Asymptotic behavior of the interface for entire vector minimizers in phase transitions (Q2149497)

This paper studies globally bounded entire minimizers \(u: {\mathbb{R}}^n \rightarrow {\mathbb{R}}^m \) of Allen-Cahn systems for phase transition potentials \(W\ge 0,\) \(W=0\) on a finite set of distinct points \(a_1, \dots, a_N\) and \(W(u)\sim \vert u-a_i\vert^\alpha\) near \(u=a_i,\) with \(\alpha\in (0,2).\) Some estimates are established on the diffuse interface \(I_0=\{x\in {\mathbb{R}}^n\,:\, \min_{i=1,\dots,N} \vert u(x)-a_i\vert >0\}\) and on the free boundary \(\partial I_0.\) In particular, the authors prove that there exists a positive constant \(r_0\) such that for \(r\ge r_0,\) \[ {\mathcal L}^n (I_0\cap B_r(0^n))\le c_1 r^{n-1} \quad \mbox{and}\quad {\mathcal H}^{n-1}(\partial^*I_0\cap B_r(0^n))\ge c_2r^{n-1}, \] where \(\partial^*\) denotes the De Giorgi reduced boundary. Moreover, if \(\alpha=1,\) then the upper bound \[ {\mathcal H}^{n-1}(\partial^*I_0\cap B_r(0^n))\le c_3r^{n-1} \] holds. The constants \(c_1, c_2, c_3\) are independent of \(r.\)