   ISSN 1088-6850(online) ISSN 0002-9947(print)

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

Authors: Joachim Michel and Mei-Chi Shaw
Journal: Trans. Amer. Math. Soc. 351 (1999), 4365-4380
MSC (1991): Primary 35N05, 35N10, 32F10
DOI: https://doi.org/10.1090/S0002-9947-99-02519-2
Published electronically: July 9, 1999
MathSciNet review: 1675218
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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.

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Joachim Michel
Affiliation: Université du Littoral, Centre Universitaire de la Mi-Voix, F-62228 Calais, France
Email: michel@lma.univ-littoral.fr

Mei-Chi Shaw
Affiliation: Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556
MR Author ID: 160050
Email: mei-chi.shaw.l@nd.edu

Keywords: Cauchy-Riemann equations, piecewise smooth boundary, tangential Cauchy-Riemann equations.
Received by editor(s): August 11, 1997
Received by editor(s) in revised form: May 7, 1998
Published electronically: July 9, 1999
Additional Notes: Partially supported by NSF grant DMS 98-01091