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A boundary value problem for the nonlinear Dirac equation on compact spin manifold

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Abstract

The positive energy theorem is a significant subject in general relativity theory. In Witten’s proof of this theorem, the solution of a free Dirac equation which is a spinor filed plays an important role. In order to prove the positive energy theorem for black holes, Gibbons, Hawking, Horowitz and Perry imposed a local boundary condition on the apparent horizon of the black hole. Then the Dirac equation under this boundary condition forms an elliptic boundary value problem. In fact, this kind of local boundary condition can be generally defined by a Chirality operator on the Dirac bundle over a spin manifold. In this paper, by establishing a proper analysis setting and developing variational arguments, we study a nonlinear Dirac equation on a compact spin manifold (M, g) which satisfies the local boundary condition with respect to a Chirality operator.

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Acknowledgements

The authors thank the referee for the useful suggestions. The work was supported partially by the National Science Foundation of China (NSFC 11331010).

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Authors and Affiliations

  1. Institute of Mathematics AMSS, School of Mathematical Sciences UCAS, Chinese Academy of Sciences, Beijing, 100190, China

    Yanheng Ding

  2. Yau Mathematical Sciences Center, Tsinghua University, Beijing, 100084, China

    Jiongyue Li

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  1. Yanheng Ding
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  2. Jiongyue Li
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Correspondence to Jiongyue Li.

Additional information

Communicated by A. Malchiodi.

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Ding, Y., Li, J. A boundary value problem for the nonlinear Dirac equation on compact spin manifold. Calc. Var. 57, 72 (2018). https://doi.org/10.1007/s00526-018-1350-x

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