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Harnack inequality for nonlocal operators on manifolds with nonnegative curvature

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Abstract

We establish the Krylov–Safonov Harnack inequalities and Hölder estimates for fully nonlinear nonlocal operators of non-divergence form on Riemannian manifolds with nonnegative sectional curvatures. To this end, we first define the nonlocal Pucci operators on manifolds that give rise to the concept of non-divergence form operators. We then provide the uniform regularity estimates for these operators which recover the classical estimates for second order local operators as limits.

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Acknowledgements

The research of Jongmyeong Kim is supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (2016R1E1A1A01941893). Minhyun Kim gratefully acknowledges funding by the Deutsche Forschungsgemeinschaft (GRK 2235/2 2021 - 282638148). The research of Ki-Ahm Lee is supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP): NRF-2020R1A2C1A01006256.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Seoul National University, Seoul, 08826, Republic of Korea

    Jongmyeong Kim & Ki-Ahm Lee

  2. Fakultät für Mathematik, Universität Bielefeld, 33615, Bielefeld, Germany

    Minhyun Kim

  3. Korea Institute for Advanced Study, Seoul, 02455, Republic of Korea

    Ki-Ahm Lee

Authors
  1. Jongmyeong Kim
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  2. Minhyun Kim
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Correspondence to Jongmyeong Kim.

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Communicated by O. Savin.

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Kim, J., Kim, M. & Lee, KA. Harnack inequality for nonlocal operators on manifolds with nonnegative curvature. Calc. Var. 61, 22 (2022). https://doi.org/10.1007/s00526-021-02124-0

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