Concentration Phenomena, the Vanishing Mass Conjecture, and an anisotropic Michael-Simon inequality

Abstract: Concentrating (that is, $L^1$-bounded but not equi-integrable) sequences of functions satisfying a linear PDE constraint, such as curl- or divergence-freeness, arise in many situations within PDE theory, the calculus of variations, and geometric measure theory. I will present recent work proving that every Young measure generated by a concentrating sequence admits a Choquet-type decomposition into simpler concentration Young measures whose barycenters lie in the Tartar wave cone associated to the differential constraint. This resolves Bouchitté’s 2001 Vanishing Mass Conjecture and also yields a number of applications, including to compensated compactness, the structure of the singular part of PDE-constrained measures, and Morrey’s Conjecture in the calculus of variations. The main new ingredients of the proof are "barrier functions" for convex cones and a smoothing argument based on the heat flow. I will also discuss a further application to scale-invariant bounds for measures on "thin" sets, which furthermore implies a general anisotropic Michael-Simon inequality for varifolds (the first version of this last application was found by AI based on the ideas in the proof of the Vanishing Mass Conjecture).

This is joint work with Luca Gennaioli (for the Vanishing Mass Conjecture) and Guido De Philippis, Luca Gennaioli, Alessandro Pigati (for the scale-invariant bounds and the Michael-Simon inequality).

Preprints: arxiv.org/abs/2608.20899 and arxiv.org/abs/2609.34573

analysis of PDEsdifferential geometry

Audience: researchers in the topic

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NCTS international Geometric Measure Theory seminar

Series comments: We envisage an event built around virtual presentations on progress in geometric measure theory by external speakers. Every researcher is free to register as a participant and thus gain access to a virtual facility which is complete with lobby, lecture hall, and areas with boards for discussion. Thus, it shall recreate the exchange possibilities found at international conferences.

Focus: regularity and singularity theories for submanifolds of Riemannian manifolds and some of its applications.

Frequency: one presentation every other month.

Registration: required for new participants, go to the seminar website (allow at least one working day for processing).

Virtual venue: HyHyve space NCTS iGMT seminar (only for registered participants, opened one hour before the events).

You might want to consult the description of the premises and instructions.

Former organiser: Guido De Philippis (till March 2022).

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