Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

Request Permissions   Purchase Content 
 
 

 

A class of domains with noncompact $ \overline{\partial}$-Neumann operator


Author: Debraj Chakrabarti
Journal: Proc. Amer. Math. Soc. 141 (2013), 2351-2359
MSC (2010): Primary 32W05
DOI: https://doi.org/10.1090/S0002-9939-2013-11504-2
Published electronically: March 6, 2013
MathSciNet review: 3043016
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The $ \overline {\partial }$-Neumann operator (the inverse of the complex Laplacian) is shown to be noncompact on certain domains in complex Euclidean space. These domains are either higher-dimensional analogs of the Hartogs triangle or have such a generalized Hartogs triangle imbedded appropriately in them.


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

  • 1. Çelik, Mehmet and Straube, Emil J; Observations regarding compactness in the $ \overline {\partial }$-Neumann problem. Complex Var. Elliptic Equ. 54 (2009), no. 3-4, 173-186. MR 2513533 (2010j:32059)
  • 2. Chen, So-Chin and Shaw, Mei-Chi; Partial differential equations in several complex variables. AMS/IP Studies in Advanced Mathematics, 19. International Press, Boston, MA, 2001. MR 1800297 (2001m:32071)
  • 3. Chakrabarti, Debraj; Spectrum of the complex Laplacian on product domains. Proc. Amer. Math. Soc. 138 (2010), no. 9, 3187-3202. MR 2653944 (2011k:32058)
  • 4. Folland, G. B. and Kohn, J. J.; The Neumann problem for the Cauchy-Riemann complex. Annals of Mathematics Studies, No. 75. Princeton University Press, 1972. MR 0461588 (57:1573)
  • 5. Fu, Siqi and Straube, Emil J.; Compactness in the $ \overline {\partial }$-Neumann problem. Complex analysis and geometry (Columbus, OH, 1999), 141-160, Ohio State Univ. Math. Res. Inst. Publ. 9, de Gruyter, Berlin, 2001. MR 1912737 (2004d:32053)
  • 6. Fu, Siqi; Spectrum of the $ \overline {\partial }$-Neumann Laplacian on polydiscs. Proc. Amer. Math. Soc. 135 (2007), no. 3, 725-730. MR 2262868 (2007h:32056)
  • 7. Harvey, Reese and Polking, John; Removable singularities of solutions of linear partial differential equations. Acta Math. 125 (1970), 39-56. MR 0279461 (43:5183)
  • 8. Khanh, Tran Vu; A general method of weights in the $ \overline {\partial }$-Neumann problem. Ph.D. Thesis submitted to the University of Padova, Italy, in December 2009. Available online at arxiv.org, Article 1001.5093.
  • 9. Krantz, Steven G.; Compactness of the $ \overline {\partial }$-Neumann operator. Proc. Amer. Math. Soc. 103 (1988), no. 4, 1136-1138. MR 954995 (89f:32032)
  • 10. Ligocka, Ewa; The regularity of the weighted Bergman projections. Seminar on Deformations, Lecture Notes in Mathematics, 1165, Springer, Berlin, 1985, 197-203. MR 825756 (87d:32044)
  • 11. Straube, Emil J.; Lectures on the $ L^2$-Sobolev theory of the $ \overline {\partial }$-Neumann problem. ESI Lectures in Mathematics and Physics. European Mathematical Society, Zürich, 2010. MR 2603659 (2011b:35004)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 32W05

Retrieve articles in all journals with MSC (2010): 32W05


Additional Information

Debraj Chakrabarti
Affiliation: TIFR Centre for Applicable Mathematics, Sharadanagara, Chikkabommasandra, Bengaluru-560 065, India
Email: debraj@math.tifrbng.res.in

DOI: https://doi.org/10.1090/S0002-9939-2013-11504-2
Keywords: $\overline{\partial}$-Neumann operator, Hartogs triangle
Received by editor(s): July 13, 2011
Received by editor(s) in revised form: October 13, 2011
Published electronically: March 6, 2013
Communicated by: Franc Forstneric
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society