# On some fully nonlinear subelliptic PDEs

### (Università di Padova)

**Event:** ERC Workshop on Geometric Analysis on sub-Riemannian and Metric Spaces

**Date:** Oct 11, 2011,
**time:** 11:40

**Place:** Centro De Giorgi, Scuola Normale Superiore

**Abstract.** We present some joint work with Paola Mannucci on two classes of Hessian equations involving a given family of vector fields such as the generators of a Carnot group. The first class contains Monge-Ampere equations, possibly with terms involving the gradient of the solution. The second type of equations prescribes the minimal eigenvalue of the symmetrized Hessian matrix with respect to the given vector fields. Under general assumptions we prove comparison principles for viscosity solutions and uniqueness for the Dirichlet problem. Existence of solutions is established under stronger conditions. The notion of convexity with respect to vector fields plays an important role for both classes of equations.