Traces of functions of bounded A-variation and variational problems with linear growth

Dominic Breit, Lars Diening, Franz Gmeineder

Published 2017-07-21Version 1

In this paper, we consider the space $BV^{A}(\Omega)$ of functions of bounded $A$-variation. For a given first order linear homogeneous differential operator with constant coefficients $A$, this is the space of $L^1$--functions $u:\Omega\rightarrow R^N$ such that the distributional differential expression $A u$ is a finite (vectorial) Radon measure. We show that for Lipschitz domains $\Omega\subset R^{n}$, $BV^{A}(\Omega)$--functions have an $L^1(\partial\Omega)$--trace if and only if $A$ is $C$-elliptic (or, equivalently, if the kernel of $A$ is finite dimensional). The existence of an $L^1(\partial\Omega)$--trace was previously only known for the special cases that $A u$ coincides either with the full or the symmetric gradient of the function $u$ (and hence covered the special cases $BV$ or $BD$). As an application we study quasiconvex variational functionals with linear growth depending on $A u$ and show the existence of a minimiser in $BV^{A}(\Omega)$.