The Eigenvalue Problem for the Complex Monge-Ampere Operator

We prove the existence of the first eigenvalue and an associated eigenfunction with Dirichlet condition for the complex Monge-Ampere operator on a bounded strongly pseudoconvex domain in C-n. We show that the eigenfunction is plurisubharmonic, smooth with bounded Laplacian in Omega and boundary values 0. Moreover it is unique up to a positive multiplicative constant. To this end, we follow the strategy used by P. L. Lions in the real case. However, we have to prove a new theorem on the existence of solutions for some special complex degenerate Monge-Ampere equations. This requires establishing new a priori estimates of the gradient and Laplacian of such solutions using methods and results of Caffarelli et al. (Commun Pure Appl Math 38(2):209-252, 1985) and Guan (Commun Anal Geom 6(4):687-703, 1998). Finally we provide a Pluripotential variational approach to the problem and using our new existence theorem, we prove a Rayleigh quotient type formula for the first eigenvalue of the complex Monge-Ampere operator.