On the singular set of $\operatorname{BV}$ minimizers for non-autonomous functionals

TL;DR

This paper studies the regularity and singular set of minimizers for non-autonomous convex variational problems with linear growth, establishing higher integrability and bounds on the singular set's Hausdorff dimension.

Contribution

It provides new regularity results for BV minimizers under ellipticity and H"older continuity assumptions, including bounds on the singular set.

Findings

Higher integrability of minimizer gradients

Bounds on the Hausdorff dimension of the singular set

Regularity results for non-autonomous convex functionals

Abstract

We investigate regularity properties of minimizers for non-autonomous convex variational integrands $F(x, \mathrm{D} u)$ with linear growth, defined on bounded Lipschitz domains $\Omega \subset \mathbb{R}^n$. Assuming appropriate ellipticity conditions and H\"older continuity of $\mathrm{D}_zF(x,z)$ with respect to the first variable, we establish higher integrability of the gradient of minimizers and provide bounds on the Hausdorff dimension of the singular set of minimizers.