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.

Preface

List of Symbols


Chapter I. Some Examples to Begin with


Section 1. Various Notations. Linear Equations


Section 2. Nonlinear Equations


Section 3. Nonlinear Systems


Section 4. Further Nonlinear Problems


Section 5. A Free Boundary Problem. The Plate Equation


Chapter II. Introduction


Section 6. Second Order Equations


Section 7. Higher Order Equations


Section 8. Spaces of Continuous Functions. Solution of a Differential Equation


Section 9. Boundary Conditions


Section 10. Solution of a Boundary Value Problem


Section 11. On an Integral Identity


Chapter III. The Weak Solution of a Boundary Value Problem


Section 12. The Carathéodory Property and the Nemyckii Operators


Section 13. Sobolev Spaces


Section 14. Differential Operators


Section 15. Boundary Value Problems


Section 16. Various Generalizations


Section 17. Regularity of the Weak Solution


Chapter IV. The Variational Method


Section 18. First Derivative of a Functional


Section 19. Potentials of Boundary Value Problems


Section 20. The Euler Necessary Condition


Section 21. Second Derivative of a Functional


Section 22. Lagrange Conditions


Section 23. Convex Functionals


Section 24. Weak Convergence and Weak Compactness


Section 25. Reflexive Spaces


Section 26. Existence Theorems


Section 27. Minimal Surfaces


Section 28. Excursion on Numerical Methods


Chapter V. The Topological Method


Section 29. Existence Theorems


Section 30. The Brouwer and the Leray-Schauder Degree of a Mapping


Section 31. General Boundary Conditions for Second Order Ordinary Differential Equations


Section 32. Summary of Chapters IV and V. Some Additional Remarks


Chapter VI. Noncoercive Problems


Section 33. Vanishing Nonlinearities. Regular Case


Section 34. Vanishing Nonlinearities. Singular Case


Section 35. Jumping Nonlinearities with Finite Jumps


Section 36. Jumping Nonlinearities with Infinite Jumps


Section 37. Rapid Nonlinearities


Section 38. Periodic Problems


Chapter VII. Variational Inequalities


Section 39. Formulation of the Problem


Section 40. More on the Definition of the Solution of a Variational Inequality


Section 41. Examples


Section 42. Some Special Results


Section 43. Existence Theorems


References


Index