A curvature flow in the plane with a nonlocal term

  • Luis A. Caffarelli
  • Hui Yu


We study the geometric flow of a planar curve driven by its curvature and the normal derivative of its capacity potential. Under a convexity condition that is natural to our problem, we establish long term existence and large time asymptotics of this flow.

Mathematics Subject Classification

35K10 53A04 



Hui Yu would like to thank many colleagues and friends, in particular Hongjie Dong, Dennis Kriventsov and Tianling Jin, for fruitful discussions concerning this project, especially for the discussion about parabolic equations in one spatial dimension. He is also grateful to Yanyan Li and Jingang Xiong for their invitation to Beijing Normal University, where part of this work was conducted.


  1. 1.
    Athanasopoulos, I., Caffarelli, L.A., Kenig, C., Salsa, S.: An area-Dirichlet minimization problem. Commun. Pure Appl. Math. 54(4), 479–499 (2001)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Caffarelli, L.A., Córdoba, A.: An elementary regularity theory of minimal surfaces. Differ. Integral Equ. 6, 1–13 (1993)MathSciNetMATHGoogle Scholar
  3. 3.
    Gage, M., Hamilton, R.S.: The heat equation shrinking convex plane curves. J. Differ. Geom. 23, 69–96 (1986)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Huisken, G.: Flow by mean curvature of convex surfaces into spheres. J. Differ. Geom. 20, 237–266 (1984)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Yu, H.: Motion of sets by curvature and derivative of capacity potential, preprintGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsThe University of Texas at AustinAustinUSA
  2. 2.Department of MathematicsColumbia University in the City of New YorkNew YorkUSA

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