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 (Formula presented.), 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.