The <IMG ALIGN=BOTTOM HEIGHT=16 WIDTH=12> problem on domains with piecewise smooth boundaries with applications期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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The problem on domains with piecewise smooth boundaries with applications

Authors:	Joachim Michel Mei-Chi Shaw

Institution:	Université du Littoral, Centre Universitaire de la Mi-Voix, F-62228 Calais, France ; Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556

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 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 smooth up to the boundary (for appropriate degree if $\Omega$ satisfies one of the following conditions: i) $\Omega$ is the transversal intersection of bounded smooth pseudoconvex domains. 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. 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.

Keywords:	Cauchy-Riemann equations piecewise smooth boundary tangential Cauchy-Riemann equations

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