One-dimensionality of the minimizers in the large volume limit for a diffuse interface attractive/repulsive model in general dimension (Q2062975)

In this paper the authors study the diffuse interface generalized antiferromagnetic model with local/nonlocal attractive/repulsive terms in competition studied in [the authors et al., ``One-dimensionality of the minimizers for a diffuse interface generalized antiferromagnetic model in general dimension'', Preprint, \url{arXiv:1907.06419}]. Such energy has two parameters: \(\tau\), which represents the relative strength of the local term with respect to the nonlocal one, and \(\varepsilon\), which describes the transition scale in the Modica-Mortola type term. Restricting to the periodic box of length \(L\), in [loc. cit.] it has been shown that, in dimension \(d\ge 1\), for \(0<\tau,\varepsilon\ll1\), the minimizers are non-constant one-dimensional periodic functions. In this paper the authors show that periodicity and one-dimensionality of minimizers occur also in the zero temperature analogue of the thermodynamic limit, namely as \(L\to\infty\). The main energy is \[ \begin{multlined} \tilde{\mathcal{F}}_{J,L,\varepsilon} (u):= \frac{J}{L^d} \Big[ 3\varepsilon \int_{[0,L)^d} \|\nabla u\|_1^2 dx +\frac{3}{\varepsilon} \int_{[0,L)^d}W(u) dx\Big] \\ -\frac{1}{L^d}\int_{\mathbb{R}^d}\int_{[0,L)^d} |u(x+\zeta) -u(x)|^2 K(\zeta)dxd\zeta, \end{multlined} \tag{1} \] where \(\|y\|_1:=\sum_{i=1}^d|y_i|\) denotes the 1-norm, \(W(t)=t^2(1-t)^2\) is a double-well potential, and \(K_p(\zeta)=(\|\zeta\|_1+1)^{-p}\) is a radial kernel. For the energy (1), there exists a critical threshold \[ J_C = \int |\zeta_1| K(\zeta)d\zeta, \] such that minimizers are either \(u\equiv 0\) or \(u\equiv 1\) whenever \(J>J_C\). For \(\varepsilon>0\) and \(0<\tau :=J_C-J\ll 1\), on the other hand, non trivial minimizers are expected. In [loc. cit.] it has been shown that for small \(\varepsilon\) and \(\tau\), there exist periodic functions of finite period \(2h\) (such period might be nonunique) such that minimizers of (1) are all one-dimensional \(2h\)-periodic. How small the parameters \(\varepsilon\) and \(\tau\) have to be, can, however, depend on \(L\) itself. In this paper, the authors show that this is not the case, i.e. there exist \(\varepsilon_0\) and \(\tau_0\), independent of \(L\), such that the above result holds for all \(\varepsilon<\varepsilon_0\) and \(\tau<\tau_0\), provided that the kernel \(K_p\) satisfies \(p\ge d+2\). The main difficulty is that rigidity estimates for sets of finite energy in the asymptotic limit as \(\tau\) and \(\varepsilon\) tend to \(0\) degenerate as \(L\to \infty\). That is, the bound on the \(\Gamma\)-limit of \( \tilde{\mathcal{F}}_{\tau,L,\varepsilon}\) tends to be weaker and weaker as \(L\to \infty\).