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Maximum principles and the method of moving planes for the uniformly elliptic nonlocal Bellman operator and applications

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Abstract

In this paper, we establish various maximum principles and develop the method of moving planes for equations involving the uniformly elliptic nonlocal Bellman operator. As a consequence, we derive multiple applications of these maximum principles and the moving planes method. For instance, we prove symmetry, monotonicity and uniqueness results and asymptotic properties for solutions to various equations involving the uniformly elliptic nonlocal Bellman operator in bounded domains, unbounded domains, epigraph or \({\mathbb {R}}^{n}\). In particular, the uniformly elliptic nonlocal Monge–Ampère operator introduced by Caffarelli and Charro (Ann PDE 1:4, 2015) is a typical example of the uniformly elliptic nonlocal Bellman operator.

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Acknowledgements

The authors are grateful to the referees for their careful reading and valuable comments and suggestions that improved the presentation of the paper.

Funding

Wei Dai is supported by the NNSF of China (No. 11971049) and the Fundamental Research Funds for the Central Universities.

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Authors and Affiliations

  1. School of Mathematical Sciences, Beihang University (BUAA), Beijing, 100083, People’s Republic of China

    Wei Dai

  2. Institute of Applied Mathematics, Chinese Academy of Sciences, Beijing, 100190, People’s Republic of China

    Guolin Qin

  3. University of Chinese Academy of Sciences, Beijing, 100049, People’s Republic of China

    Guolin Qin

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  1. Wei Dai
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  2. Guolin Qin
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Dai, W., Qin, G. Maximum principles and the method of moving planes for the uniformly elliptic nonlocal Bellman operator and applications. Annali di Matematica 200, 1085–1134 (2021). https://doi.org/10.1007/s10231-020-01027-9

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