Partial Differential Equations

Content: Preface
 Part I. Sobolev Spaces: 1. Notation, basic properties, distributions
 2. Geometric assumptions for the domain
 3. Definitions and density properties for the Sobolev-Slobodeckii spaces 
 4. The transformation theorem and Sobolev spaces on differentiable manifolds
 5. Definition of Sobolev spaces by the Fourier transformation and extension theorems
 6. Continuous embeddings and Sobolev\'s lemma
 7. Compact embeddings
 8. The trace operator
 9. Weak sequential compactness and approximation of derivatives by difference quotients
 Part II. Elliptic Differential Operators: 10. Linear differential operators
 11. The Lopatinskil-Sapiro condition and examples
 12. Fredholm operators
 13. The main theorem and some theorems on the index of elliptic boundary value problems
 14. Green\'s formulae
 15. The adjoint boundary value problem and the connection with the image space of the original operator
 16. Examples
 Part III. Strongly Elliptic Differential Operators and the Method of Variations: 17. Gelfand triples, the Law-Milgram, V-elliptic and V-coercive operators
 18. Agmon\'s condition
 19. Agmon\'s theorem: conditions for the V-coercion of strongly elliptic differential operators
 20. Regularity of the solutions of strongly elliptic equations
 21. The solution theorem for strongly elliptic equations and examples
 22. The Schauder fixed point theorem and a non-linear problem
 23. Elliptic boundary value problemss for unbounded regions
 Part IV. Parabolic Differential Operators: 24. The Bochner integral
 25. Distributions with values in a Hilbert space H and the space W
 26. The existence and uniqueness of the solution of a parabolic differential equation
 27. The regularity of solutions of the parabolic differential equation
 28. Examples
 Part V. Hyperbolic Differential Operators: 29. Existence and uniqueness of the solution
 30. Regularity of the solutions of the hyperbolic differential equation
 Part VI. Difference Processes for the Calculation of the Solution of the Partial Differential Equation: 32. Functional analytic concepts for difference processes
 33. Difference processes for elliptic differential equations and for the wave equation
 34. Evolution equations
 References
 Function and distribution spaces
 Index.