Partial Differential Equations

Partial Differential Equations

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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.
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Table of contents

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.
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