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Second order estimates for Hessian type fully nonlinear elliptic equations on Riemannian manifolds

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Abstract

We derive a priori estimates for second order derivatives of solutions to a wide class of fully nonlinear elliptic equations on Riemannian manifolds. There had been significant work in this direction, especially in connection with important geometric problems and other applications, but one had to make use of the special structures or needed extra assumptions which are more technical in nature to overcome various difficulties. In this paper we are able to remove most of the technical assumptions and derive the estimates under conditions which are close to optimal. These estimates enable one to prove existence results which are new even for bounded domains in Euclidean space.

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Author information

Authors and Affiliations

  1. Department of Mathematics, Ohio State University, Columbus, OH, 43210, USA

    Bo Guan

  2. Department of Mathematics, Harbin Institute of Technology, Harbin, 150001, China

    Heming Jiao

Authors
  1. Bo Guan
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  2. Heming Jiao
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Correspondence to Bo Guan.

Additional information

Communicated by A. Chang.

Dedicate to Professor Wu Congxin on his 80th birthday.

Research of B. Guan was supported in part by NSF grants. Research of H. Jiao was supported in part by a CRC graduate fellowship.

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Guan, B., Jiao, H. Second order estimates for Hessian type fully nonlinear elliptic equations on Riemannian manifolds. Calc. Var. 54, 2693–2712 (2015). https://doi.org/10.1007/s00526-015-0880-8

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  • DOI: https://doi.org/10.1007/s00526-015-0880-8

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