The Eigenvalue Problem for the Complex Monge–Ampère Operator

We prove the existence of the first eigenvalue and an associated eigenfunction with Dirichlet condition for the complex Monge–Ampère operator on a bounded strongly pseudoconvex domain in Cn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {C}}^n$$\end{document}. We show that the eigenfunction is plurisubharmonic, smooth with bounded Laplacian in Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega $$\end{document} 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–Ampère 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–Ampère operator.