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Stationary Sets and Asymptotic Behavior of the Mean Curvature Flow with Forcing in the Plane

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Abstract

We consider the flat flow solutions of the mean curvature equation with a forcing term in the plane. We prove that for every constant forcing term the stationary sets are given by a finite union of disks with equal radii and disjoint closures. On the other hand for every bounded forcing term tangent disks are never stationary. Finally in the case of an asymptotically constant forcing term we show that the only possible long time limit sets are given by disjoint unions of disks with equal radii and possibly tangent.

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Acknowledgements

The research of N.F. has been funded by PRIN Project 2015PA5MP7. The research of V.J. was supported by the Academy of Finland grant 314227. N.F. and M.M. are members of Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of INdAM.

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Authors and Affiliations

  1. Dipartimento di Matematica e Applicazioni, Università di Napoli ”Federico II”, Naples, Italy

    Nicola Fusco

  2. Matematiikan ja Tilastotieteen Laitos, Jyväskylän Yliopisto, Jyväskylä, Finland

    Vesa Julin

  3. Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Università di Parma, Parma, Italy

    Massimiliano Morini

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  1. Nicola Fusco
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  2. Vesa Julin
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  3. Massimiliano Morini
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Correspondence to Nicola Fusco.

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Fusco, N., Julin, V. & Morini, M. Stationary Sets and Asymptotic Behavior of the Mean Curvature Flow with Forcing in the Plane. J Geom Anal 32, 53 (2022). https://doi.org/10.1007/s12220-021-00806-x

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