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

## TL;DR

This paper establishes general lower semicontinuity and relaxation theorems for linear-growth integral functionals under PDE constraints, extending known results for BV, BD, and higher-order PDEs, with proofs based on recent advances in measure singularities and convexity notions.

## Contribution

It generalizes lower semicontinuity and relaxation results to vector measures with arbitrary order PDE constraints, broadening the scope of existing theorems.

## Key findings

- Proves lower semicontinuity for measure functionals under PDE constraints.
- Develops relaxation theorems for linear-growth integral functionals.
- Utilizes recent advances in measure singularities and convexity theory.

## 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.