Nonlinear Subelliptic Equations
on Carnot Groups
The main objective
of this course is to present an extension of Jensen's uniqueness theorem
for viscosity solutions of second order uniformly elliptic equations to
Carnot groups. This is done via an extension of the comparison
principle
for semicontinuous functions of Crandall-Ishii-Lions. We first present
the details for the Heisenberg group, where the ideas and the calculations
are easier to appreciate. We then consider the general case of Carnot groups
and present applications to the theory of convex functions and to minimal
Lipschitz extensions.