On the Levi Monge-Ampére Equation

Abstract

We are concerned with some notions of curvatures associated with pseudoconvexity and the Levi form as the classical Gauss and Mean curvatures are related to the convexity and to the Hessian matrix. In particular, given a prescribed non negative function K, the Levi Monge Ampère equation for the graph of a function $u: {\mathbb{R}}^{2n+1} \rightarrow \mathbb{R}$ is

$$\displaystyle{\det \mathcal{L} = K(x,u){(1 + \vert Du{\vert }^{2})}^{\frac{n+1} {2} },}$$

where $\mathcal{L}$ is the Levi form of the graph u and D u is the Euclidean gradient of u. More generally, we shall consider elementary symmetric functions of the eigenvalues of the Levi form $\mathcal{L}$ and we shall first show that these curvature equations contain information about the geometric feature of a closed hypersurface. Then, we shall show that the curvature operators lead to a new class of second order fully nonlinear equations whose characteristic form, when computed on generalized pseudoconvex functions, is nonnegative definite with kernel of dimension one. Thus, the equations are not elliptic at any point. However, they have the following redeeming feature: the missing ellipticity direction can be recovered by suitable commutation relations. We shall use this property to study existence, uniqueness and regularity of viscosity solutions of the Dirichlet problem for graphs with prescribed Levi curvature.