Morse theory for some lower-\(C^ 2\) functions in finite dimension
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.
scientific article; zbMATH DE number 4054434
scientific article