1. The Mountain Cliff Theorem
- Author
-
Martin Schechter
- Subjects
Combinatorics ,Set (abstract data type) ,Lemma (mathematics) ,geography ,symbols.namesake ,geography.geographical_feature_category ,Climbing ,Path (graph theory) ,Cliff ,Hilbert space ,symbols ,Mountain pass ,Mathematics - Abstract
Publisher Summary This chapter focuses on the mountain cliff theorem. Many variational problems can be formulated in such a way that one is seeking solutions for a C' functional G defined on a real Hilbert space H. One successful method is the mountain pass lemma of Ambrosetti–Rabinowitz. The basic idea is to consider the set S of all continuous paths P connecting 0 and e hoping to find one on which the maximum of G is no greater than its maximum on any other path in S. If such a path exists, it is a true mountain pass over which a traveler can go without climbing higher than necessary. The basic idea is to consider the set S of all continuous paths P connecting 0 and e hoping to find one on which the maximum of G is no greater than its maximum on any other path in S.
- Published
- 1992