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

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 in measure solutions to linear PDE’s and of the corresponding generalized convexity classes. (with G. De Philippis, F. Rindler)