Partial Differential Equations

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