Published Paper

Inserted: 19 oct 2023
Last Updated: 11 jun 2025

Journal: Nonlinear Differential Equations and Applications NoDEA
Volume: 32
Year: 2025
Doi: https://doi.org/10.1007/s00030-025-01063-5

Abstract:

In 18, the authors introduced the space of scalar-valued functions $GBV_\star(A)$ 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 18 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 $A$.