RaDiCHAPDE · Rectifiability and Density in Carnot and Homogeneous Groups, and Applications to Partial Differential Equations

Rectifiability and Density in Carnot and Homogeneous Groups, and Applications to Partial Differential Equations

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.

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