Density estimates for a variational model driven by the Gagliardo norm

Abstract

We prove density estimates for level sets of minimizers of the energy 2s\|u\|Hs()2+∫ W(u)\,dx, with s ∈ (0,1), where \|u\|Hs() denotes the total contribution from in the Hs norm of u, and W is a double-well potential. As a consequence we obtain, as 0, the uniform convergence of the level sets of u to either a Hs-nonlocal minimal surface if s∈(0, 1 2), or to a classical minimal surface if s ∈[ 1 2,1).