Density estimates for a variational model driven by the Gagliardo norm

We prove density estimates for level sets of minimizers of the energy\u3b52s{norm of matrix}u{norm of matrix}Hs(\u3a9)2+ 2b\u3a9W(u)dx, with s 08(0, 1), where {norm of matrix}u{norm of matrix}Hs(\u3a9) denotes the total contribution from \u3a9 in the Hs norm of u, and W is a double-well potential.As a consequence we obtain, as \u3b5\u21920+, the uniform convergence of the level sets of u to either an Hs-nonlocal minimal surface if s 08(0,12), or to a classical minimal surface if s 08[12,1)

Γ-convergence for nonlocal phase transitions

We discuss the \u393-convergence, under the appropriate scaling, of the energy functional 25u 25 Hs(\u3a9) 2+ 2b \u3a9W(u)dx, with s 08(0,1), where 25u 25H s (\u3a9) denotes the total contribution from \u3a9 in the H s norm of u, and W is a double-well potential. When s 08[1/2,1), we show that the energy \u393-converges to the classical minimal surface functional - while, when s 08(0,1/2), it is easy to see that the functional \u393-converges to the nonlocal minimal surface functional

All functions are locally ss-harmonic up to a small error

We show that we can approximate every function f 08 Ck (B1) by an s-harmonic function in B1 that vanishes outside a compact set. That is, s-harmonic functions are dense in Cloc k. This result is clearly in contrast with the rigidity of harmonic functions in the classical case and can be viewed as a purely nonlocal feature

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 \u3c3 08 (0,1). We obtain regularity results for the minimizers and for their free boundaries 02u>0 using blow-up analysis. We will also give related results about density estimates, monotonicity formulas, Euler-Lagrange equations and extension problems