Rectifiability and approximate differentiability of higher order for sets

Abstract

The main goal of this paper is to develop a concept of approximate differentiability of higher order for subsets of the Euclidean space that allows to characterize higher order rectifiable sets, extending somehow well known facts for functions. We emphasize that for every subset A of the Euclidean space and for every integer k ≥ 2 we introduce the approximate differential of order k of A and we prove it is a Borel map whose domain is a (possibly empty) Borel set. This concept could be helpful to deal with higher order rectifiable sets in applications.MSC-classes 2010. 28A75 (Primary); 49Q15 (Secondary).

Fine properties of the curvature of arbitrary closed sets

“…Second order approximate differentiability. Finally in section 6 we analyse the relation of the present notion of curvature with the notion of approximate curvature for second-order rectifiable sets introduced by the author in [San17].…”

“…Basic facts on approximate differentiability for functions are collected in [San17,§2]. Here we point out some additional remarks.…”

Pointwise differentiability of higher order for sets

“…The utility to consider the validity of Theorem D became apparent during the author's ongoing investigation of a special case of the varifold regularity problem formulated jointly with Scharrer in [MS17, Question 3]. Furthermore, the present paper is the third in a sequence of studies (initiated by the author in [Men19] and continued by Santilli in [San19]) that is ultimately directed towards possible higher order pointwise differentiability properties of stationary integral varifolds. For approximate differentiability of second order, the central elliptic partial differential equation involves as inhomogeneous term precisely a distribution, that is ν 1 B(0,1) pointwise differentiable of order 0 at all points in a set, that is compact and has positive L n measure but does not possess further regularity properties (see [Men13,4.4 (6)]).…”

“…In the author's view, a theory of higher order pointwise differentiability for a class of objects should consist of at least four results: Borel regularity of the differentials, rectifiability of the family of k jets, a Rademacher-Stepanov type theorem, and a Lusin type approximation theorem by functions of class k. Such theories (possibly with Borel regularity replaced by appropriate measurability) have been developed for approximate differentiation of functions (successively, by Whitney in [Whi51], Isakov in [Isa87a], 2 and Liu and Tai in [LT94]), for differentiation in Lebesgue spaces with respect to L n (by Calderón and Zygmund in [CZ61]), for pointwise differentiation of sets (by the author in [Men19]), and for approximate differentiation of sets (by Santilli in [San19]).…”

A geometric second-order-rectifiable stratification for closed subsets of Euclidean space

“…The key to reduce this criterion to the nonparametric case is the construction (in 2.1) of a countable collection G of m rectifiable subsets P of W with H m (Z ∼ f [ G]) = 0 such that, for each P ∈ G, the restriction f |P is univalent and (f |P ) −1 is Lipschitzian. The nonparametric case was comprehensively studied in [San17]; however, for the present purpose, also [Sch09] would be sufficient (see 2.6).…”