Book contents
- Frontmatter
- Contents
- Introduction
- 1 Retrospective
- I First Steps Toward the Mountains
- 2 Palais-Smale Condition: Definitions and Examples
- 3 Obtaining “Almost Critical Points” – Variational Principle
- 4 Obtaining “Almost Critical Points” – The Deformation Lemma
- II Reaching the Mountain Pass Through Easy Climbs
- III A Deeper Insight in Mountains Topology
- IV The Landscape Becoming Less Smooth
- V Speculating about the Mountain Pass Geometry
- VI Technical Climbs
- A Background Material
- Bibliography
- Index
4 - Obtaining “Almost Critical Points” – The Deformation Lemma
Published online by Cambridge University Press: 04 September 2009
- Frontmatter
- Contents
- Introduction
- 1 Retrospective
- I First Steps Toward the Mountains
- 2 Palais-Smale Condition: Definitions and Examples
- 3 Obtaining “Almost Critical Points” – Variational Principle
- 4 Obtaining “Almost Critical Points” – The Deformation Lemma
- II Reaching the Mountain Pass Through Easy Climbs
- III A Deeper Insight in Mountains Topology
- IV The Landscape Becoming Less Smooth
- V Speculating about the Mountain Pass Geometry
- VI Technical Climbs
- A Background Material
- Bibliography
- Index
Summary
Locating critical values for a smooth functional / on a manifold X essentially reduces to capturing the changes in the topology of the sublevel sets Ia = {x ∊ X; I(x) < a} as a varies in ℝ.
I. Ekeland and N. Ghoussoub, New aspects of the calculus of variations in the large. Bull. Am. Math. Soc., 39, no. 2, 207–265 (2001)This chapter is a continuation of the previous one. The “deformation lemma” we will study is very important because, as we will see later, an important part of the inherent topological aspects of critical point theorems is always expressed in terms of deformations.
Another important tool, older than Ekeland's principle and widely used (in his quantitative form) to get “almost critical points,” is the deformation lemma. Its form used nowadays seems to be due to Clark [242] following some ideas of Rabinowitz [734]. But the idea of deforming level sets near regular values to cross these values following the steepest descent direction of the function was already known before and is a basic tool in Morse theory. We will see it in action even in the original proof of the finite dimensional mountain pass theorem (MPT) of Courant that goes back to 1950. (See the notes of the next chapter.)
- Type
- Chapter
- Information
- The Mountain Pass TheoremVariants, Generalizations and Some Applications, pp. 33 - 44Publisher: Cambridge University PressPrint publication year: 2003