A remark on nonlinear Dirac equations

Wang, Changyou

doi:10.1090/S0002-9939-10-10438-9

A remark on nonlinear Dirac equations
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Proc. Amer. Math. Soc. 138 (2010), 3753-3758 Request permission

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.

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