Microlocal compactness forms (MCFs) are a new tool to study oscillations and concentrations in L^p-bounded sequences of functions. Decisively, MCFs retain information about the location, value distribution, and direction of oscillations and concentrations, thus extending both the theory of (generalized) Young measures and the theory of H-measures. Since in L^p-spaces oscillations and concentrations discriminate between weak and strong compactness, MCFs allow to precisely quantify the difference between these two notions of compactness.
The definition involves both a “real-space” variable and a Fourier variable, whereby both pointwise restrictions and differential constraints on the sequence can be investigated at the same time - paving the way for applications to Tartar's framework of compensated compactness. Consequently, we establish a new weak-to-strong compactness theorem in a "geometric" way. Finally, MCFs enable a systematic study of the hierarchy of phase mixtures (oscillations) in solids (this is ongoing work), which lately has attracted a lot of attention in the engineering literature.