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 ${\mathcal{E}}_s^p(u,\mathrm{\Omega }):=(1-s)(\frac{1}{2}{\int }_{\mathrm{\Omega }}{\int }_{\mathrm{\Omega }}\frac{\mathrm{\mid }u(x)-u(y){\mathrm{\mid }}^p}{\mathrm{\mid }x-y{\mathrm{\mid }}^{n+sp}}dxdy+{\int }_{\mathrm{\Omega }}{\int }_{{\mathbb{R}}^n\setminus \mathrm{\Omega }}\frac{\mathrm{\mid }u(x)-u(y){\mathrm{\mid }}^p}{\mathrm{\mid }x-y{\mathrm{\mid }}^{n+sp}}dxdy)+{\int }_{\mathrm{\Omega }}W(u(x))dx,$ where $p\in (1,+\mathrm{\infty })$, $s\in (0,1)$ and $W$ is a double-well potential with polynomial growth $m\in [p,+\mathrm{\infty })$ from the minima. The nonlocal estimates obtained are uniform as $s\to 1$. Moreover, making use of a $\mathrm{\Gamma }$-convergence result for ${\mathcal{E}}_s^p$ as $s\to 1$, we obtain density estimates for the minimizers of the limit energy functional, which takes the form ${\mathcal{E}}_1^p(u,\mathrm{\Omega }):=\frac{K_{n,p}}{2p}{\int }_{\mathrm{\Omega }}{\mid \mathrm{\nabla }u(x)\mid }^p+{\int }_{\mathrm{\Omega }}W(u(x))dx,$ for a suitable $K_{n,p}\in (0,+\mathrm{\infty })$.