Calculus of Variations and Geometric Measure Theory
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M. Bardi - P. Mannucci

Comparison Principles and Dirichlet problem for fully nonlinear degenerate equations of Monge-Ampère type

created by bardi on 18 Dec 2009
modified on 08 Jan 2014


Published Paper

Inserted: 18 dec 2009
Last Updated: 8 jan 2014

Journal: Forum Mathematicum
Volume: 25
Pages: 1291--1330
Year: 2013
Doi: 10.1515/forum-2013-0067


We study partial differential equations of Monge-Ampère 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-Ampère equations.

Keywords: Carnot groups, Viscosity solutions, subelliptic equations, Monge-Ampère equation


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