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Short time existence and smoothness of the nonlocal mean curvature flow of graphs

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Abstract

We consider the geometric evolution problem of entire graphs moving by fractional mean curvature. For this, we study the associated nonlocal quasilinear evolution equation satisfied by the family of graph functions. We establish, using an analytic semigroup approach, short time existence, uniqueness and optimal Hölder regularity in time and space of classical solutions of the nonlocal equation, depending on the regularity of the initial graph. The method also yields \(C^\infty \)-smoothness estimates of the evolving graphs for positive times.

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Acknowledgements

This work is supported by the Alexander von Humboldt foundation and the German Academic Exchange Service (DAAD). The authors would like to thank Sven Jarohs, Esther Cabezas-Rivas and Ousmane Seydi for many fruitful discussions.

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Authors and Affiliations

  1. African Institute for Mathematical Sciences in Senegal, KM 2, Route de Joal, B.P. 14 18, Mbour, Senegal

    Anoumou Attiogbe & Mouhamed Moustapha Fall

  2. Institut für Mathematik, Goethe-Universität, Frankfurt, Robert-Mayer-Str. 10, 60629, Frankfurt, Germany

    Tobias Weth

Authors
  1. Anoumou Attiogbe
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  2. Mouhamed Moustapha Fall
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  3. Tobias Weth
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Correspondence to Mouhamed Moustapha Fall.

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Attiogbe, A., Fall, M.M. & Weth, T. Short time existence and smoothness of the nonlocal mean curvature flow of graphs. Math. Ann. (2023). https://doi.org/10.1007/s00208-023-02737-0

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