Mathematics

Summary: H-measures, also called microlocal defect measures, were introduced independently by Luc Tartar and Patrick Gérard. They have been successfully applied in many problems involving asymptotic limits of quadratic quantities of L2 sequences. However, they turned insufficient for nonlinear problems with solutions in Lp spaces, which prompted the introduction of H-distributions, made by Antonić and Mitrović, as an extension of H-measures to the Lp-Lq setting. The two basic steps used in the construction of H-measures and H-distributions are the First Commutation lemma (which gives the compactness of the commutator of the Fourier multiplier and the multiplication operator) and the Schwartz Kernel theorem (allowing for the identification the bilinear functional with the distribution), which we discuss in order to refine some results on H-distributions. In the last part of the talk, we will show and apply a variant of the compensated compactness using a variant of H-distributions. This talk is based on joint works with Nenad Antonić, Marko Erceg, and Darko Mitrović.