Compactness of Special Functions of Bounded Higher Variation

Abstract

Given an open set Ω ⊂ R m and n > 1, we introduce the new spaces GBnV (Ω) of Generalized functions of bounded higher variation and GSBnV (Ω) of Generalized special functions of bounded higher variation that generalize, respectively, the space BnV introduced by Jerrard and Soner in [43] and the corresponding SBnV space studied by De Lellis in [24]. In this class of spaces, which allow as in [43] the description of singularities of codimension n, the distributional jacobian Ju need not have finite mass: roughly speaking, finiteness of mass is not required for the (m − n)-dimensional part of Ju, but only finiteness of size. In the space GSBnV we are able to provide compactness of sublevel sets and lower semicontinuity of Mumford-Shah type functionals, in the same spirit of the codimension 1 theory [5,6].

Multi-Value Microstructural Descriptors for Complex Materials: Analysis of Ground States

“…In the case of metric spaces with the properties above, all such definitions give rise to the same space of functions (see [9]). For our purposes, the most convenient definition is the one proposed in [2] and then generalized in [54][55][56].…”

“…Let (X, d) be a complete, separable and locally compact metric space-the metric is indicated by d. Different definitions of weakly differentiable functions with values in a metric space have been proposed in the literature (see, example, [2,33,35,54]). In the case of metric spaces with the properties above, all such definitions give rise to the same space of functions (see [9]).…”

BV-norm continuity of sweeping processes driven by a set with constant shape

“…For S ⊆ I we write Lip(f, S) := sup{d(f (s), f (t))/|t − s| : s, t ∈ S, s = t}, Lip(f ) := Lip(f, I), the Lipschitz constant of f , and Lip(I; X) := {f ∈ X I : Lip(f ) < ∞}, the set of X-valued Lipschitz continuous functions on I. We recall now the notion of BV function with values in a metric space (see, e.g., [1,48]). …”

Estimates for the topological degree and related topics

“…Various definitions of Sobolev spaces and the variants of well-known properties of Sobolev spaces have been studied by many authors, e.g., Ambrosio [3], Korevaar and Schoen [55], Reshetnyak [78], Haj laz and Koskela [46], Bourgain, Brezis, and Mironescu [15] and the references therein. In this section, we will discuss several characterizations of Sobolev spaces which are motivated by the quantity used in the estimates for the degree in Section 2.…”