A remark on nonlinear Dirac equations
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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 \[ \not \partial \psi = H_{jkl}\langle \psi ^j, \psi ^k\rangle \psi ^l,\] 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.References
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Additional Information
- Changyou Wang
- Affiliation: Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506
- Email: cywang@ms.uky.edu
- 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
- © Copyright 2010 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 138 (2010), 3753-3758
- MSC (2010): Primary 58J05
- DOI: https://doi.org/10.1090/S0002-9939-10-10438-9
- MathSciNet review: 2661574