Gradient integrability for bounded $\mathrm{BD}$-minimizers

Abstract

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

3 figures, comments welcome