Partial Differential Equations

by Friedman, Avner

• ISBN13: 9780486469195
• ISBN10: 0486469190
• Format: Paperback
• Copyright: 2008-11-24
• Publisher: Dover Publications

Summary

This three-part treatment focuses on elliptic and evolution equations. Largely self-contained, this volume concludes with a series of independent topics related to the methods and results of preceding sections and introduces advanced topics for further study. Geared toward graduate and postgraduate students, it also constitutes a valuable reference for professionals. 1969 edition.

Table of Contents

Part 1. Elliptic Equations

1. Definitions

2. Green's Identity

3. Fundamental Solutions

4. Construction of Fundamental Solutions

5. Partition of Unity

6. Weak and Strong Derivatives

7. Strong Derivative as a Local Property

8. Calculus Inequalities

9. Extended Sobolev Inequalities in R(superscript n)

10. Extended Sobolev Inequalities in Bounded Domains

11. Imbedding Theorems

12. Gärding's Inequality

13. The Dirichlet Problem

14. Existence Theory

15–16. Regularity in the Interior

17. Regularity on the Boundary

18. A Priori Inequalities

19. General Boundary Conditions

20. Problems

Part 2. Evolution Equations

1. Strongly Continuous Semigroups

2. Analytic Semigroups

3. Fundamental Solutions and the Cauchy Problems

4–5. Construction of Fundamental Solutions

6. Uniqueness of Fundamental Solutions

7. Solution of the Cauchy Problem

8. Differentiability of Solutions

9. The Initial-Boundary Value Problem for Parabolic Equations

10. Smoothness of the Solutions of the Initial-Boundary Value Problem

11. A Differentiability Theorem in Hilbert Space

12. A Uniqueness Theorem in Hilbert Space

13. Convergence of Solutions as t --> infinity

14. Fractional Powers of Operators

15. Proof of Lemma 14.5

16. Nonlinear Evolution Equations

17. Nonlinear Parabolic Equations

18. Uniqueness for Backward Equations

19. Lower Bounds on Solutions as t --> infinity

20. Problems

Part 3. Selected Topics

1. Analyticity of Solutions of Elliptic Equations

2. Analyticity of Solutions of Evolution Equations

3. Analyticity of Solutions of Parabolic Equations

4. Lower Bounds for Solutions of Evolution Inequalities

5. Weighted Elliptic Equations

6. Asymptotic Expansions of Solutions of Evolution Equations

7. Asymptotic Behavior of Solutions of Elliptic Equations

8. Integral Equations in Banach Space

9. Optimal Control in Banach Space

Bibliographical Remarks

Bibliography