Comparison principles and Dirichlet problem for equations of Monge-Ampere type associated to vector fields

TL;DR

This paper establishes a comparison principle and proves the existence and uniqueness of solutions for a class of Monge-Ampere type equations involving vector fields of Carnot type, extending classical Euclidean results to subelliptic settings.

Contribution

It introduces a comparison principle for viscosity solutions of Monge-Ampere equations associated with Carnot group vector fields, ensuring uniqueness and existence under broader conditions.

Findings

Proves a comparison principle for viscosity solutions in Carnot groups.

Establishes existence of solutions via Perron method under growth or subsolution conditions.

Generalizes classical Euclidean Monge-Ampere results to subelliptic equations.

Abstract

We study the Dirichlet problem for subelliptic partial differential equations of Monge-Ampere type involving the derivates with respect to a family X of vector fields of Carnot type. The main result is a comparison principle among viscosity subsolutions, convex with respect to X, and viscosity supersolutions (in a weaker sense than usual), which implies the uniqueness of solution to the Dirichlet problem. Its assumptions include the equation of prescribed horizontal Gauss curvature in Carnot groups. By Perron method we also prove the existence of a solution either under a growth condition of the nonlinearity with respect to the gradient of the solution, or assuming the existence of a subsolution attaining continuously the boundary data, therefore generalizing some classical result for Euclidean Monge-Ampere equations.