Partial Differential Equations

Description

This book is a rigorous introduction to the abstract theory of partial differential equations. The main prerequisite is familiarity with basic functional analysis: more advanced topics such as Fredholm operators, the Schauder fixed point theorem and Bochner integrals are introduced when needed, and the book begins by introducing the necessary material from the theory of distributions and Sobolev spaces. Using such techniques, the author presents different methods available for solving elliptic, parabolic and hyperbolic equations. He also considers the difference process for the practical solution of a partial differential equation, emphasising that it is possible to solve them numerically by simple methods. Many examples and exercises are provided throughout, and care is taken to explain difficult points. Advanced undergraduates and graduate students will appreciate this self-contained and practical introduction 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.