Linear and quasilinear elliptic equations, Volume 46 (Mathematics in Science and Engineering)

✍ Scribed by Ladyzhenskaya (editor)

• Publisher: Academic Press
• Year: 1968
• Tongue: English
• Leaves: 515
• Category: Library

✦ Table of Contents

Front Cover
Linear and Quasilinear Elliptic Equations
Copyright Page
Contents
Preface
CHAPTER 1. INTRODUCTION
1. The Basic Notation and Terminology
2. Admissible Extensions of the Concept of Solution of Linear and Quasilinear Equations
3. The Basic Results and Their Possible Development
CHAPTER 2. AUXILIARY PROPOSITIONS
1. Some Simple Inequalities


4. Other Auxiliary Propositions
5. Bounds for max |u(x)|


8. The Classes BmN1
CHAPTER 3. LINEAR EQUATIONS

2. Schauder’s A Priori Estimate



6. The Second and Third Boundary-Value Problems
7. Interior Estimates in L2 of the Second Derivatives of an Arbitrary Function in Terms of the Values of an Elliptic Operator Applied to It
8. The Second Fundamental Inequality for Elliptic Operators


11. Other Ways of Proving the Second Fundamental Inequality




16. Diffraction Problems
17. The Case of Two Independent Variables
18. Two-Dimensional Saddle-Shaped Surfaces
CHAPTER 4. QUASILINEAR EQUATIONS WITH PRINCIPAL PART IN DIVERGENCE FORM
1. Bounded Generalized Solutions. Continuity in the Sense of Holder
2. Uniqueness in the Small


5. Generalized Second-Order Derivatives

7. Bounds for the Maximum Absolute Values of Generalized Solutions
8. Existence Theorems
CHAPTER 5. VARIATIONAL PROBLEMS
1. Statement of the Problems
2. Finding Functions That Minimize the Functional l (u)
3. Finding a Bound for the Maximum Absolute Value of Solutions of Variational Problems
4. Proof of Hölderness of Generalized Solutions
5. The Theorem on Uniqueness in the Small of Generalized Solutions
6. Further Investigation of the Differentiability Properties of Generalized Solutions
CHAPTER 6. QUASILINEAR EQUATIONS OF THE GENERAL FORM


3. Existence Theorems
4. Two-Dimensional Problems
CHAPTER 7. LINEAR SYSTEMS OF ELLIPTIC EQUATIONS





6. Differentiability Properties of Generalized Solutions
CHAPTER 8. QUASILINEAR SYSTEMS




5. Existence Theorems
CHAPTER 9. OTHER DEVICES FOR OBTAINING BOUNDS FOR THE DERIVATIVES HÖLDER CONSTANTS FOR SOLUTIONS AND THEIR DERIVATIVES
1. The Case of the Simplest Equation
2. Bounds on Hölder Constants for Solutions of Equations (Linear and Quasilinear) with Principal Part in Divergence Form
3. Bounds on the Oscillations of the Derivatives of Solutions of Equations with Principal Part in Divergence Form
4. Equations Not in Divergence Form

6. Nirenberg’s Estimate
7. Morrey’s Method for Finding a Bound for the Hölder Constant for Solutions of Two-Dimensional Variational Problems
CHAPTER 10. OTHER BOUNDARY-VALUE PROBLEMS
1. Formulation of the Problems and the General Procedure for Solving Them
2. Quasilinear Equations with Principal Part in Divergence Form
3. Quasilinear Systems
BIBLIOGRAPHY
INDEX