On $\mathrm{BV}^{\mathbb{A}}$-Minimisers in two Dimensions

TL;DR

This paper studies the regularity properties of minimizers in a specialized function space associated with elliptic differential operators in two dimensions, revealing gradient integrability within a known ellipticity range.

Contribution

It extends regularity results for $ ext{BV}^ ext{A}$-minimisers to a broader class of elliptic operators in two dimensions, leveraging their structural properties.

Findings

Gradient integrability established within the sharp ellipticity range.

Regularity results depend on the special structure of the differential operators.

Advances understanding of minimizers in $ ext{BV}^ ext{A}$ spaces for elliptic operators.

Abstract

We investigate into the regularity of $\mathrm{BV}^{\mathbb{A}}$-minimisers for $\mathbb{C}$-elliptic differential operators $\mathbb{A}$ in $2$ dimensions. Our studies strongly rely on the special structure of such differential operators. The gradient integrability is established for the sharp ellipticity range known from the (symmetric) gradient case.