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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$.

References [Enhancements On Off] (What's this?)

  • [Bar92] David E. Barrett, Behavior of the Bergman projection on the Diederich-Fornæss worm, Acta Math. 168 (1992), nos. 1-2, 1-10. MR 1149863 (93c:32033)
  • [BS90] Harold P. Boas and Emil J. Straube, Equivalence of regularity for the Bergman projection and the $ \overline \partial$-Neumann operator, Manuscripta Math. 67 (1990), no. 1, 25-33. MR 1037994 (90k:32057)
  • [CS01] So-Chin Chen and Mei-Chi Shaw, Partial differential equations in several complex variables, AMS/IP Studies in Advanced Mathematics, vol. 19, American Mathematical Society, Providence, RI; International Press, Boston, MA, 2001. MR 1800297 (2001m:32071)
  • [DF77] Klas Diederich and John Erik Fornæss, Pseudoconvex domains: An example with nontrivial Nebenhülle, Math. Ann. 225 (1977), no. 3, 275-292. MR 0430315 (55:3320)
  • [FS77] John Erik Fornæss and Edgar Lee Stout, Spreading polydiscs on complex manifolds, Amer. J. Math. 99 (1977), no. 5, 933-960. MR 0470251 (57:10009)
  • [HW68] L. Hörmander and J. Wermer, Uniform approximation on compact sets in $ C\sp{n}$, Math. Scand. 23 (1968), 5-21 (1969). MR 0254275 (40:7484)
  • [KN65] J. J. Kohn and L. Nirenberg, Non-coercive boundary value problems, Comm. Pure Appl. Math. 18 (1965), 443-492. MR 0181815 (31:6041)
  • [Ran86] R. Michael Range, Holomorphic functions and integral representations in several complex variables, Graduate Texts in Mathematics, vol. 108, Springer-Verlag, New York, 1986. MR 847923 (87i:32001)

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Additional Information

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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