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Metric critical point theory. I: Morse regularity and homotopic stability of a minimum. (English) Zbl 0852.58018
The authors discuss the concepts of regular and critical points for a continuous function on a metric space. The basic definitions of regular and critical points are closely related to the notion of the weak slope for continuous functions introduced by M. Degiovanni and M. Marzocchi [Ann. Mat. Pura. Appl., IV Ser. 167, 73-100 (1994; Zbl 0828.58006)], but the general approach is quite different and sometimes new even for smooth functions.
Among the results presented here are: a quantitative deformation lemma, a “potential well” theorem giving an a priori lower estimate for the size of the potential well associated with the given local minimum, and a theorem on stability of the local and global minimum under some “regular” perturbation of the function. Nonsmooth extensions (and even) strengthening of some well-known facts, including certain mountain-pass and bifurcation theorems, a criterion for a global minimum, a Lyusternik’s second order characterization of the tangent space to a level set are given. As application of the approach, the authors give an extension of the Weierstrass sufficient condition for a strong minimum to functionals in the calculus of variations with \(C^{1,1}\) integrands.
Reviewer: V.Moroz (Minsk)

MSC:
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
49J52 Nonsmooth analysis
49K05 Optimality conditions for free problems in one independent variable
54C30 Real-valued functions in general topology
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