Rigidity and compensated compactness for measures

This talk will review a number of recent rigidity and compensated compactness results for measures or L^1-maps. In the first part I will discuss the "stationary" question of how additional restrictions on the polar density of a vector- or matrix-valued measure force this measure to have additional dimensionality and rectifiability properties. In the second part I will investigate sequences of measures (or L^1-maps) under differential and pointwise constraints. The central question here is when these restrictions force the sequence to have additional (and sometimes unexpected) compactness properties. In comparison to the classical L^p-compensated compactness theory for p>1, here one also needs to consider concentration-of-mass effects, which necessitates the development of new techniques.

This is joint work with Arroyo-Rabasa, De Philippis, Hirsch, Palmieri, Skorobogatova. See the group website www.ercsingularity.org for preprints and links to published papers.