About the Morse theory for certain variational problems
type: Item
instance of: scholarly article
title: About the Morse theory for certain variational problems
zbMATH Open document ID: 0537.35006
DOI: 10.1007/BF01456186
author: Gerhard Ströhmer
published in: Mathematische Annalen
publication date: 1985-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/182958
review text: A short time after its inception Morse theory was carried over to the calculus of variations for single integrals. Multiple integrals, however, presented obstacles overcome only decades later by Palais and Smale. They assume a sort of weakened compactness property called the Palais-Smale condition. In this paper we show that no such condition is necessary for the class of problems we consider. By some a-priori estimates recently obtained by the author [see Math. Z. 186, 179-199 (1984)] we are able to artificially endow the space of admissible functions with the necessary compactness properties. We do this for variational problems with uniformly elliptic Euler equations of roughly the same type as those considered by \textit{O. A. Ladyzhenskaya} and \textit{N. N. Ural'tseva} [cf. Linear and quasilinear elliptic equations (1968; Zbl 0164.130), pp, 318- 336].
Mathematics Subject Classification ID: 35A15
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 49J20
Mathematics Subject Classification ID: 35B45
zbMATH DE Number: 3853578
zbMATH Keywords: multiple integrals
zbMATH Keywords: Morse theory
zbMATH Keywords: calculus of variations
zbMATH Keywords: compactness
zbMATH Keywords: Palais-Smale condition
zbMATH Keywords: Euler equations
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3853578