Metric critical point theory. I: Morse regularity and homotopic stability of a minimum
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 \textit{M. Degiovanni} and \textit{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.
scientific article; zbMATH DE number 891058