Gradient integrability for bounded $\mathrm{B}\mathrm{D}$-minimizers

We establish that locally bounded relaxed minimizers of degenerate elliptic symmetric gradient functionals on $\mathrm{B}\mathrm{D}(\mathrm{\Omega })$ have weak gradients in ${\mathrm{L}}_{\mathrm{l}\mathrm{o}\mathrm{c}}^1(\mathrm{\Omega };{\mathbb{R}}^{n\times n})$. This is achieved for the sharp ellipticity range that is presently known to yield ${\mathrm{W}}_{\mathrm{l}\mathrm{o}\mathrm{c}}^{1,1}$-regularity in the full gradient case on $\mathrm{B}\mathrm{V}(\mathrm{\Omega };{\mathbb{R}}^n)$. As a consequence, we also obtain the first Sobolev regularity results for minimizers of the area-type functional on $\mathrm{B}\mathrm{D}(\mathrm{\Omega })$.