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 ${\mathcal{E}}_s^p(u,\hat{I}\text{c◯}):=(1-s)\left(\frac{1}{2}{\int }_{\hat{I}}\text{c◯}{\int }_{\hat{I}}\text{c◯}\frac{|u(x)-u(y){|}^p}{|x-y{|}^{n+sp}}dxdy+{\int }_{\hat{I}}\text{c◯}{\int }_{{\mathbb{R}}^n\setminus \hat{I}\text{c◯}}\frac{|u(x)-u(y){|}^p}{|x-y{|}^{n+sp}}dxdy\right)+{\int }_{\hat{I}}\text{c◯}W(u(x))dx,$ 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 $s\to 1$. Moreover, making use of a $\hat{I}“$-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 ${\mathcal{E}}_1^p(u,\hat{I}\text{c◯}):=\frac{K_{n,p}}{2p}{\int }_{\hat{I}}\text{c◯}{\left|\nabla u(x)\right|}^p+{\int }_{\hat{I}}\text{c◯}W(u(x))dx,$ for a suitable $K_{n,p}\in (0,+\infty )$.

56 pages