Minimization of a fractional perimeter-Dirichlet integral functional

  • Luis Caffarelli

    University of Texas at Austin, Department of Mathematics, 1 University Station, C1200 Austin, TX 78712-1082, USA
  • Ovidiu Savin

    Columbia University, Mathematics Department, 2990 Broadway, New York, NY 10027, USA
  • Enrico Valdinoci

    Weierstraß Institut für Angewandte Analysis und Stochastik, Mohrenstraße 39, D-10117 Berlin, Germany


We consider a minimization problem that combines the Dirichlet energy with the nonlocal perimeter of a level set, namely

with . We obtain regularity results for the minimizers and for their free boundaries using blow-up analysis. We will also give related results about density estimates, monotonicity formulas, Euler–Lagrange equations and extension problems.

Cite this article

Luis Caffarelli, Ovidiu Savin, Enrico Valdinoci, Minimization of a fractional perimeter-Dirichlet integral functional. Ann. Inst. H. Poincaré Anal. Non Linéaire 32 (2015), no. 4, pp. 901–924

DOI 10.1016/J.ANIHPC.2014.04.004