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- W1994302110 abstract "SEVERAL years ago the author, and independently Smale, generalized the Morse theory of critical points to cover certain functions on hilbert manifolds [5,6 and 91. Shortly thereafter J. Schwartz showed how the same techniques allowed one also to extend the LusternikSchnirelman theory of critical points to functions on hilbert manifolds [7]. Now while Morse theory, by its nature, is more or less naturally restricted to hilbert manifolds, the natural context for Lusternik-Schnirelman theory is a smooth real valued function on a Banach manifold. In this paper we shall extend Schwartz’s results to cover this more general situation. It is not however simply a desire for generality in an abstract theorem that prompted the present work. The motivation for considering critical point theory in the context of infinite dimensional manifolds comes from the immediate applicability of results to proving existence theorems in the Calculus of Variations. As long as the integrands one considers are of a basically quadratic nature, hilbert manifolds of maps belonging to some Sobolev space Lf are the natural manifolds in which to formulate the problems. However for more general integrands it becomes necessary to use Banach manifolds, for example manifolds of maps belonging to one of the more general Sobolev spaces Lkp. Applications in this direction will be found in a forthcoming paper of Browder [2]." @default.
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- W1994302110 date "1966-05-01" @default.
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- W1994302110 title "Lusternik-Schnirelman theory on Banach manifolds" @default.
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- W1994302110 doi "https://doi.org/10.1016/0040-9383(66)90013-9" @default.
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