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

 

A remark on nonlinear Dirac equations


Author: Changyou Wang
Journal: Proc. Amer. Math. Soc. 138 (2010), 3753-3758
MSC (2010): Primary 58J05
Published electronically: April 22, 2010
MathSciNet review: 2661574
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Abstract: For an $ n$-dimensional spin manifold $ M$ with a fixed spin structure and a spinor bundle $ \Sigma M$, we prove an $ \epsilon$-regularity theorem for weak solutions to the nonlinear Dirac equation

$\displaystyle \slashed\partial\psi= H_{jkl}\langle \psi^j, \psi^k\rangle \psi^l,$

of cubic nonlinearity. In particular, it implies that any weak solution is smooth when $ n=2$, which answers a question raised by Chen, Jost, and Wang.


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Additional Information

Changyou Wang
Affiliation: Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506
Email: cywang@ms.uky.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-10-10438-9
PII: S 0002-9939(10)10438-9
Received by editor(s): October 13, 2008
Received by editor(s) in revised form: January 20, 2009
Published electronically: April 22, 2010
Additional Notes: The author was partially supported by NSF grant 0601162
Communicated by: Matthew J. Gursky
Article copyright: © Copyright 2010 American Mathematical Society