### Abstract

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-settings) but rather compare projections onto the nullspace of A as we approach the boundary. As a sample application, we study the Dirichlet problem for quasiconvex variational functionals with linear growth depending on Au.

Original language | English |
---|---|

Pages (from-to) | 559–594 |

Number of pages | 36 |

Journal | Analysis and PDE |

Volume | 13 |

Issue number | 2 |

DOIs | |

Publication status | Published - 19 Mar 2020 |

### Keywords

- Functions of bounded A-variation
- Linear growth functionals
- Trace operator

### ASJC Scopus subject areas

- Analysis
- Numerical Analysis
- Applied Mathematics

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## Cite this

*Analysis and PDE*,

*13*(2), 559–594. https://doi.org/10.2140/apde.2020.13.559