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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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