Book contents
- Frontmatter
- Contents
- Introduction
- 1 Retrospective
- I First Steps Toward the Mountains
- II Reaching the Mountain Pass Through Easy Climbs
- III A Deeper Insight in Mountains Topology
- IV The Landscape Becoming Less Smooth
- 14 The Semismooth MPT
- 15 The Nonsmooth MPT
- 16 The Metric MPT
- V Speculating about the Mountain Pass Geometry
- VI Technical Climbs
- A Background Material
- Bibliography
- Index
16 - The Metric MPT
Published online by Cambridge University Press: 04 September 2009
- Frontmatter
- Contents
- Introduction
- 1 Retrospective
- I First Steps Toward the Mountains
- II Reaching the Mountain Pass Through Easy Climbs
- III A Deeper Insight in Mountains Topology
- IV The Landscape Becoming Less Smooth
- 14 The Semismooth MPT
- 15 The Nonsmooth MPT
- 16 The Metric MPT
- V Speculating about the Mountain Pass Geometry
- VI Technical Climbs
- A Background Material
- Bibliography
- Index
Summary
Mathematical discoveries, small or great are never born of spontaneous generation. They always presuppose a soil seeded with preliminary knowledge and well prepared by labour, both conscious and subconscious.
Bertrand RusselGoing further in nonsmoothness, we present now a variant of the MPT for continuous functionals on metric spaces. Appropriate notions of critical point and Palais-Smale condition are defined to handle this more general situation that still contains as particular cases the previous results stated when more smoothness and regularity on the functional were supposed.
The metric MPT was discovered independently by Degiovanni and Marzocchi [310] and Katriel [516]. They both use Ekeland's variational principle but in two different ways. The method of Degiovanni and Marzocchi has become widely known these days. Nevertheless, Katriel's approach will be more familiar to those who have read the previous chapter devoted to the nonsmooth MPT.
Preliminaries
In both papers [310, 516], the notions of critical point and Palais-Smale condition are defined in a very similar way although they use different terminology.
Critical Points of Continuous Functions in Metric Spaces
The definition of a critical point for a continuous function defined on a metric space reduces to the usual one known in the smooth case when the functional is smooth.
- Type
- Chapter
- Information
- The Mountain Pass TheoremVariants, Generalizations and Some Applications, pp. 186 - 204Publisher: Cambridge University PressPrint publication year: 2003