We review recent developments in the theory of weak convergence of pde-constrained sequences. We consider the weak lower semicontinuity problem along weakly convergent A-free sequences, where A is a linear pde system of constant rank, and provide improvements to the A-quasiconvexity theory of Fonseca--Müller and the compensated compactness theory of Murat--Tartar. Special emphasis will be placed on concentration effects of weak convergence, in particular by presenting the resolution of a question due to Coifman--PL Lions--Meyer--Semmes and a recent connection between quasiconcavity and higher integrability, generalizing an old result of Müller. Time permitting, we will present the characterization of Young measures generated by A-free sequences by duality with A-quasiconvex functions and recent advances in the regularity theory for A-quasiconvex variational problems. 

Joint work with Christopher Irving, André Guerra, Jan Kristensen, Zhuolin Li, and Matthew Schrecker.

Recently a new inhabitant entered the zoo of convexity notions for vectorial variational problems: functional convexity. I would like to report of progress in understanding the corresponding integrands, but also new insight into fine properties of most general class of related integrands: It turns out that rank-one convex functions share surprisingly many pointwise differentiablity properties with ordinary convex functions.