Periodic Reporting for period 1 - RaDiCHAPDE (Rectifiability and Density in Carnot and Homogeneous Groups, and Applications to Partial Differential Equations)
Periodo di rendicontazione: 2022-10-01 al 2024-09-30
The project RaDiCHAPDE links the geometric properties of singular objects with their analytical properties: the existence of density with local flatness, and the differentiability of Lipschitz functions with diffuseness.
First, the project aims to provide the community working on partial differential equations (PDEs) in parabolic spaces with a reliable set of geometric measure theory tools. Parabolic spaces are crucial because they form the natural metric setting for solving the Dirichlet problem for the heat equation on time-varying domains. So far, only characterizations of regular surfaces through properties related to the solvability of PDEs are available in parabolic spaces. RaDiCHAPDE aims to offer alternative descriptions of such surfaces. Moreover, area formulae and sufficient criteria for determining that a set is "a surface" are completely missing. RaDiCHAPDE has developed such tools.
Second, RaDiCHAPDE seeks to deepen our understanding of the relationship between the differentiability properties of Lipschitz functions and the regularity of measures in Carnot groups and general metric spaces. This problem is significant because Lipschitz functions are ubiquitous in mathematical analysis, and their differentiability properties are fundamental to modern analysis. They are essential in subfields such as the regularity theory of PDEs, the calculus of variations, and the structure of metric spaces. A better understanding of these maps leads to a deeper comprehension of analysis as a whole.
First, the project aims to provide the community working on partial differential equations (PDEs) in parabolic spaces with a reliable set of geometric measure theory tools. Parabolic spaces are crucial because they form the natural metric setting for solving the Dirichlet problem for the heat equation on time-varying domains. So far, only characterizations of regular surfaces through properties related to the solvability of PDEs are available in parabolic spaces. RaDiCHAPDE aims to offer alternative descriptions of such surfaces. Moreover, area formulae and sufficient criteria for determining that a set is "a surface" are completely missing. RaDiCHAPDE has developed such tools.
Second, RaDiCHAPDE seeks to deepen our understanding of the relationship between the differentiability properties of Lipschitz functions and the regularity of measures in Carnot groups and general metric spaces. This problem is significant because Lipschitz functions are ubiquitous in mathematical analysis, and their differentiability properties are fundamental to modern analysis. They are essential in subfields such as the regularity theory of PDEs, the calculus of variations, and the structure of metric spaces. A better understanding of these maps leads to a deeper comprehension of analysis as a whole.
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.
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.
The study of the density problem in the parabolic space has hinted at a potentially more general result in metric spaces. More specifically, it was possible to construct a metric on the parabolic space such that surfaces for which the surface measure has density do not exhibit any property of local regularity when viewed as immersed objects, despite being isometric to a lower-dimensional parabolic plane. Further research based on this observation could lead to a characterization of "locally regular metric spaces." This would be extremely useful for PDE problems in homogeneous and Carnot groups. Additionally, during the resolution of the density problem, tools such as area-type formulas were developed, which are expected to be highly useful for solving boundary value problems.
The proof of the converse of Pansu's theorem, as described above, involved the development of many new techniques. However, some questions remain open. For instance, determining whether the constructed differentiability bundle contains all the directions along which Lipschitz functions are differentiable almost everywhere could lead to improved structural results for intrinsic Lipschitz maps and finite perimeter sets in Carnot groups.
In both cases, analogous results established in the Euclidean setting have had far-reaching consequences. For example, the solution of the density problem in Euclidean spaces by D. Preiss led to the resolution of Plateau's problem and provided the first characterization of rectifiability of the mutual absolute continuity set for the two-phase problem for the harmonic measure. Similarly, the solution of the converse of Rademacher's theorem resulted in significant breakthroughs in the structure of Lipschitz differentiability spaces, the codimension 0 case of the flat chain conjecture, and found applications even in applied fields such as elasticity theory. It is expected that the results obtained by RaDiCHAPDE could lead to similarly impactful breakthroughs in parabolic spaces and Carnot groups.
The proof of the converse of Pansu's theorem, as described above, involved the development of many new techniques. However, some questions remain open. For instance, determining whether the constructed differentiability bundle contains all the directions along which Lipschitz functions are differentiable almost everywhere could lead to improved structural results for intrinsic Lipschitz maps and finite perimeter sets in Carnot groups.
In both cases, analogous results established in the Euclidean setting have had far-reaching consequences. For example, the solution of the density problem in Euclidean spaces by D. Preiss led to the resolution of Plateau's problem and provided the first characterization of rectifiability of the mutual absolute continuity set for the two-phase problem for the harmonic measure. Similarly, the solution of the converse of Rademacher's theorem resulted in significant breakthroughs in the structure of Lipschitz differentiability spaces, the codimension 0 case of the flat chain conjecture, and found applications even in applied fields such as elasticity theory. It is expected that the results obtained by RaDiCHAPDE could lead to similarly impactful breakthroughs in parabolic spaces and Carnot groups.