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

Abstract

In this paper we provide density estimates for a class of functions which includes all the minimizers of the energy $\mathcal{E}_s^p(u,\Omega):=(1-s)\left(\frac{1}{2}\int_{\Omega}\int_{\Omega}\frac{|u(x)-u(y)|^p}{|x-y|^{n+sp}}\,dx\,dy +\int_{\Omega}\int_{\mathbb{R}^n \setminus \Omega}\frac{|u(x)-u(y)|^p}{|x-y|^{n+sp}}\,dx\,dy\right)+\int_{\Omega}W(u(x))\,dx,$ where $p\in (1,+\infty)$, $s \in \left(0,1\right)$ and $W$ is a double-well potential with polynomial growth $m\in \left[p,+\infty\right)$ from the minima. The nonlocal estimates obtained are uniform as $s\to1$. Moreover, making use of a $\Gamma$-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 $\mathcal{E}_1^p(u,\Omega):=\frac{K_{n,p}}{2p}\int_{\Omega} \left|\nabla u(x)\right|^p+\int_{\Omega} W(u(x))\,dx,$ for a suitable $K_{n,p}\in (0,+\infty)$.