Asymptotic behavior of the free interface for entire vector minimizers in phase transitions

Abstract

We study globally bounded entire minimizers $u:\mathbb{R}^n\rightarrow\mathbb{R}^m$ of Allen-Cahn systems for potentials $W\geq 0$ with $\{W=0\}=\{a_1,...,a_N\}$ and $W(u)\sim |u-a_i|^\alpha$ near $u=a_i$, $0<\alpha<2$. Such solutions are, over large regions, identically equal to some zeroes of the potential $a_i$'s. We establish the estimates \begin{equation*} \mathcal{L}^n(I_0\cap B_r(x_0))\leq c_1r^{n-1},\quad \mathcal{H}^{n-1}(\partial^* I_0\cap B_r(x_0))\geq c_2r^{n-1}, \quad r\geq r_0(x_0) \end{equation*} for the diffuse interface $I_0:=\{x\in\mathbb{R}^n: \min_{1\leq i\leq N}|u(x)-a_i|>0\}$ and the free boundary $\partial I_0$. Furthermore, if $\alpha=1$ we establish the upper bound \begin{equation*} \mathcal{H}^{n-1}(\partial^* I_0\cap B_r(x_0))\leq c_3r^{n-1}, \quad r\geq r_0(x_0). \end{equation*}