Hearing pseudoconvexity in Lipschitz domains with holes via $\overline{\partial }$

Abstract

Let $\Omega =\tilde{\Omega }\setminus \overline{D}$ where $\tilde{\Omega }$ is a bounded domain with connected complement in ${\mathbb{C}}^n$ (or more generally in a Stein manifold) and $D$ is relatively compact open subset of $\tilde{\Omega }$ with connected complement in $\tilde{\Omega }$. We obtain characterizations of pseudoconvexity of $\tilde{\Omega }$ and $D$ through the vanishing or Hausdorff property of the Dolbeault cohomology groups on various function spaces. In particular, we show that if the boundaries of $\tilde{\Omega }$ and $D$ are Lipschitz and $C^2$-smooth respectively, then both $\tilde{\Omega }$ and $D$ are pseudoconvex if and only if $0$ is not in the spectrum of the $\overline{\partial }$-Neumann Laplacian on $(0,q)$-forms for $1\le q\le n-2$ when $n\ge 3$; or $0$ is not a limit point of the spectrum of the $\overline{\partial }$-Neumannn Laplacian on $(0,1)$-forms when $n=2$.

accepted for publication in Math. Zeitschrift