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

The core of this project is Geometric Measure Theory (GMT) in Homogeneous Groups.

The PI suggests exploring exciting original research avenues regarding the interplay between the concepts of flatness, density, and regularity of measures, and their applications to the theory of Partial Differential Equations (PDEs) and Free Boundary Problems (FBPs) in non-Euclidean spaces.

The projects potential for groundbreaking discovery is achieved by focusing on the investigation of i) the extension of the very classical density problem, whose solution in Euclidean spaces is codified in the celebrated Preiss' rectifiability theorem, to parabolic and Kolmogorov spaces; ii) the quantitative Reifenberg Theorem for measures in the parabolic space and quantitative dimensional estimates of the mutual singular set for the caloric measure in a two-phase problem; iii) the interplay between differentiability of Lipschitz functions and fine geometric properties of Radon measures in general Homogeneous Groups.

As a byproduct of the study of iii), it will be obtained a converse to Pansu's Differentiability Theorem.