In this paper, we consider the space BV A (Ω) 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 : Ω → R N such that the distributional differential expression Au is a finite (vectorial) Radon measure. We show that for Lipschitz domains Ω ⊂ R n , BV A (Ω)-functions have an L 1 (∂Ω)-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 (∂Ω)-trace was previously only known for the special cases that Au 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-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 Au.

We establish an ε-regularity result for the derivative of a map of bounded variation that minimizes a strongly quasiconvex variational integral of linear growth, and, as a consequence, the partial regularity of such BV minimizers. This result extends the regularity theory for minimizers of quasiconvex integrals on Sobolev spaces to the context of maps of bounded variation. Previous partial regularity results for BV minimizers in the linear growth set-up were confined to the convex situation.

We prove that the inhomogeneous estimate of vector fields on balls in R n B |D k−1 u| n/(n−1) dxholds if and only if the linear, constant coefficient differential operator A of order k has finite dimensional null-space (FDN). This generalizes the Gagliardo-Nirenberg-Sobolev inequality on domains and provides the local version of the analogous homogeneous embedding in full-spaceproved by Van Schaftingen precisely for elliptic and cancelling (EC) operators, building on fundamental L 1 -estimates from the works of Bourgain and Brezis. We prove that FDN strictly implies EC and discuss the contrast between homogeneous and inhomogeneous estimates on both algebraic and analytic level.2010 Mathematics Subject Classification. Primary: 46E35; Secondary: 26D10.provided that A is EC and u ∈ C ∞ (Ω, V ) satisfy B j u = 0 on ∂Ω, where B j is a (finite collection of) linear differential operator(s) defined on ∂Ω that satisfy the Lopatinskiȋ-Shapiro Complementing Conditions. Such a result would provide a reasonable analogue of the results in [33,1,2,27] to the case p = 1, in spite of Ornstein's Non-inequality. The aim of this paper is to confirm this expectation in the case when B j ≡ 0 ("no boundary condition") and Ω is a ball (whereas Van Schaftingen's result [56, Thm. 1.3] essentially deals with the antipodal case when B j = ∂ j ν , j = 0 . . . k − 1, i.e., "all boundary conditions"). We emphasize that in the present situation the geometry of ∂Ω is not the foremost problem, as is the extendibility of functions u : Ω → V to some v : R n → V while ensuring that Av ∈ L 1 (R n , V ) boundedly. In fact, as we shall see below, this property