A new space of generalised vector-valued functions of bounded variation

TL;DR

The paper extends the scalar generalised functions of bounded variation with star-restrictions to vector-valued maps $GBV_ullet(A;\mathbb{R}^k)$, addressing cohesive fracture energies that combine bulk, Cantor, and surface terms. It develops a robust vector-valued theory: a componentwise definition, a composition-based characterisation using smooth truncations, and a Cantor-part Rank-One property, together with stability under linear changes of coordinates. It proves a lower semicontinuity result for energy functionals $E^{f,g}$ under convergence in measure and provides vector-valued compactness results analogous to the scalar theory, including a direct-method framework for minimisation via modified sequence convergence $\,\mathcal{L}^d$-a.e. in $A$. These results pave the way for homogenisation and variational fracture analyses in multi-dimensional, vector-valued settings using cohesive-type energies.

Abstract

In [Dal Maso and Toader, NoDEA 2022], the authors introduced the space $GBV_\star(A)$ to minimise a class of functionals whose study is motivated by fracture mechanics. In this paper, we extend the definition of $GBV_\star(A)$ to the vectorial case, introducing the space $GBV_\star(A;\mathbb{R}^k)$. We study the main properties of $GBV_\star(A;\mathbb{R}^k)$ and prove a lower semicontinuity result useful for minimisation purposes. With the Direct Method in mind, we adapt the arguments of [Dal Maso and Toader, NoDEA 2022] to show that minimising sequences in $GBV_\star(A;\mathbb{R}^k)$ can be modified to obtain a minimising sequence converging $\mathcal{L}^d$-a.e in $A$.

Theorems & Definitions (41)

• Proposition 2.1
• Theorem 2.2: Chain Rule in $BV(A;{{\mathbb R}}^k)$
• Definition 2.3
• Proposition 2.4
• Proposition 2.5: DalToa22
• Definition 3.1
• Remark 3.2
• Proposition 3.3
• Definition 3.4
• Lemma 3.5