On the trace operator for functions of bounded A-variation

Dominic Breit, Lars Diening, Franz Gmeineder

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1 Citation (Scopus)
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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 →R N such that the distributional differential expression Au is a finite (vectorial) Radon measure. We show that for Lipschitz domains Ω ⊂ R n, BV A(Ω)-functions have an L 1(∂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(∂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.

Original languageEnglish
Pages (from-to)559–594
Number of pages36
JournalAnalysis and PDE
Volume13
Issue number2
DOIs
Publication statusPublished - 19 Mar 2020

Keywords

  • Functions of bounded A-variation
  • Linear growth functionals
  • Trace operator

ASJC Scopus subject areas

  • Analysis
  • Numerical Analysis
  • Applied Mathematics

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