Comparison principles and Dirichlet problem for fully nonlinear degenerate equations of Monge–Ampère type

Abstract

Abstract. We study fully nonlinear partial differential equations of Monge–Ampère type involving the derivatives with respect to a family of vector fields. The main result is a comparison principle among viscosity subsolutions, convex with respect to , 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 the 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.

The Dirichlet problem for prescribed principal curvature equations

“…This is reminescent of the definition of sub/supersolution for Monge-Ampere type equations in [2] and [14].…”

“…We now state a general existence result for the Dirichlet problem for (2). Our abstract assumptions are (h1) There exist a C-semiconvex subsolution u of (G i ) * = 0 in Ω and a bounded C-supersolutionū of (G i ) * = 0 in Ω such that u ≤ū on Ω.…”

“…Problems showing non-degeneracy in one direction arise frequently in the subelliptic environment, see [2,19] and references therein. Many ideas used here are borrowed from [3], a seminal paper for equations without zeroth order terms.…”

“…Another approach, proposed in the User's Guide, is to exploit the structure of the operator to perturb subsolutions into strict subsolutions (or supersolutions into strict supersolutions). We will implement the latter method, pointing out that a direct perturbation technique (as in [1,2,14]) does not seem to work due to poor regularity and ellipticity properties of F i . Indeed, if u is any twice differentiable subsolution of −F i = −f and φ ∈ C 2 , setting u = u + φ, nondegenerate ellipticity and Lipschitz regularity implies that for some δ, L > 0…”

“…It is wellknown that the geometry of the boundary plays an important role, as nonexistence of solutions or loss of boundary data is very likely to occur when dealing with spectral equations (see for example [9,21]). We propose in Theorem 4.1 an existence argument which is based on the Perron's method introduced by Ishii; we adapt the version presented in [2], as the restriction to semiconvex subsolutions requires some modifications of the standard technique. An additional assumption on the geometry of ∂Ω guarantees that the solution obtained by means of Perron's method does not suffer of loss of boundary data; to show this we adapt an argument of Da Lio [9].…”