Singular integrals, harmonic functions and boundary regularity in Heisenberg groups


Main funder


Funds granted by main funder (€)

438 874,00


Funding program

Research post as Academy Research Fellow, AoF (Academy of Finland)


Project timetable

Project start date: 01/09/2019

Project end date: 31/08/2024


Summary

This project addresses foundations and applications of geometric measure theory in the setting of a non-commutative Lie group, the Heisenberg group H^n, endowed with a non-Euclidean metric. The algebraic structure of H^n is a rich source of research problems in the theory of partial differential equations and sub-Riemannian geometry. Many of these problems ultimately concern the boundary regularity of sets:

Which domains in H^n are admissible for the sub-Laplace Dirichlet problem with L^p boundary data?
How regular are the boundaries of isoperimetric sets in H^n?

The project aims to contribute to these and related questions by using singular integral operators (SIO) and concepts of quantitative rectifiability. In R^n, quantitatively rectifiable sets are e.g. sets with big pieces of k-dimensional Lipschitz graphs, and their study is an active research area dating back to the work of G. David and S. Semmes in the 90s. In H^n, a theory of quantitative rectifiability has only emerged in recent years.

The first goal of the project is to unify different approaches to (quantitative) rectifiability in H^n and advance them in connection with SIO and certain "intrinsic" Lipschitz graphs.
The second part concerns applications in H^n of quantitative rectifiability and SIO to harmonic functions (removability, Dirichlet problem) and parametrizations (isoperimetric sets, boundary regularity of quasiballs).


Principal Investigator


Primary responsible unit


Last updated on 2019-27-05 at 14:37