Global Regularity and Spectra of Laplace-Beltrami Operators on Pseudoconvex Domains

Abstract

In the theory of elliptic differential operators, the result on Laplace-Beltrami operators defined on compact complex manifolds is a remarkable one and has many important applications to the cohomology theory on compact complex manifolds. On the other hand, in recent years, the property of Laplace-Beltrami operators on non-compact complex manifolds has been investigated from various aspects. In particular, the Kohn's solution to 5-Neumann problem is one of the most remarkable results (see [2] [6]). Looking back to our situation i. e. the cohomology theory on weakly 1-complete manifolds (for example, [10] [11] [13]), it seems that the Kohn's argument, which is based on //-estimates for the d operator, is applicable to the study of the cohomological property of weakly 1complete manifolds. In this paper, having this motivation in mind, and on the other hand, purely from the point of view of partial differential equations, we study the global boundary regularity and the behavior of spectra of LaplaceBeltrami operators on pseudoconvex domains. We apply the result to the cohomology theory of weakly 1-complete manifolds by showing an upper semi-continuity theorem for the dimension of the cohomology groups on a family of weakly 1complete manifolds. The plan of this paper is as follows. In Section 2, we prepare the notations needed in the latter sections and give a sufficient condition for the solvability of the L 9-Neumann problem. In Section 3, we state our main results. In Section 4, we show the basic estimate which is crucial to prove the regularization