| Summary: | <p>In the first part of this work we investigate lower semi-continuity of integral functionals defined on vector fields that satisfy linear pde constraints that satisfy the so-called constant rank condition. In particular, we simplify the definition of A-quasiconvexity and characterize a class of A-free generalized Young measures by duality with A-quasiconvex functions.</p> <p>In the second part we investigate linear L<sup>1</sup>-estimates, i.e., study linear (overdetermined) elliptic systems with L<sup>1</sup>-data. We extend Van Schaftingen's inequalities to constant rank operators, give a new embedding into bounded functions at the endpoint of the theory, study pointwise properties and integrability of restrictions of solutions, and give a generalization of the Gagliardo-Nirenberg inequality on domains.</p>
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