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).

Defining the m-th stratum of a closed subset of an n dimensional Euclidean space to consist of those points, where it can be touched by a ball from at least n − m linearly independent directions, we establish that the m-th stratum is second-order rectifiable of dimension m and a Borel set. This was known for convex sets, but is new even for sets of positive reach. The result is based on a new criterion for second-order rectifiability.MSC-classes 2010. 52A20 (Primary); 28A78, 49Q15 (Secondary)

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We consider a two phase elastic thin film with soft inclusions subject to bending dominated deformations. The soft (void) phase may comprise asymptotically small droplets within the elastic matrix. We perform a dimension reduction analysis and obtain a novel 'Blake-Zisserman-Kirchhoff' functional on a natural space of 'flat and fractured' two-dimensional isometric immersions that combines Kirchhoff's classical plate theory with Blake-Zisserman type surface energy contributions at cracks, folds and the boundary of voids.