Density estimates for a (non)local variational model with degenerate double-well potential

In this paper we provide density estimates for a class of functions which includes all the minimizers of the energy (Formula presented.) where p∈(1,+∞), s∈0,1 and W is a double-well potential with polynomial growth m∈p,+∞ from the minima. The nonlocal estimates obtained are uniform as s→1. Moreover, making use of a Γ-convergence result for Esp as s→1, we obtain density estimates for the minimizers of the limit energy functional, which takes the form (Formula presented.) for a suitable Kn,p∈(0,+∞).