Research.Pdf (526.4Kb)
NONLINEAR EQUATIONS WITH NATURAL GROWTH TERMS A Thesis presented to the Faculty of the Graduate School at the University of Missouri In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy by BENJAMIN JAMES JAYE Professor Igor E. Verbitsky, Thesis Supervisor MAY 2011 The undersigned, appointed by the Dean of the Graduate School, have examined the dissertation entitled: NONLINEAR EQUATIONS WITH NATURAL GROWTH TERMS presented by Benjamin James Jaye, a candidate for the degree of Doctor of Philosophy and hereby certify that, in their opinion, it is worthy of acceptance. Professor Igor E. Verbitsky Professor Steven Hofmann Professor Stephen Montgomery-Smith Professor David Retzloff ACKNOWLEDGMENTS First and foremost, I would like to express my gratitude to my Ph.D. supervisor, Professor Igor E. Verbitsky, for his support and for the generosity with which he shares his ideas. I would also like to thank the mathematicians who I have learnt from during my graduate studies, in particular I am grateful to Professors J. Bennett, S. Hofmann, D. Hundertmark, N. Kalton, J. Kristensen, J. L. Lewis, V. G. Maz'ya, G. Mingione, M. Mitrea, S. Montgomery-Smith, F. Nazarov, N. C. Phuc, and M. Rudelson. Finally, I am also grateful to the mathematics departments at the University of Missouri, and also the University of Birmingham in the United Kingdom. ii TABLE OF CONTENTS ACKNOWLEDGMENTS . ii ABSTRACT . v CHAPTER . 1 Introduction . 1 2 The homogeneous problem . 14 2.1 Preliminaries . 19 2.2 Proof of the main result . 23 2.3 A remark on higher integrability . 42 2.4 The proof of Theorem 2.0.4 .
[Show full text]