Regularity for subelliptic PDE through uniform estimates in multi scale geometries
Luca Capogna (Worcester Polytechnic Institute)
Thursday, November 12, 2015
SubRiemannian geometry and subelliptic PDE provide a useful framework for modeling a broad range of systems, ranging from motion of robot arms, to some mechanisms of perception in the visual cortex and beyond. In most cases the mathematical challenges associated to these objects are due to the lack of ellipticity of the operators involved and on the degeneracy of the ambient metric structure. A natural approach to tackle such difficulties is by approximating the subRiemannian metric through a family of collapsing Riemannian approximates. This scheme also immediately provides elliptic approximates for the subelliptic operators in play.
In this talk I will describe some recent results concerning stability of doubling properties, Poincare inequalities, Gaussian estimates on heat kernels, Schauder estimates, Etc. etc. in this approximation scheme. I will also address some concrete applications (that provided the initial motivation for this study), including the study of motion by mean curvature, and Liouville type theorems for quasiconformal mappings in the subRiemannian setting.