Nonlinear Elliptic Equations of the Second Order (Graduate Studies in Mathematics) (Graduate Studies in Mathematics, 171)

Nonlinear elliptic differential equations are a diverse subject with important applications to the physical and social sciences and engineering. They also arise naturally in geometry. In particular, much of the progress in the area in the twentieth century was driven by geometric applications, from the Bernstein problem to the existence of Kähler–Einstein metrics. This book, designed as a textbook, provides a detailed discussion of the Dirichlet problems for quasilinear and fully nonlinear elliptic differential equations of the second order with an emphasis on mean curvature equations and on Monge–Ampère equations. It gives a user-friendly introduction to the theory of nonlinear elliptic equations with special attention given to basic results and the most important techniques. Rather than presenting the topics in their full generality, the book aims at providing self-contained, clear, and “elementary” proofs for results in important special cases. This book will serve as a valuable resource for graduate students or anyone interested in this subject. Nonlinear elliptic differential equations are a diverse subject with important applications to the physical and social sciences and engineering. They also arise naturally in geometry. In particular, much of the progress in the area in the twentieth century was driven by geometric applications, from the Bernstein problem to the existence of Kahler-Einstein metrics. This book, designed as a textbook, provides a detailed discussion of the Dirichlet problems for quasilinear and fully nonlinear elliptic differential equations of the second order with an emphasis on mean curvature equations and on Monge-Ampere equations. It gives a user-friendly introduction to the theory of nonlinear elliptic equations with special attention given to basic results and the most important techniques. Rather than presenting the topics in their full generality, the book aims at providing self-contained, clear, and "elementary" proofs for results in important special cases. This book will serve as a valuable resource for graduate students or anyone interested in this subject. Cover Title page Dedication Contents Preface Introduction Chapter 1. Linear elliptic equations Part 1. Quasilinear elliptic equations Chapter 2. Quasilinear uniformly elliptic equations Chapter 3. Mean curvature equations Chapter 4. Minimal surface equations Part 2. Fully nonlinear elliptic equations Chapter 5. Fully nonlinear uniformly elliptic equations Chapter 6. Monge-Ampère equations Chapter 7. Complex Monge-Ampère equations Chapter 8. Generalized solutions of Monge-Ampère equations Bibliography Index Back Cover Qing Han. Includes Bibliographical References And Index.