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A singular Monge-Ampère equation on unbounded domains

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Abstract

In this paper, we study the Dirichlet problem for a singular Monge-Ampère type equation on unbounded domains. For a few special kinds of unbounded convex domains, we find the explicit formulas of the solutions to the problem. For general unbounded convex domain Ω, we prove the existence for solutions to the problem in the space C∞(Ω) ∩ C(Ω̅). We also obtain the local C1/2-estimate up to the ∂Ω and the estimate for the lower bound of the solutions.

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Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant Nos. 11771237 and 41390452).

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Authors and Affiliations

  1. Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, China

    Huaiyu Jian & You Li

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  1. Huaiyu Jian
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  2. You Li
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Correspondence to Huaiyu Jian.

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Jian, H., Li, Y. A singular Monge-Ampère equation on unbounded domains. Sci. China Math. 61, 1473–1480 (2018). https://doi.org/10.1007/s11425-018-9351-1

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  • DOI: https://doi.org/10.1007/s11425-018-9351-1

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