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 |