Geodesic
Minimization of a fractional perimeter-Dirichlet integral functional
Annales de l'I.H.P. Analyse non linéaire, Tome 32 (2015) no. 4, pp. 901-924
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We consider a minimization problem that combines the Dirichlet energy with the nonlocal perimeter of a level set, namely

∫ Ω|∇u(x)| 2 dx+ Per σ ({u>0},Ω),
with σ∈(0,1). We obtain regularity results for the minimizers and for their free boundaries ∂{u>0} using blow-up analysis. We will also give related results about density estimates, monotonicity formulas, Euler–Lagrange equations and extension problems.

DOI : 10.1016/j.anihpc.2014.04.004
Keywords: Free boundary problems, Fractional minimal surfaces, Regularity theory 
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     author = {Caffarelli, Luis and Savin, Ovidiu and Valdinoci, Enrico},
     title = {Minimization of a fractional {perimeter-Dirichlet} integral functional},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {901--924},
     year = {2015},
     publisher = {Elsevier},
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Caffarelli, Luis; Savin, Ovidiu; Valdinoci, Enrico. Minimization of a fractional perimeter-Dirichlet integral functional. Annales de l'I.H.P. Analyse non linéaire, Tome 32 (2015) no. 4, pp. 901-924. doi: 10.1016/j.anihpc.2014.04.004

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