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

TL;DR

This paper proves that bounded minimizers of certain degenerate elliptic functionals in the space of functions with bounded deformation have gradients in L^1, leading to new Sobolev regularity results for these minimizers.

Contribution

It establishes the first Sobolev regularity results for minimizers of area-type functionals on BD, extending known regularity in the full gradient case to the BD setting.

Findings

Locally bounded BD-minimizers have weak gradients in L^1.

First Sobolev regularity results for area-type functional minimizers on BD.

Regularity results are achieved within the sharp ellipticity range.

Abstract

We establish that locally bounded relaxed minimizers of degenerate elliptic symmetric gradient functionals on $\mathrm{BD}(\Omega)$ have weak gradients in $\mathrm{L}_{\mathrm{loc}}^{1}(\Omega;\mathbb{R}^{n\times n})$. 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 $\mathrm{BV}(\Omega;\mathbb{R}^{n})$. As a consequence, we also obtain the first Sobolev regularity results for minimizers of the area-type functional on $\mathrm{BD}(\Omega)$.