On the compensated compactness method for concentration effects

We present some new and surprising examples of uniform integrability of sequences satisfying both a linear pde constraint and a (nonlinear) pointwise constraint. One example that we will explore would be higher integrability of hessian sequences that does not follow from the theory of quasiregular mappings. Earlier examples can be found in the works of D. Serre and G. De Philippis et al. We identify a new phenomenon that gives a counterexample to a conjecture in G. De Philippis et al. Some of our proofs rely on the theory of k-hessian equations, which we complement with new Sobolev regularity results. This is joint work with A. Guerra and M. Schrecker.