Minimization of a fractional perimeter-Dirichlet integral functional

We consider a minimization problem that combines the Dirichlet energy with the nonlocal perimeter of a level set, namely∫Ω|∇u(x)|2dx+Perσ({u>0},Ω), with σ∈(0,1)σ∈(0,1). We obtain regularity results for the minimizers and for their free boundaries ∂{u>0}∂{u>0} using blow-up analysis. We will also give related results about density estimates, monotonicity formulas, Euler–Lagrange equations and extension problems.