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Partial Differential Equations [NOOK Book]

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.
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Partial Differential Equations

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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.
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Product Details

  • ISBN-13: 9780486151595
  • Publisher: Dover Publications
  • Publication date: 11/30/2011
  • Series: Dover Books on Mathematics
  • Sold by: Barnes & Noble
  • Format: eBook
  • Pages: 272
  • File size: 14 MB
  • Note: This product may take a few minutes to download.

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
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