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

## TL;DR

This paper investigates holomorphic approximation via boundary value problems for the $ar ext{ extdegree}$ operator on annuli, characterizing domain properties through $L^2$ cohomology in a Hilbert space framework.

## Contribution

It introduces a novel approach to holomorphic approximation using mixed boundary value problems for $ar ext{ extdegree}$ on annuli, linking domain properties to $L^2$ cohomology.

## Key findings

- Characterization of pseudoconvexity via $L^2$ cohomology vanishing.
- Establishment of Runge type property for annular domains.
- Development of boundary conditions for $ar ext{ extdegree}$ in Hilbert spaces.

## 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.