Holomorphic Approximation and Mixed Boundary Value Problems for $\partial$

Abstract

In this paper we study holomorphic approximation using boundary value problems for $\bar{\partial }$ on an annulus in the Hilbert space setting. The associated boundary conditions for $\bar{\partial }$ are the mixed boundary problems on an annulus. We characterize pseudoconvexity and Runge type property of the domain by the vanishing of related $L^2$ cohomology groups.