Vol. 13, No. 2, 2020

Download this article
Download this article For screen
For printing
Recent Issues

Volume 13
Issue 6, 1605–1954
Issue 5, 1269–1603
Issue 4, 945–1268
Issue 3, 627–944
Issue 2, 317–625
Issue 1, 1–316

Volume 12, 8 issues

Volume 11, 8 issues

Volume 10, 8 issues

Volume 9, 8 issues

Volume 8, 8 issues

Volume 7, 8 issues

Volume 6, 8 issues

Volume 5, 5 issues

Volume 4, 5 issues

Volume 3, 4 issues

Volume 2, 3 issues

Volume 1, 3 issues

The Journal
About the Journal
Editorial Board
Editors’ Interests
Subscriptions
 
Submission Guidelines
Submission Form
Policies for Authors
Ethics Statement
 
ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
Author Index
To Appear
 
Other MSP Journals
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