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.
Table of ContentsPart 1. Elliptic Equations1. Definitions2. Green's Identity3. Fundamental Solutions4. Construction of Fundamental Solutions5. Partition of Unity6. Weak and Strong Derivatives7. Strong Derivative as a Local Property8. Calculus Inequalities9. Extended Sobolev Inequalities in R(superscript n)10. Extended Sobolev Inequalities in Bounded Domains11. Imbedding Theorems12. Gärding's Inequality13. The Dirichlet Problem14. Existence Theory15–16. Regularity in the Interior17. Regularity on the Boundary18. A Priori Inequalities19. General Boundary Conditions20. ProblemsPart 2. Evolution Equations1. Strongly Continuous Semigroups2. Analytic Semigroups3. Fundamental Solutions and the Cauchy Problems4–5. Construction of Fundamental Solutions6. Uniqueness of Fundamental Solutions7. Solution of the Cauchy Problem8. Differentiability of Solutions9. The Initial-Boundary Value Problem for Parabolic Equations10. Smoothness of the Solutions of the Initial-Boundary Value Problem11. A Differentiability Theorem in Hilbert Space12. A Uniqueness Theorem in Hilbert Space13. Convergence of Solutions as t --> infinity14. Fractional Powers of Operators15. Proof of Lemma 14.516. Nonlinear Evolution Equations17. Nonlinear Parabolic Equations18. Uniqueness for Backward Equations19. Lower Bounds on Solutions as t --> infinity20. ProblemsPart 3. Selected Topics1. Analyticity of Solutions of Elliptic Equations2. Analyticity of Solutions of Evolution Equations3. Analyticity of Solutions of Parabolic Equations4. Lower Bounds for Solutions of Evolution Inequalities5. Weighted Elliptic Equations6. Asymptotic Expansions of Solutions of Evolution Equations7. Asymptotic Behavior of Solutions of Elliptic Equations8. Integral Equations in Banach Space9. Optimal Control in Banach SpaceBibliographical RemarksBibliography