The Neumann Problem for the Cauchy-Riemann Complex. (AM-75), Volume 75

✍ Scribed by Gerald B. Folland; Joseph John Kohn

• Publisher: Princeton University Press
• Year: 2016
• Tongue: English
• Leaves: 156
• Series: Annals of Mathematics Studies; 75
• Category: Library

✦ Synopsis

Part explanation of important recent work, and part introduction to some of the techniques of modern partial differential equations, this monograph is a self-contained exposition of the Neumann problem for the Cauchy-Riemann complex and certain of its applications. The authors prove the main existence and regularity theorems in detail, assuming only a knowledge of the basic theory of differentiable manifolds and operators on Hilbert space. They discuss applications to the theory of several complex variables, examine the associated complex on the boundary, and outline other techniques relevant to these problems. In an appendix they develop the functional analysis of differential operators in terms of Sobolev spaces, to the extent it is required for the monograph.

✦ Table of Contents

TABLE OF CONTENTS
FOREWORD
CHAPTER I: Formulation of the Problem
1. Introduction
2. Almost-complex manifolds and differential operators
3. Operators on Hilbert space
CHAPTER II: The Main Theorem
1. Statement of the theorem
2. Estimates and regularity in the interior
3. Elliptic regularization
4. Estimates at the boundary
5. Proof of the Main Theorem
CHAPTER III: Interpretation of the Main Theorem
1. Existence and regularity theorems for the ∂ complex
2. Pseudoconvexity and the basic estimate
CHAPTER IV: Applications
1. The Newlander-Nirenberg theorem
2. The Levi problem
3. Remarks on ∂ cohomology
4. Multiplier operators on holomorphic functions
CHAPTER V: The Boundary Complex
1. Duality theorems
2. The induced boundary complex
3. ∂-closed extensions of forms
4. The abstract model
CHAPTER VI: Other Methods and Results
1. The method of weight functions
2. Hölder and L^p estimates for ∂
3. Miscellaneous remarks and questions
APPENDIX: The Functional Analysis of Differential Operators
1. Sobolev norms on Euclidean space
2. Sobolev norms on manifolds
3. Tangential Sobolev norms
4. Difference operators
5. Operators constructed from Λ^s and Λt^s
REFERENCES
TERMINOLOGICAL INDEX
NOTATIONAL INDEX