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Transactions of the American Mathematical Society

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An obstacle problem arising in large exponent limit of power mean curvature flow equation


Authors: Qing Liu and Naoki Yamada
Journal: Trans. Amer. Math. Soc. 372 (2019), 2103-2141
MSC (2010): Primary 35K93, 53C44, 35D40, 35B40
DOI: https://doi.org/10.1090/tran/7717
Published electronically: March 25, 2019
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Abstract: We study limit behavior for the level-set power mean curvature flow equation as the exponent tends to infinity. Under Lipschitz continuity, quasiconvexity, and coercivity of the initial condition, we show that the limit of the viscosity solutions can be characterized as the minimal supersolution of an obstacle problem involving the $ 1$-Laplacian. Such behavior is closely related to applications of power mean curvature flow in image denoising. We also discuss analogous behavior for other evolution equations with related applications.


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Additional Information

Qing Liu
Affiliation: Department of Applied Mathematics, Fukuoka University, Fukuoka 814-0180, Japan
Email: qingliu@fukuoka-u.ac.jp

Naoki Yamada
Affiliation: Department of Applied Mathematics, Fukuoka University, Fukuoka 814-0180, Japan
Email: nyamada@math.sci.fukuoka-u.ac.jp

DOI: https://doi.org/10.1090/tran/7717
Keywords: Power mean curvature flow, viscosity solutions, asymptotic behavior
Received by editor(s): August 30, 2017
Received by editor(s) in revised form: July 31, 2018, and September 2, 2018
Published electronically: March 25, 2019
Additional Notes: The work of the first author was supported by Japan Society for the Promotion of Science (JSPS), Grant-in-Aid for Young Scientists, No. 16K17635, and by the grant from Central Research Institute of Fukuoka University, No. 177102.
Article copyright: © Copyright 2019 American Mathematical Society