On the trace operator for functions of bounded $\mathbb{A}$-variation

Abstract

We consider the space ${BV}^{\mathbb{𝔸}}\left(\mathrm{\Omega }\right)$ of functions of bounded $\mathbb{𝔸}$-variation. For a given first-order linear homogeneous differential operator with constant coefficients $\mathbb{𝔸}$, this is the space of $L^1$-functions $u:\mathrm{\Omega }\to {ℝ}^N$ such that the distributional differential expression $\mathbb{𝔸}u$ is a finite (vectorial) Radon measure. We show that for Lipschitz domains $\mathrm{\Omega }\subset {ℝ}^n$, ${BV}^{\mathbb{𝔸}}\left(\mathrm{\Omega }\right)$-functions have an $L^1\left(\partial \mathrm{\Omega }\right)$-trace if and only if $\mathbb{𝔸}$ is $ℂ$-elliptic (or, equivalently, if the kernel of $\mathbb{𝔸}$ is finite-dimensional). The existence of an $L^1\left(\partial \mathrm{\Omega }\right)$-trace was previously only known for the special cases that $\mathbb{𝔸}u$ coincides either with the full or the symmetric gradient of the function $u$ (and hence covered the special cases BV or BD). As a main novelty, we do not use the fundamental theorem of calculus to construct the trace operator (an approach which is only available in the BV- and BD-settings) but rather compare projections onto the nullspace of $\mathbb{𝔸}$ as we approach the boundary. As a sample application, we study the Dirichlet problem for quasiconvex variational functionals with linear growth depending on $\mathbb{𝔸}u$.