Singularities, Rectifiability, and PDE-constraints

Abstract: Surprisingly many different problems of Analysis naturally lead to questions about singularities in (vector) measures. These problems come from both "pure" Analysis, such as the question for which measures Rademacher's theorem on the differentiability of Lipschitz functions holds, and its non-Euclidean analogues, as well as from "applied" Analysis, for example the problem to determine the fine structure of slip lines in elasto-plasticity. It is a remarkable fact that many of the (vector) measures that naturally occur in these questions satisfy an (under-determined) PDE constraint, e.g., divergence- or curl-freeness. The crucial task is then to analyse the fine properties of these PDE-constrained measures, in particular to determine the possible singularities that may occur. It turns out that the PDE constraint imposes strong restrictions on the shape of these singularities, for instance that they can only occur on a set of bounded Hausdorff-dimension, or even that the measure is k-rectifiable where its upper k-density is positive. The essential difficulty in the analysis of PDE-constrained measures is that many standard methods from harmonic analysis are much weaker in an L$^1$-context and thus new strategies are needed. In this talk, I will survey recent and ongoing work on this area of research.

analysis of PDEsdifferential geometry

Audience: researchers in the topic

---

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).

---