Comparison Principles for subelliptic equations of Monge-Ampere type.pdfVIP

a r X i v : 0 8 0 2 .1 4 5 6 v 1 [ m a t h .A P ] 1 1 F e b 2 0 0 8 Comparison principles for subelliptic equations of Monge-Ampe?re type Martino Bardi, Paola Mannucci Dipartimento di Matematica P. e A., Universita? di Padova via Trieste 63, 35121 Padova, Italy Abstract We present two comparison principles for viscosity sub- and superso- lutions of Monge-Ampe?re-type equations associated to a family of vector fields. In particular, we obtain the uniqueness of a viscosity solution to the Dirichlet problem for the equation of prescribed horizontal Gauss cur- vature in a Carnot group. Introduction We consider fully nonlinear partial differential equations of the form (0.1) ? det(D2 X u) +H(x, u,DXu) = 0, in ?, where ? ? IRn is open and bounded, DXu denotes the gradient of u with respect to a given family of C1,1 vector fields X1, ..., Xm, DXu := (X1u, ..., Xmu), D 2 X u denotes the symmetrized Hessian matrix of u with respect to the same vector fields (D2Xu)ij := (XiXju+XjXiu) /2, and H is a given Hamiltonian, at least continuous and nondecreasing in u. Our main examples are the vector fields that generate the homogeneous Carnot groups [4, 7], and in that case DXu and D 2 X u are called, respectively, the hori- zontal gradient and the horizontal Hessian. A theory of fully nonlinear subelliptic equations was started recently by Bieske [5, 6] and Manfredi [17, 3], and Monge-Ampe?re equations of the form (0.1) with H = f(x) are listed among the main examples. For such equations on the Heisenberg group Gutierrez and Montanari [12] proved, among other things, a comparison principle among smooth sub- and supersolutions (see also [11] for related results). An example that motivates the dependence on the gradient DXu in H is the prescribed horizontal Gauss curvature equation in Carnot groups, as defined by Danielli, Garofalo and Nhieu [10], (0.2) ? det(D2 X u) + k(x) ( 1 + |DXu| 2 )m+2 2 = 0, in ?, 1 for a given continuous k : ? →]0,+∞[. In this paper we begin a study