Research Projects

The main target is to develop methods that will allow us to visualize and prove, in the subelliptic setting, regularity results which are known in the Euclidean case. $\mathbb{R}^{n}$ is a commutative group under vector addition and is equipped with the isotropic dilation (scaling) which is a group automorphism. The Heisenberg groups and Carnot groups in general, on the other hand, are not commutative groups and they are homogeneous groups under an anisotropic dilation. As a result, the correct analogue for the $p$-Laplace equation and system are no longer elliptic, but only subelliptic. Since only components of the horizontal gradient appear in the equations, it is very difficult to obtain good estimates for the vertical derivatives. We hope to show that in spite of these rather serious challenges, many of the regularity results in the Euclidean case have suitable analogues or modifications in the subelliptic setting. Often the correct statements to aim for needs to be discovered first and can be significantly different from the Euclidean case.