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Partial Differential Equations

Dover Publications
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This is a Dover reprint of a book originally published in 1969. As evidenced by its inclusion in the MAA's list of recommendations for undergraduate libraries, it's a good book. At the time of publication, V. Komkov wrote in Mathematical Reviews:

The approach is thoroughly modern. The prerequisites are rather high: at least two semesters of functional analysis plus some prior course in elementary differential equations. On the other hand the reviewer, who has taught a course using this text, found out that students who studied this text have acquired a high level of maturity in this field.

The text blends exposition and some insight into research problems in a lucid manner. This book should be on the shelf of every mathematician working on differential equations.

The prerequisites in question are described in the preface as "the theory of Lebesgue integration" and "the very basic concepts of Banach spaces." The author points out that some familiarity with "classical methods for solving the Laplace equation and the heat equation under the usual boundary conditions" might be useful as motivation.

As always, the Dover edition is fairly priced and well produced.

Date Received: 
Tuesday, December 30, 2008
Include In BLL Rating: 
Avner Friedman
Publication Date: 
Fernando Q. Gouvêa
BLL Rating: 


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
Publish Book: 
Modify Date: 
Wednesday, May 13, 2009