Concentration phenomena and the Vanishing Mass Conjecture

Abstract

Concentrating (that is, non-equi-integrable) sequences of functions satisfying linear PDE constraints arise in a wide variety of problems in PDE, the calculus of variations, and geometric measure theory. In 2001, Bouchitté conjectured that such concentrations can always be represented as superpositions of ``simple'' concentrations, a claim he termed the Vanishing Mass Conjecture. We fully resolve this conjecture for all first-order constant-coefficient linear differential operators, proving that the value-distribution (Young) measure of any such sequence admits a Choquet-type decomposition into probability measures whose barycenters lie in the associated Tartar wave cone. In fact, we prove a significantly stronger statement than originally conjectured, namely that the constituent measures are themselves generated by concentrating sequences satisfying the same PDE constraint. This provides a complete structural description of concentrating sequences and yields several applications, including a new proof of a theorem of De~Philippis and the second author on the singular polar of PDE-constrained measures, two types of compensated compactness results, and a surprising result on the support cardinality of extremal concentration Young measures. We also place our results in the context of Morrey's Conjecture, showing that its analogue for pure concentrations is false. Our proofs introduce several new techniques, most notably convexity arguments involving ``barrier functions'' and a careful smoothing procedure via the heat flow that replaces classical Fourier methods.