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

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 $\mathrm{\Omega }\subset {\mathbb{R}}^n$. Assuming appropriate ellipticity conditions and Hölder continuity of ${\mathrm{D}}_{z}F(x,z)$ 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.