A regularity result for $BV^{\mathcal{A}}(\Omega)$

TL;DR

This paper extends the regularity result known for symmetrized gradients to a broader class of first-order linear elliptic operators with the rank-one property, showing that bounded $ ext{A}$-variation implies membership in $BV^{ ext{A}}$.

Contribution

It generalizes the known regularity result from the symmetrized gradient case to a wider class of elliptic operators satisfying the rank-one property.

Findings

Distributions with bounded $ ext{A}$-variation belong to $BV^{ ext{A}}$ on Lipschitz domains.

The result broadens the class of operators for which regularity results hold.

Extension of classical $BD$ regularity to more general elliptic operators.

Abstract

It is well known that distributions whose symmetrized gradient is a bounded Radon measure belong to the space $BD$ on bounded domains with $\mathcal{C}^1$ boundary. In this work, we extend this result to a broader class of first-order linear elliptic operators. More precisely, let $\mathcal{A}$ be a first-order linear elliptic operator satisfying the rank-one property. We prove that if a distribution defined on a Lipschitz domain has bounded $\mathcal{A}$-variation, then it belongs to the space $BV^{\mathcal{A}}$.