The $\overline {\partial }$ problem on domains with piecewise smooth boundaries with applications

Abstract:

Let $\Omega$ be a bounded domain in $\mathbb C^n$ such that $\Omega$ has piecewise smooth boudnary. We discuss the solvability of the Cauchy-Riemann equation \begin{equation*} \overline {\partial }u=\alpha \quad \text {in}\quad \Omega \tag {0.1} \end{equation*} where $\alpha$ is a smooth $\overline {\partial }$-closed $(p,q)$ form with coefficients $C^\infty$ up to the bundary of $\Omega$, $0\le p\le n$ and $1\le q\le n$. In particular, Equation (0.1) is solvable with $u$ smooth up to the boundary (for appropriate degree $q)$ if $\Omega$ satisfies one of the following conditions:

1. [i)] $\Omega$ is the transversal intersection of bounded smooth pseudoconvex domains.

2. [ii)] $\Omega =\Omega _1\setminus \overline \Omega _2$ where $\Omega _2$ is the union of bounded smooth pseudoconvex domains and $\Omega _1$ is a pseudoconvex convex domain with a piecewise smooth boundary.

3. [iii)] $\Omega =\Omega _1\setminus \overline {\Omega }_2$ where $\Omega _2$ is the intersection of bounded smooth pseudoconvex domains and $\Omega _1$ is a pseudoconvex domain with a piecewise smooth boundary.

The solvability of Equation (0.1) with solutions smooth up to the boundary can be used to obtain the local solvability for $\overline {\partial }_b$ on domains with piecewise smooth boundaries in a pseudoconvex manifold.