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.
