / Chemistry / Doctoral / Doctor of Philosophy

Fong Ching-yee. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2002. / Includes bibliographical references (leaves 72-76). / Abstracts in English and Chinese. / ABSTRACT (English Version) --- p.iii / ABSTRACT (Chinese Version) --- p.v / ACKNOWLEGEMENT --- p.vi / TABLE OF CONTENTS --- p.vii / LIST OF TABLES --- p.ix / LIST OF FIGURES --- p.xi / LIST OF APPENDICES --- p.xiii / Chapter CHAPTER ONE --- Introduction / Chapter 1.1 --- Importance and Recent Development in Metalloporphyrin --- p.1 / Chapter 1.2 --- Structure of Metalloporphyrin --- p.2 / Chapter 1.3 --- General Properties of Metalloporphyrin --- p.4 / Chapter 1.4 --- Linkage Patterns of Metalloporphyrin Polymers --- p.7 / Chapter 1.5 --- Reasons for Studying Co-Por-Au Polymer --- p.10 / Chapter CHAPTER TWO --- Cobalt Porphyrin Gold (Co-Por-Au) Polymer Model / Chapter 2.1 --- Synthetic Scheme --- p.13 / Chapter 2.2 --- Experimental Results and Related Properties --- p.14 / Chapter 2.3 --- The Structure of Co-Por-Au Polymer --- p.18 / Chapter CHAPTER THREE --- Structure Characterization of Co-Por-Au Polymer by DFT Method / Chapter 3.1 --- Quantum Chemical Calculations --- p.21 / Chapter 3.2 --- Density Functional Theory --- p.22 / Chapter 3.3 --- Computational Details --- p.24 / Chapter 3.4 --- Justification for using Gaussian 98 and VASP --- p.27 / Chapter 3.5 --- Results and Discussions --- p.30 / Chapter 3.5.1 --- Monomers of TPHP and TPCNP --- p.30 / Chapter 3.5.1.1 --- The Geometry of the Monomer Structures of TPhP and TPCNP --- p.30 / Chapter 3.5.1.2 --- The Axial Coordination Mode of Monomer in TPhP and TPCNP --- p.34 / Chapter 3.5.1.3 --- Comparison between Hybrid DFT and Pure DFT method --- p.38 / Chapter 3.5.1.4 --- Comparison with Other Porphyrin System --- p.40 / Chapter 3.5.1.5 --- Summary --- p.43 / Chapter 3.5.2 --- Polymers of TPhP and TPcnP --- p.44 / Chapter 3.5.2.1 --- (μ-pyrazine)(octaethylporphyrinato)iron(II) {[Fe(OEP)pyz]}n --- p.44 / Chapter 3.5.2.2 --- Energetic Comparison of TPHP and TPCNP with Different Axial Coordination Modes --- p.46 / Chapter 3.5.2.3 --- Geometry of the Repeating Units in the Polymer of TPhP and TPCNP --- p.48 / Chapter 3.5.2.4 --- Comparison with Other Porphyrin System --- p.52 / Chapter 3.5.2.5 --- The Electronic Structures of TPHP and TPCNP --- p.55 / Chapter 3.5.2.6 --- Summary --- p.58 / Chapter CHAPTER FOUR --- Conclusion --- p.59 / APPENDIX I-III --- p.61 / REFERENCES --- p.72

Ma Nap Tak. / Thesis submitted in: December 2004. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2005. / Includes bibliographical references (leaves 109-114). / Abstracts in English and Chinese. / ABSTRACT (ENGLISH) --- p.iii / ABSTRACT (CHINESE) --- p.iv / ACKNOWLEDGMENTS --- p.v / ABBREVIATION --- p.vi / TABLE OF CONTENTS --- p.vii / LIST OF FIGURES --- p.x / LIST OF TABLES AND GRAPHS --- p.xii / Chapter CHAPTER 1 --- Introduction and Background --- p.1 / Chapter 1.1 --- Introduction --- p.1 / Chapter 1.2 --- Hydrogen bonds (H-bonds) in DNA --- p.2 / Chapter 1.2.1 --- Experimental Evidences of Hydrogen Bonding --- p.3 / Chapter 1.3 --- Three-centered hydrogen bond (TCHB) --- p.4 / Chapter 1.3.1 --- Definition of Three-centered hydrogen bond (TCHB) --- p.5 / Chapter 1.3.2 --- Significance of Three-centered hydrogen bond (TCHB) --- p.6 / Chapter 1.3.3 --- Characterization of Three-centered hydrogen bond (TCHB) --- p.7 / Chapter 1.3.4 --- Classification of Three-centered hydrogen bond (TCHB) --- p.7 / Chapter 1.4 --- Charge transfer in DNA --- p.9 / Chapter 1.4.1 --- Theory of DNA charge transfer --- p.9 / Chapter 1.4.2 --- Short and long range hole transfer in DNA --- p.10 / Chapter 1.4.3 --- Electron transfer in DNA --- p.12 / Chapter 1.4.4 --- Summary of DNA charge transfer --- p.12 / Chapter 1.5 --- Thesis Scope --- p.13 / Chapter CHAPTER 2 --- Theory and methodology --- p.16 / Chapter 2.1 --- Introduction --- p.16 / Chapter 2.2 --- Theory --- p.17 / Chapter 2.2.1 --- Density Functional Theory (DFT) --- p.17 / Chapter 2.2.2 --- Basis set selection --- p.18 / Chapter 2.2.3 --- Natural Bond Orbital (NBO) --- p.19 / Chapter 2.2.3.1 --- Natural Population Analysis (NPA) --- p.20 / Chapter 2.2.3.2 --- E(2) --- p.20 / Chapter 2.2.3.2 --- bond index --- p.21 / Chapter 2.2.4 --- spin-spin coupling constants --- p.22 / Chapter 2.2.5 --- Molecular Orbital (MO) --- p.23 / Chapter 2.3 --- Methodology --- p.24 / Chapter 2.3.1 --- Test calculation for TCHBs by NBO --- p.24 / Chapter 2.3.2 --- Geometry Optimization --- p.24 / Chapter 2.3.3 --- NBO analysis --- p.25 / Chapter 2.3.4 --- J-coupling constants (lhJnx) and MO calculations --- p.26 / Chapter 2.4 --- Summary --- p.26 / Chapter CHAPTER 3 --- Results and Discussion --- Hydrogen bonding in DNA --- p.27 / Chapter 3.1 --- Introduction --- p.27 / Chapter 3.2 --- Method for extracting DNA dimer models --- p.27 / Chapter 3.3 --- Computed results of Inter H-bonds of the ten dimer models --- p.33 / Chapter 3.3.1 --- Geometric parameters --- p.33 / Chapter 3.3.2 --- Natural Bond Orbital (NBO) Analysis --- p.36 / Chapter 3.3.2.1 --- E(2) and Wiberg Bond index --- p.36 / Chapter 3.3.2.2 --- The relationship of E(2) and bond distance --- p.40 / Chapter 3.3.2.3 --- The relationship of E(2) and bond angle --- p.42 / Chapter 3.3.2.4 --- The relationship of E(2) and bond index --- p.44 / Chapter 3.3.3 --- Spin-Spin Coupling Constants of inter-H bonds --- p.46 / Chapter 3.3.3.1 --- The relationship of spin-spin coupling constant and distance --- p.49 / Chapter 3.3.3.2 --- The relationship of spin-spin coupling constant and bond angle --- p.50 / Chapter 3.3.3.3 --- The relationship of spin-spin coupling constant and E(2) energy --- p.52 / Chapter 3.4 --- Experimental Characterization of Three-centered H-bonds --- p.54 / Chapter 3.5 --- Theoretical Characterization of Three-centered H-bonds --- p.55 / Chapter 3.5.1 --- Geometry properties (360°C Rule) --- p.55 / Chapter 3.5.2 --- NMR properties (Spin-Spin Coupling Constants) --- p.55 / Chapter 3.5.3 --- NBO properties (E(2) and Wiberg bond index) --- p.56 / Chapter 3.6 --- Computed results of Three-centered hydrogen bonds (TCHBs) of the ten dimer models --- p.56 / Chapter 3.6.1 --- Natural Bond Orbital (NBO) Analysis --- p.56 / Chapter 3.6.1.1 --- Determination of TCHBs in the ten dimer models --- p.56 / Chapter 3.6.1.2 --- Analysis of TCHB interactions (E(2) and bond index) --- p.62 / Chapter 3.6.1.3 --- The relationship between E(2) and bond distance of TCHBs --- p.63 / Chapter 3.6.1.4 --- The relationship between E(2) and bond angle of TCHBs --- p.65 / Chapter 3.6.2 --- Spin-Spin Coupling Constants of TCHBs --- p.66 / Chapter 3.6.2.1 --- The relationship between spin-spin coupling constant and bond distance of TCHBs --- p.68 / Chapter 3.6.2.2 --- The relationship between spin-spin coupling constant and E(2) energy of TCHBs --- p.69 / Chapter 3.6.3 --- Geometry of TCHBs --- p.71 / Chapter 3.7 --- Summary --- p.72 / Chapter CHAPTER 4 --- Results and Discussion --- Charge location and charge transfer in DNA --- p.77 / Chapter 4.1 --- Introduction --- p.77 / Chapter 4.2 --- Method --- p.78 / Chapter 4.3 --- Computed results of the charge location of the trimer models --- p.81 / Chapter 4.3.1 --- Location of excess positive charge --- p.81 / Chapter 4.3.2 --- Location of excess negative charge --- p.90 / Chapter 4.4 --- Role of TCHBs in charge transfer --- p.95 / Chapter 4.4.1 --- Introduction --- p.95 / Chapter 4.4.2 --- "Analysis of G1G2C3, C3A4A5 and A8A9C10 trimers" --- p.96 / Chapter 4.4.3 --- Analysis of A7A8A9 and A8A9C10 trimers --- p.103 / Chapter 4.5 --- Summary --- p.105 / Chapter CHAPTER 5 --- Concluding Remarks --- p.107 / REFERENCES / APPENDIX

by Lo Yiu Ming. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1998. / Includes bibliographical references (leaves 89-90). / Abstract also in Chinese. / Abstract --- p.iii / Acknowledgements --- p.iv / Chapter Chapter 1. --- Introduction --- p.1 / Chapter 1.1. --- Main Idea --- p.5 / Chapter 1.2. --- Organization --- p.8 / Chapter Chapter 2. --- Geometric and Analytic Preliminaries --- p.9 / Chapter 2.1. --- A Review on Measure Theory --- p.9 / Chapter 2.2. --- Submanifolds in Rn --- p.11 / Chapter 2.3. --- Several Results from PDEs --- p.17 / Chapter 2.4. --- Biharmonic Comparison Lemma --- p.20 / Chapter Chapter 3. --- Approximate Graphical Decomposition --- p.24 / Chapter 3.1. --- Some Preliminaries --- p.24 / Chapter 3.2. --- Approximate Graphical Decomposition --- p.30 / Chapter Chapter 4. --- Existence & Regularity of Measure-theoretic Limits of Minimizing Sequence --- p.41 / Chapter 4.1. --- Willmore Functional and Area --- p.41 / Chapter 4.2. --- Existence of Measure-theoretic Limit of Minimizing Sequence --- p.45 / Chapter 4.3. --- Higher Regularity at Good Points --- p.54 / Chapter 4.4. --- Convergence in Hausdorff Distance Sense --- p.62 / Chapter 4.5. --- Regularity near Bad Points --- p.64 / Chapter Chapter 5. --- Existence of Genus 1 Minimizers in Rn --- p.83 / References --- p.89

Thesis (Ph. D.)--University of Missouri-Columbia, 2007. / The entire dissertation/thesis text is included in the research.pdf file; the official abstract appears in the short.pdf file (which also appears in the research.pdf); a non-technical general description, or public abstract, appears in the public.pdf file. Title from title screen of research.pdf file (viewed on December 17, 2007) Vita. Includes bibliographical references.

In this thesis, we shall study disjointness preserving linear functionals of the Wiener ring. It is clear that Wiener ring is a dense subalgebra of C(T)in the usual supremum norm .However, Wiener ring is also isomorphic to L1(Z). So it has an 1 norm . By studying the structure of ideals of the Wiener ring, we discover that disjointness preserving linear functionals are the same under different norms. Bounded disjointness preserving linear functionals of the Wiener ring is a multiple of the point mass in both cases. Finally, we establish the existence of unbounded disjointness preserving linear functionals of the Wiener ring.

Thesis (Ph. D.)--University of Washington, 2002. / Vita. Includes bibliographical references (p. 242-256).