On critical $L^p$-differentiability of BD-maps

Abstract

We prove that functions of locally bounded deformation on ${\mathbb{R}}^n$ are ${\mathrm{L}}^{n/(n-1)}$-differentiable ${\mathcal{L}}^n$-almost everywhere. More generally, we show that this critical ${\mathrm{L}}^p$-differentiability result holds for functions of locally bounded $\mathbb{A}$-variation, provided that the first order, homogeneous differential operator $\mathbb{A}$ has finite dimensional null-space.