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
- V Speculating about the Mountain Pass Geometry
- 17 The MPT on Convex Domains
- 18 MPT in Order Intervals
- 19 The Linking Principle
- 20 The Intrinsic MPT
- 21 Geometrically Constrained MPT
- VI Technical Climbs
- A Background Material
- Bibliography
- Index
20 - The Intrinsic 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
- V Speculating about the Mountain Pass Geometry
- 17 The MPT on Convex Domains
- 18 MPT in Order Intervals
- 19 The Linking Principle
- 20 The Intrinsic MPT
- 21 Geometrically Constrained MPT
- VI Technical Climbs
- A Background Material
- Bibliography
- Index
Summary
Intrinsicadj inherent, born, built-in, congenital, connate, constitutional, deepseated, elemental, essential, inborn, inbred, indwelling, ingenerate, ingrained, innate, intimate.
From Webster's Electronic ThesaurusUsing the concept of linking of two subsets A and B, seen in Chapter 19, Schechter proved an intrinsic version of the MPT where an estimate for ||Φ′(u)|| appears, as a function of the difference between the supremum of Φ on A and its infimum on B, and of the distance between B and the proper subset of A where Φ assumes greater values than on B.
We will present Schechter's result and some of its immediate consequences, but we will focus on its metric extension due to Corvellec, which presents nicely and clearly its principles and basic ideas.
The main references for the subject of this chapter are the papers [257, 770, 771, 808]. You may also consult the chapter of notes and remarks at the end of Schechter's book [816].
The aim of Schechter, in [808], was a new statement of the MPT without the aid of “auxiliary sets” (the local minimum and the lower point e in the statement of the original MPT or the compact set K and its closed subset K* that appear in the statements of [153, 623, 628, 835], for example).
- Type
- Chapter
- Information
- The Mountain Pass TheoremVariants, Generalizations and Some Applications, pp. 240 - 247Publisher: Cambridge University PressPrint publication year: 2003