#### Vol. 13, No. 2, 2020

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

### Dominic Breit, Lars Diening and Franz Gmeineder

Vol. 13 (2020), No. 2, 559–594
##### Abstract

We consider the space ${BV}^{\mathbb{𝔸}}\left(\Omega \right)$ 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 $u:\Omega \to {ℝ}^{N}$ such that the distributional differential expression $\mathbb{𝔸}u$ is a finite (vectorial) Radon measure. We show that for Lipschitz domains $\Omega \subset {ℝ}^{n}$, ${BV}^{\mathbb{𝔸}}\left(\Omega \right)$-functions have an ${L}^{1}\left(\partial \Omega \right)$-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 ${L}^{1}\left(\partial \Omega \right)$-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 $\mathbb{𝔸}u$.

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##### Keywords
trace operator, functions of bounded $\mathbb{A}$-variation, linear growth functionals
##### Mathematical Subject Classification 2010
Primary: 46E35, 26D10, 26B30
Secondary: 46E30, 49J45
##### Milestones
Received: 8 December 2018
Accepted: 23 February 2019
Published: 19 March 2020
##### Authors
 Dominic Breit Department of Mathematics Heriot-Watt University Riccarton Edinburgh United Kingdom Lars Diening Fakultät für Mathematik Universität Bielefeld Bielefeld Germany Franz Gmeineder Department of Applied Mathematics University of Bonn Bonn Germany