# Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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## The $\overline {\partial }$ problem on domains with piecewise smooth boundaries with applicationsHTML articles powered by AMS MathViewer

by Joachim Michel and Mei-Chi Shaw
Trans. Amer. Math. Soc. 351 (1999), 4365-4380 Request permission

## 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.

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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
• 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