Book contents
- Frontmatter
- Contents
- Preface
- Introduction
- Chapter 1 Lipschitz and Smooth Perturbed Minimization Principles
- Chapter 2 Linear and Plurisubharmonic Perturbed Minimization Principles
- Chapter 3 The Classical Min-Max Theorem
- Chapter 4 A Strong Form of the Min-Max Principle
- Chapter 5 Relaxed Boundary Conditions in the Presence of a Dual Set
- Chapter 6 The Critical Set in the Mountain Pass Theorem
- Chapter 7 Group Actions and Multiplicity of Critical Points
- Chapter 8 The Palais-Smale Condition Around a Dual Set – Examples
- Chapter 9 Morse Indices of Min-Max Critical Points – The Non Degenerate Case
- Chapter 10 Morse Indices of Min-Max Critical Points – The Degenerate Case
- Chapter 11 Morse-type Information on Palais-Smale Sequences
- Appendices by David Robinson
- A Relevant function spaces and inequalities
- B Variational formulations of some boundary value problems
- C The blowing-up of singularities
- D Elements of degree theory
- E Basic properties of martingales
- References
- Index
B - Variational formulations of some boundary value problems
Published online by Cambridge University Press: 18 December 2009
Book contents
- Frontmatter
- Contents
- Preface
- Introduction
- Chapter 1 Lipschitz and Smooth Perturbed Minimization Principles
- Chapter 2 Linear and Plurisubharmonic Perturbed Minimization Principles
- Chapter 3 The Classical Min-Max Theorem
- Chapter 4 A Strong Form of the Min-Max Principle
- Chapter 5 Relaxed Boundary Conditions in the Presence of a Dual Set
- Chapter 6 The Critical Set in the Mountain Pass Theorem
- Chapter 7 Group Actions and Multiplicity of Critical Points
- Chapter 8 The Palais-Smale Condition Around a Dual Set – Examples
- Chapter 9 Morse Indices of Min-Max Critical Points – The Non Degenerate Case
- Chapter 10 Morse Indices of Min-Max Critical Points – The Degenerate Case
- Chapter 11 Morse-type Information on Palais-Smale Sequences
- Appendices by David Robinson
- A Relevant function spaces and inequalities
- B Variational formulations of some boundary value problems
- C The blowing-up of singularities
- D Elements of degree theory
- E Basic properties of martingales
- References
- Index
Summary
Weak Solutions
In the variational approach to solving problems, we associate with the given problem a C1 functional, ϕ, on a suitable Banach space (more generally, a manifold) in such a way that the critical points of ϕ are exactly the solutions to the original problem. Before proceeding with the variational formulation of some differential equations, we need to settle on an appropriate notion of differentiability for functions defined on a Banach space.
Definition B.1. Suppose that X is a Banach space and a function ϕ : X → ℝ is given.
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- Information
- Duality and Perturbation Methods in Critical Point Theory , pp. 233 - 235Publisher: Cambridge University PressPrint publication year: 1993