Partial Differential Equations

by Avner Friedman

Overview

This three-part treatment of partial differential equations focuses on elliptic and evolution equations. Largely self-contained, it concludes with a series of independent topics directly related to the methods and results of the preceding sections that helps introduce readers to advanced topics for further study. Geared toward graduate and postgraduate students of mathematics, this volume also constitutes a valuable reference for mathematicians and mathematical theorists.
Starting with the theory of elliptic equations and the solution of the Dirichlet problem, the text develops the theory of weak derivatives, proves various inequalities and imbedding problems, and derives smoothness theorems. Part Two concerns evolution equations in Banach space and develops the theory of semigroups. It solves the initial-boundary value problem for parabolic equations and covers backward uniqueness, asymptotic behavior, and lower bounds at infinity. The final section includes independent topics directly related to the methods and results of the previous material, including the analyticity of solutions of elliptic and parabolic equations, asymptotic behavior of solutions of elliptic equations near infinity, and problems in the theory of control in Banach space.

Product Details

ISBN-13:	9780486469195
Publisher:	Dover Publications
Publication date:	11/24/2008
Series:	Dover Books on Mathematics
Pages:	272
Product dimensions:	6.00(w) x 9.10(h) x 0.60(d)

About the Author

Avner Friedman taught mathematics at Indiana University, the University of California at Berkeley, the University of Minnesota, Stanford, and Northwestern University. He received Sloan and Guggenheim Fellowships and is the author of six other Dover books.

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