Fucik / Kufner Nonlinear Differential Equations

1. Auflage 2014
ISBN: 978-1-4832-7837-7
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band Volume 2, 360 Seiten, Web PDF

Reihe: Studies in Applied Mechanics

Studies in Applied Mathematics, 2: Nonlinear Differential Equations focuses on modern methods of solutions to boundary value problems in linear partial differential equations. The book first tackles linear and nonlinear equations, free boundary problem, second order equations, higher order equations, boundary conditions, and spaces of continuous functions. The text then examines the weak solution of a boundary value problem and variational and topological methods. Discussions focus on general boundary conditions for second order ordinary differential equations, minimal surfaces, existence theorems, potentials of boundary value problems, second derivative of a functional, convex functionals, Lagrange conditions, differential operators, Sobolev spaces, and boundary value problems. The manuscript examines noncoercive problems and vibrational inequalities. Topics include existence theorems, formulation of the problem, vanishing nonlinearities, jumping nonlinearities with finite jumps, rapid nonlinearities, and periodic problems. The text is highly recommended for mathematicians and engineers interested in nonlinear differential equations.

Autoren/Hrsg.

Fucik, Svatopluk

Kufner, Alois

Inhaltsverzeichnis

1;Front Cover;1
2;Nonlinear Differential Equations;4
3;Copyright Page;5
4;Table of Contents;6
5;PREFACE;8
6;LIST OF SYMBOLS;12
7;CHAPTER I. SOME EXAMPLES TO BEGIN WITH;16
7.1;Section 1. Various Notations. Linear Equations;16
7.2;Section 2. Nonlinear Equations;20
7.3;Section 3. Nonlinear Systems;25
7.4;Section 4. Further Nonlinear Problems;28
7.5;Section 5. A Free Boundary Problem. The Plate Equation;31
8;CHAPTER II. INTRODUCTION;34
8.1;Section 6. Second Order Equations;34
8.2;Section 7. Higher Order Equations;39
8.3;Section 8. Spaces of Continuous Functions. Solution of a Differential Equation;44
8.4;Section 9. Boundary Conditions;47
8.5;Section 10. Solution of a Boundary Value Problem;61
8.6;Section 11. On an Integral Identity;70
9;CHAPTER III. THE WEAK SOLUTION OF A BOUNDARY VALUE PROBLEM;74
9.1;Section 12. The Carathéodory Property and the N.myckiéi Operators;74
9.2;Section 13. Sobolev Spaces;86
9.3;Section 14. Differential Operators;95
9.4;Section 15. Boundary Value Problems;102
9.5;Section 16. Various Generalizations;132
9.6;Section 17. Regularity of the Weak Solution;161
10;CHAPTER IV. THE VARIATIONAL METHOD;170
10.1;Section 18. First Derivative of a Functional;170
10.2;Section 19. Potentials of Boundary Value Problems;176
10.3;Section 20. The Euler Necessary Condition;180
10.4;Section 21. Second Derivative of a Functional;183
10.5;Section 22. Lagrange Conditions;185
10.6;Section 23. Convex Functionals;188
10.7;Section 24. Weak Convergence and Weak Compactness;192
10.8;Section 25. Reflexive Spaces;196
10.9;Section 26. Existence Theorems;199
10.10;Section 27. Minimal Surfaces;219
10.11;Section 28. Excursion on Numerical Methods;226
11;CHAPTER V. THE TOPOLOGICAL METHOD;240
11.1;Section 29. Existence Theorems;240
11.2;Section 30. The Brouwer and the Leray-Schauder Degree of a Mapping;249
11.3;Section 31. General Boundary Conditions for Second Order Ordinary Differential Equations;263
11.4;Section 32. Summary of Chapters IV and V. Some Additional Remarks;268
12;CHAPTER VI. NONCOERCIVE PROBLEMS;275
12.1;Section 33. Vanishing Nonlinearities. Regular Case;275
12.2;Section 34. Vanishing Nonlinearities. Singular Case;282
12.3;Section 35. Jumping Nonlinearities with Finite Jumps;291
12.4;Section 36. Jumping Nonlinearities with Infinite Jumps;300
12.5;Section 37. Rapid Nonlinearities;308
12.6;Section 38. Periodic Problems;310
13;CHAPTER VII. VARIATIONAL INEQUALITIES;313
13.1;Section 39. Formulation of the Problem;313
13.2;Section 40. More on the Definition of the Solution of a Variational Inequality;319
13.3;Section 41. Examples;325
13.4;Section 42. Some Special Results;339
13.5;Section 43. Existence Theorems;344
14;REFERENCES;354
15;INDEX;358