The first objective of the project was to determine the type of regularity a Radon measure with density exhibits in parabolic spaces. This is highly relevant for applications, as parabolic spaces serve as the natural setting for studying the heat equation on time-varying domains. It has been shown that, in such domains, the solvability of the Dirichlet problem requires a form of strong flatness in the time direction. In other words, the domain in which the heat equation is solved cannot change too rapidly over time. The resolution of the density problem in parabolic spaces revealed that regular domains, where the Dirichlet problem for the heat equation is solvable, possess a level of regularity that exceeds that of domains with merely "smooth" boundaries. Despite this, it was shown that if the surface measure of the domains satisfies a quantified density condition, a quantitative flatness property—known as the bilateral weak geometric lemma, a weak form of Reifenberg flatness—can be established.
The second major objective achieved by the project was the proof of the converse of Pansu's theorem. Pansu's theorem states that a Lipschitz function between two Carnot groups is differentiable almost everywhere with respect to the Haar measure of the group. The RaDiCHAPDE project tackled the converse question: What can be said about measures for which Lipschitz functions are differentiable almost everywhere? Specifically, are these measures absolutely continuous with respect to the Haar measure of the domain group? Addressing this question required three key ingredients.
The first was the introduction of a sub-Riemannian differentiability bundle for Lipschitz functions with respect to general Radon measures. Given any Radon measure on a Carnot group, this bundle consists of directions along which all Lipschitz functions are differentiable, thereby extending Pansu's theorem to any Radon measure.
The second ingredient is in the spirit of proving that flat 1-currents can be closed to normal currents. The theory of currents in Carnot groups is crippled by the scarsity of Lipschitz maps and by the fact that their polar is constrained to live on a non-involutive distribution of planes. The third and final ingredient is the proof of a De Philippis-Rindler-type theorem for differential operators of order 1. This result for vector-valued PDEs extends to Carnot groups the structure results for A-free measures, where A is a first-order differential operator. Pansu's differentiability theorem was later employed to prove that Pansu's differentiability spaces satisfy a stronger version of Cheeger's conjecture for the pushforward of their measure with respect to charts.
Finally, the project also produced an alternative proof of the flat chain conjecture in dimension 1 and a uniqueness result for Plateau's problem for typical data.
