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A remark on irregularity of the $ \overline{\partial}$-Neumann problem on non-smooth domains

Author: Sönmez Sahutoglu
Journal: Proc. Amer. Math. Soc. 136 (2008), 2529-2533
MSC (2000): Primary 32W05
Published electronically: March 4, 2008
MathSciNet review: 2390523
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Abstract: It is an observation due to J. J. Kohn that for a smooth bounded pseudoconvex domain $ \Omega$ in $ \mathbb{C}^n$ there exists $ s>0$ such that the $ \overline{\partial}$-Neumann operator on $ \Omega$ maps $ W^s_{(0,1)}(\Omega)$ (the space of $ (0,1)$-forms with coefficient functions in $ L^2$-Sobolev space of order $ s$) into itself continuously. We show that this conclusion does not hold without the smoothness assumption by constructing a bounded pseudoconvex domain $ \Omega$ in $ \mathbb{C}^{2}$, smooth except at one point, whose $ \overline{\partial}$-Neumann operator is not bounded on $ W^s_{(0,1)}(\Omega)$ for any $ s>0$.

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Sönmez Sahutoglu
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109-1043

Keywords: $\overline {\partial }$-Neumann problem, worm domains
Received by editor(s): August 21, 2006
Received by editor(s) in revised form: April 23, 2007
Published electronically: March 4, 2008
Communicated by: Mei-Chi Shaw
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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