On the trace operator for functions of bounded 𝔸-variation

Breit, Dominic; Diening, Lars; Gmeineder, Franz

doi:10.2140/apde.2020.13.559

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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

DOI: 10.2140/apde.2020.13.559

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 $u : Ω \to ℝ^{N}$ such that the distributional differential expression $𝔸 u$ is a finite (vectorial) Radon measure. We show that for Lipschitz domains $Ω \subset ℝ^{n}$ , ${BV}^{𝔸} (Ω)$ -functions have an $L^{1} (\partial Ω)$ -trace if and only if $𝔸$ is $ℂ$ -elliptic (or, equivalently, if the kernel of $𝔸$ is finite-dimensional). The existence of an $L^{1} (\partial Ω)$ -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 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 $𝔸 u$ .

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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

Vol. 13, No. 2, 2020