On the Trace Operator for Functions of Bounded $\mathbb{A}$-Variation

Abstract

In this paper, we consider the space ${\mathrm{BV}}^{\mathbb{A}}(\Omega )$ 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 $u:\Omega \to {\mathbb{R}}^N$ such that the distributional differential expression $\mathbb{A}u$ is a finite (vectorial) Radon measure. We show that for Lipschitz domains $\Omega \subset {\mathbb{R}}^n$, ${\mathrm{BV}}^{\mathbb{A}}(\Omega )$-functions have an $L^1(\partial \Omega )$-trace if and only if $\mathbb{A}$ is $\mathbb{C}$-elliptic (or, equivalently, if the kernel of $\mathbb{A}$ is finite dimensional). The existence of an $L^1(\partial \Omega )$-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 $\mathrm{BV}$ or $\mathrm{BD}$). 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 $\mathrm{BV}$- and $\mathrm{BD}$-setting) but rather compare projections onto the nullspace as we approach the boundary. As a sample application, we study the Dirichlet problem for quasiconvex variational functionals with linear growth depending on $\mathbb{A}u$.

References in corpus (1)

• On Rank-One Convex Functions that are homogeneous of Degree One

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