Minimization of a fractional perimeter-Dirichlet integral functional

Abstract: We consider a minimization problem that combines the Dirichlet energy with the nonlocal perimeter of a level set, namely $$ \int_\Om |\nabla u(x)|^2\,dx+\Per\Big({u > 0},\Om \Big),$$ with $\sigma\in(0,1)$. We obtain regularity results for the minimizers and for their free boundaries $\p {u>0}$ using blow-up analysis. We will also give related results about density estimates, monotonicity formulas, Euler-Lagrange equations and extension problems.