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Viscosity solutions for the crystalline mean curvature flow with a nonuniform driving force term

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  • Yoshikazu Giga

    (University of Tokyo)

  • Norbert Požár

    (Kanazawa University)

Abstract

A general purely crystalline mean curvature flow equation with a nonuniform driving force term is considered. The unique existence of a level set flow is established when the driving force term is continuous and spatially Lipschitz uniformly in time. By introducing a suitable notion of a solution a comparison principle of continuous solutions is established for equations including the level set equations. An existence of a solution is obtained by stability and approximation by smoother problems. A necessary equi-continuity of approximate solutions is established. It should be noted that the value of crystalline curvature may depend not only on the geometry of evolving surfaces but also on the driving force if it is spatially inhomogeneous.

Suggested Citation

  • Yoshikazu Giga & Norbert Požár, 2020. "Viscosity solutions for the crystalline mean curvature flow with a nonuniform driving force term," Partial Differential Equations and Applications, Springer, vol. 1(6), pages 1-26, December.
    • Handle: RePEc:spr:pardea:v:1:y:2020:i:6:d:10.1007_s42985-020-00040-0
    DOI: 10.1007/s42985-020-00040-0
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    References listed on IDEAS

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    1. Cecilia De Zan & Pierpaolo Soravia, 2016. "A Comparison Principle for the Mean Curvature Flow Equation with Discontinuous Coefficients," International Journal of Differential Equations, Hindawi, vol. 2016, pages 1-6, August.
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      Keywords

      35K67; 35D40; 35K55; 35B51; 35K93;
      All these keywords.

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