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On \(L^p\)-viscosity solutions of bilateral obstacle problems with unbounded ingredients

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Abstract

The global equi-continuity estimate on \(L^p\)-viscosity solutions of bilateral obstacle problems with unbounded ingredients is established when obstacles are merely continuous. The existence of \(L^p\)-viscosity solutions is established via an approximation of given data. The local Hölder continuity estimate on the first derivative of \(L^p\)-viscosity solutions is shown when the obstacles belong to \(C^{1,\beta }\), and \(p>n\).

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Acknowledgements

The authors thank the referees for their careful reading, and several valuable comments, which help us to improve the original manuscript.

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Authors and Affiliations

  1. Deapartment of Applied Physics, Waseda University, Tokyo, 169-8555, Japan

    Shigeaki Koike

  2. Department of Mathematics, Waseda University, Tokyo, 169-8555, Japan

    Shota Tateyama

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  1. Shigeaki Koike
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  2. Shota Tateyama
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Correspondence to Shigeaki Koike.

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Communicated by Y. Giga.

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S. Koike: Supported in part by Grant-in-Aid for Scientific Research (No. 16H06339, 16H03948, 16H03946 ) of JSPS.

S. Tateyama: Supported by Grant-in-Aid for JSPS Research Fellow (No. 16J02399), and for Scientific Research (No. 16H06339).

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Koike, S., Tateyama, S. On \(L^p\)-viscosity solutions of bilateral obstacle problems with unbounded ingredients. Math. Ann. 377, 883–910 (2020). https://doi.org/10.1007/s00208-019-01854-z

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