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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Korn inequality and divergence operator: Counterexamples and optimality of weighted estimates


Authors: Gabriel Acosta, Ricardo G. Durán and Fernando López García
Journal: Proc. Amer. Math. Soc. 141 (2013), 217-232
MSC (2010): Primary 26D10; Secondary 76D07
Published electronically: May 22, 2012
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Abstract: The Korn inequality and related results on solutions of the divergence in Sobolev spaces have been widely studied since the pioneering works by Korn and Friedrichs. In particular, it is known that this inequality is valid for Lipschitz domains as well as for the more general class of John domains. On the other hand, a few known counterexamples show that those results are not valid for certain bounded domains having external cusps.

The goal of this paper is to give very simple counterexamples for a class of cuspidal domains in $ \mathbb{R}^n$. Moreover, we show that these counterexamples can be used to prove the optimality of recently obtained results involving weighted Sobolev spaces.


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Additional Information

Gabriel Acosta
Affiliation: Departamento de Matemática and IMAS, CONICET, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, 1428 Buenos Aires, Argentina
Email: gacosta@dm.uba.ar

Ricardo G. Durán
Affiliation: Departamento de Matemática and IMAS, CONICET, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, 1428 Buenos Aires, Argentina
Email: rduran@dm.uba.ar

Fernando López García
Affiliation: Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, 1428 Buenos Aires, Argentina
Address at time of publication: Worcester Polytechnic Institute, 100 Institute Road, Worcester, Massachusetts 01609-2280
Email: flopezg@dm.uba.ar, flopezgarcia@wpi.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-2012-11408-X
PII: S 0002-9939(2012)11408-X
Keywords: Korn inequality, divergence operator, bad domains
Received by editor(s): June 18, 2010
Published electronically: May 22, 2012
Additional Notes: This research was supported by ANPCyT under grants PICT 2006-01307 and 2007-910, by Universidad de Buenos Aires under grant X070, and by CONICET under grant PIP 11220090100625. The first and second authors are members of CONICET, Argentina
Communicated by: Walter Craig
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.