Compactness of the ∂ ̅ -Neumann Operator on the Intersection Domains in ℂ^(N)

Abstract

In this dissertation we are concerned with a problem which asks whether the compactness of the ∂ ̅-Neumann operator is preserved under the intersection of two bounded pseudoconvex domains in ℂ^(n) with the mild assumption that their intersection is connected. Our solutions to this problem in this dissertation can be grouped into two affirmative main results.

The first of these two main results provides a solution under the assumption that the intersection of the boundaries of the (intersecting) domains satisfies McNeal's property ( P ̃). More precisely, let Ω_(1) and Ω_(2) be bounded (not necessarily smooth) pseudoconvex domains in ℂ^(n) which intersect each other in a domain Ω. If the ∂ ̅-Neumann operators N_(q)^(Ω_(1)) and N_(q)^(Ω_(2)) are compact and the compact bΩ_(1)∩ bΩ_(2) satisfies property ( P ̃_(q)) for some 1 ≤ q ≤ n, then the ∂ ̅-Neumann operator N_(q)^(Ω) is also compact. We discuss some examples of pseudoconvex domains Ω_(1) and Ω_(2) for which the assumption "bΩ_(1) ∩ bΩ_(2) satisfies property (P ̃_(q))" actually holds.

The second main result provides a partial solution to the problem when the intersecting domains have smooth boundaries which intersect each other real transversally. More precisely, let Ω_(1) and Ω_(2) be bounded smooth pseudoconvex domains in ℂ^(n) whose boundaries intersect real transversally and let Ω be the intersection domain. If the ∂ ̅-Neumann operators N_(q)^(Ω_(1)) and N_(q)^(Ω_(2)) are compact for some 1 ≤ q ≤ n-1, then N_(n-1)^(Ω) is also compact. In particular, when n = 2, compactness of the ∂ ̅-Neumann operator is preserved under the real transversal intersection of two smooth bounded pseudoconvex domains in ℂ^(2). We also discuss some by-products of the problem when the domains are smooth and intersect real transversally.