Nonsmooth Morse–Sard theorems. (September 2017)

• Record Type: Journal Article
• Authors: Azagra, D.; Ferrera, J.; Gómez-Gil, J.
• Is Part Of: Nonlinear analysis. Volume 160(2017)
• Page Start: 53
• Page End: 69
• Publication Date: 2017-09
• Subjects: Morse–Sard theorem -- Taylor polynomial -- Subdifferential -- Nonsmooth; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2017.05.006
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We prove that every function f : R n → R satisfies that the image of the set of critical points at which the function f has Taylor expansions of order n − 1 and non-empty subdifferentials of order n is a Lebesgue-null set. As a by-product of our proof, for the proximal subdifferential ∂ P, we see that for every lower semicontinuous function f : R 2 → R the set f ( { x ∈ R 2 : 0 ∈ ∂ P f ( x ) } ) is L 1 -null.