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2020 On the trace operator for functions of bounded $\mathbb{A}$-variation
Dominic Breit, Lars Diening, Franz Gmeineder
Anal. PDE 13(2): 559-594 (2020). DOI: 10.2140/apde.2020.13.559

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:ΩN such that the distributional differential expression 𝔸u is a finite (vectorial) Radon measure. We show that for Lipschitz domains Ωn, 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.

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Dominic Breit. Lars Diening. Franz Gmeineder. "On the trace operator for functions of bounded $\mathbb{A}$-variation." Anal. PDE 13 (2) 559 - 594, 2020. https://doi.org/10.2140/apde.2020.13.559

Information

Received: 8 December 2018; Accepted: 23 February 2019; Published: 2020
First available in Project Euclid: 25 June 2020

zbMATH: 07181510
MathSciNet: MR4078236
Digital Object Identifier: 10.2140/apde.2020.13.559

Subjects:
Primary: 26B30 , 26D10 , 46E35
Secondary: 46E30 , 49J45

Keywords: functions of bounded $\mathbb{A}$-variation , linear growth functionals , Trace operator

Rights: Copyright © 2020 Mathematical Sciences Publishers

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Vol.13 • No. 2 • 2020
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