Lower semicontinuity and relaxation of linear-growth integral functionals under PDE constraints

  • 1 Mathematisches Institut, Universität Leipzig, Augustusplatz 10, 04109, Leipzig, Germany
  • 2 Scuola Internazionale Superiore di Studi Avanzati, Via Bonomea 265, 34136, Trieste, Italy
  • 3 Mathematics Institute, University of Warwick, CV4 7AL, Coventry, United Kingdom
Adolfo Arroyo-RabasaORCID iD: https://orcid.org/0000-0002-8329-1506, Guido De Philippis and Filip RindlerORCID iD: https://orcid.org/0000-0003-2126-3865

Abstract

We show general lower semicontinuity and relaxation theorems for linear-growth integral functionals defined on vector measures that satisfy linear PDE side constraints (of arbitrary order). These results generalize several known lower semicontinuity and relaxation theorems for BV, BD, and for more general first-order linear PDE side constrains. Our proofs are based on recent progress in the understanding of singularities of measure solutions to linear PDEs and of the generalized convexity notions corresponding to these PDE constraints.

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