Nonlocal Minimal Surfaces and Nonlocal Curvature

  • José M. Mazón
  • Julio Daniel Rossi
  • J. Julián Toledo
Part of the Frontiers in Mathematics book series (FM)


Recall that if a set E has minimal local perimeter in a bounded set Ω, then it has zero mean curvature at each point of ∂E ∩ Ω (see [51]), and the equation that says that the curvature is equal to zero is the Euler–Lagrange equation associated to the minimization of the perimeter of a set.


  1. 1.
    N. Abatangelo, E. Valdinoci, A notion of nonlocal curvature. Numer. Funct. Anal. Optim. 35, 793–815 (2014)MathSciNetCrossRefGoogle Scholar
  2. 24.
    A. Burchard, Cases of equality in the Riesz rearrangement inequality. Ann. Math. (2) 143(3), 499–527 (1996)MathSciNetCrossRefGoogle Scholar
  3. 29.
    L. Caffarelli, J.M. Roquejoffre, O. Savin, Nonlocal minimal surfaces. Commun. Pure Appl. Math. 63, 1111–1144 (2010)MathSciNetzbMATHGoogle Scholar
  4. 32.
    A. Chambolle, M. Morini, M. Ponsiglione, Nonlocal curvature flows. Arch. Ration. Mech. Anal. 218, 1263–1329 (2015)MathSciNetCrossRefGoogle Scholar
  5. 45.
    A. Figalli, N. Fusco, F. Maggi, V. Millot, M. Morini, Isoperimetric and stability properties of balls with respect to nonlocal energies. Commun. Math. Phys. 336, 441–507 (2015)CrossRefGoogle Scholar
  6. 47.
    R.L. Frank, R. Seiringer, Non-linear ground state representations and sharp Hardy inequalities. J. Funct. Anal. 255, 3407–3430 (2008)MathSciNetCrossRefGoogle Scholar
  7. 51.
    E. Giusti, Minimal Surface and Functions of Bounded Variation. Monographs in Mathematics, vol. 80 (Birkhäuser, Basel, 1984)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • José M. Mazón
    • 1
  • Julio Daniel Rossi
    • 2
  • J. Julián Toledo
    • 3
  1. 1.Departamento de Análisis MatemáticoUniversitat de ValènciaValenciaSpain
  2. 2.Departamento de MatemáticasUniversidad de Buenos AiresBuenos AiresArgentina
  3. 3.Departamento de Análisis MatemáticoUniversitat de ValènciaValènciaSpain

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