Minimization of a fractional perimeter-Dirichlet integral functional

Caffarelli, Luis; Savin, Ovidiu; Valdinoci, Enrico

Annales de l'I.H.P. Analyse non linéaire, Tome 32 (2015) no. 4, pp. 901-924.

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

$$\underset{\Omega }{\int }{\left|\nabla u\right(x\left)\right|}^{2}dx+{\mathrm{Per}}_{\sigma }(\{u>0\},\Omega ),$$

with $\sigma \in (0,1)$. We obtain regularity results for the minimizers and for their free boundaries $\partial \{u>0\}$ using blow-up analysis. We will also give related results about density estimates, monotonicity formulas, Euler–Lagrange equations and extension problems.

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