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 BV𝔸(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 L1-functions u : Ω N such that the distributional differential expression 𝔸u is a finite (vectorial) Radon measure. We show that for Lipschitz domains Ω n , BV𝔸(Ω)-functions have an L1(Ω)-trace if and only if 𝔸 is -elliptic (or, equivalently, if the kernel of 𝔸 is finite-dimensional). The existence of an L1(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.

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