Holomorphic Approximation and Mixed Boundary Value Problems for $\mathrm{\partial }$

In this paper we study holomorphic approximation using boundary value problems for $\bar{\mathrm{\partial }}$ on an annulus in the Hilbert space setting. The associated boundary conditions for $\bar{\mathrm{\partial }}$ 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.