Piecewise rank-one approximation of vector fields with measure derivatives

TL;DR

This paper investigates the approximation of vector fields with measure derivatives by piecewise constant or rigid functions, revealing the necessity of using a homogenized Schatten-1 norm for optimal density results.

Contribution

It introduces a novel approximation framework using a homogenized Schatten-1 norm, enabling density of piecewise functions in measure-based vector fields with bounded variation.

Findings

Piecewise constant functions are dense in vector fields with measure derivatives under the homogenized norm.

The Frobenius norm is insufficient for approximation due to oscillation and concentration phenomena.

Explicit constructions using laminates demonstrate the optimal approximation properties.

Abstract

This work adresses the question of density of piecewise constant (resp. rigid) functions in the space of vector valued functions with bounded variation (resp. deformation) with respect to the strict convergence. Such an approximation property cannot hold when considering the usual total variation in the space of measures associated to the standard Frobenius norm in the space of matrices. It turns out that oscillation and concentration phenomena interact in such a way that the Frobenius norm has to be homogenized into a (resp. symmetric) Schatten-1 norm which coincides with the Euclidean norm on rank-one (resp. symmetric) matrices. By means of explicit constructions consisting of the superposition of sequential laminates, the validity of an optimal approximation property is established at the expense of endowing the space of measures with a total variation associated with the homogenized norm in the space of matrices.