Nonlinear Elliptic Equations of the Second Order
About this Title
Qing Han, University of Notre Dame, Notre Dame, IN
Publication: Graduate Studies in Mathematics
Publication Year: 2016; Volume 171
ISBNs: 978-1-4704-2607-1 (print); 978-1-4704-2906-5 (online)
MathSciNet review: MR3468839
MSC: Primary 35-01; Secondary 32W20
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.
Graduate students and research mathematicians interested in nonlinear PDE and applications to differential geometry.
Table of Contents
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