Examples for the Infinite Dimensional Morse Lemma
- 1 November 1983
- journal article
- Published by Society for Industrial & Applied Mathematics (SIAM) in SIAM Journal on Mathematical Analysis
- Vol. 14 (6) , 1045-1055
- https://doi.org/10.1137/0514084
Abstract
Examples are presented which show how to use the Morse lemma in specific infinite dimensional examples and what can go wrong if various hypotheses are dropped. One of the examples shows that the version of the Morse lemma using singularity theory can hold, yet the hypotheses of the Morse–Palais and Morse–Tromba lemmas fail. Another example shows how to obtain a concrete normal form in infinite dimensions using the splitting lemma and hypotheses related to those in the Morse–Tromba lemma. An example of Dancer is given which shows that for the validity of the Morse lemma in Hilbert space, some hypotheses on the higher order terms must be made in addition to smoothness, if the quadratic term is only weakly nondegenerate. A general conjecture along these lines is made.Keywords
This publication has 13 references indexed in Scilit:
- The cusp catastrophe of thom in the bifurcation of minimal surfacesmanuscripta mathematica, 1984
- The Morse Lemma in Infinite Dimensions via Singularity TheorySIAM Journal on Mathematical Analysis, 1983
- A sufficient condition for a critical point of a functional to be a minimum and its application to Plateau's problemMathematische Annalen, 1983
- Symmetry and bifurcation in three-dimensional elasticity, part IArchive for Rational Mechanics and Analysis, 1982
- A theory for imperfect bifurcation via singularity theoryCommunications on Pure and Applied Mathematics, 1979
- Almost-Riemannian Structures on Banach Manifolds: The Morse Lemma and the Darboux TheoremCanadian Journal of Mathematics, 1976
- Euler bucklingPublished by Springer Nature ,1976
- The Catastrophe of a Buckling BeamPublished by Springer Nature ,1975
- The Morse lemma for Banach spacesBulletin of the American Mathematical Society, 1969
- Morse theory on Hilbert manifoldsTopology, 1963