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Morse theory for some lower-\(C^ 2\) functions in finite dimension. (English) Zbl 0646.49029
The author proves the Morse inequalities for quadratically minorized (i.e., \(f(x)\geq\) const. + const.\(| x|^ 2)\) real-valued lower- \(C^ 2\) functions (i.e. f is representable as a maximum of a family of \(C^ 2\)-smooth functions) defined on open subsets of a finite- dimensional Hilbert space under the assumption that the critical points of f are strongly nondegenerate (a difficult concept not explained here) and \(f^{-1}([a,b])\) are compact subsets for all \(a,b\in {\mathbb{R}}\). The method is based on the Moreau-Yosida approximations \(f_{\epsilon}(x)=\inf_{w}(f(w)+| x-w|^ 2/2\epsilon)\) which are \(C^ 2\)-smooth and preserve the critical points together with certain generalized Morse index (a new concept for the lower-\(C^ 2\) functions), thus reducing the problem to the classical case.
Reviewer: J.Chrastina

MSC:
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
49J45 Methods involving semicontinuity and convergence; relaxation
90C30 Nonlinear programming
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