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
- References
- Index
Chapter 2 - Linear and Plurisubharmonic Perturbed Minimization Principles
Published online by Cambridge University Press: 18 December 2009
- 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
- References
- Index
Summary
In this chapter, we present two results of Ghoussoub and Maurey: the first is concerned with linear perturbations and is applicable in reflexive Banach spaces. It will be used in Theorems 2.12 and 2.13 below to get generic minimization results in the case of critical exponents and non-zero data. The second one deals with the possibility of finding plurisubharmonic perturbations and can be applied to spaces as “bad” as L1. The proofs essentially boil down to showing that these new classes of functions are, in the terminology of Chapter 1, admissible cones of perturbations. However, the methods here are slightly more involved than the ones used earlier. Moreover, the minima for the perturbed functionals are not necessarily close to any given minimizing sequence of the original function, and the perturbations we are seeking here, can never be bounded on the whole Banach space.
Actually, our main goal for this chapter, is to introduce the reader to new and different methods for proving perturbed variational principles and especially, to the martingales techniques used in Theorem 2.17.
A minimization principle with linear perturbations
We shall first consider a situation where the perturbations can be taken to be linear.
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- Publisher: Cambridge University PressPrint publication year: 1993