Planelike Interfaces in Long-Range Ising Models and Connections with Nonlocal Minimal Surfaces

Abstract

This paper contains three types of results:

• the construction of ground state solutions for a long-range Ising model whose interfaces stay at a bounded distance from any given hyperplane,

• the construction of nonlocal minimal surfaces which stay at a bounded distance from any given hyperplane,

• the reciprocal approximation of ground states for long-range Ising models and nonlocal minimal surfaces.

In particular, we establish the existence of ground state solutions for long-range Ising models with planelike interfaces, which possess scale invariant properties with respect to the periodicity size of the environment. The range of interaction of the Hamiltonian is not necessarily assumed to be finite and also polynomial tails are taken into account (i.e. particles can interact even if they are very far apart the one from the other). In addition, we provide a rigorous bridge between the theory of long-range Ising models and that of nonlocal minimal surfaces, via some precise limit result.

Appendices

Appendix 1: Proof of Lemma 5.3

In the present appendix, we provide a proof of Lemma 5.3 in full details. As mentioned right after its statement in Sect. 5, our argument is based on the strategies already followed in e.g. [19, 35, 41].

Throughout the section, we implicitly suppose conditions (1.18) and (1.19) to be in force. Although the result may in fact hold under weaker hypotheses, we always suppose for simplicity that K satisfies both these assumptions. However, we stress that none of the steps of the proof require the periodicity hypothesis (1.20) to be valid, that we therefore do not suppose to hold.

After these introductory remarks, we may now head to the proof of Lemma 5.3.

Proof of Lemma 5.3

First, notice that, by (1.19) and the fact that F has finite K-perimeter, the characteristic function $\chi _F$ belongs to the fractional Sobolev space $W^{s, 1}(\Omega )$. Hence, by standard density results (see e.g. [36, Theorem 1.4.2.1]), there exists a sequence $\{ \varphi _n \}_{n \in \mathbb {N}} \subset W^{s, 1}(\Omega ) \cap C^\infty (\overline{\Omega })$ such that

$$\begin{aligned} {{\varphi _n \rightarrow \chi _F\; \mathrm{in}~W^{s, 1}(\Omega ), \mathrm{as}~n \rightarrow +\infty .}} \end{aligned}$$

(9.1)

By using again (1.19), this ensures that

$$\begin{aligned} \lim _{n \rightarrow +\infty } \mathscr {K}_K(\varphi _n; \Omega , \Omega ) = \mathscr {K}_K(\chi _F; \Omega , \Omega ). \end{aligned}$$

(9.2)

For $t \in (0, 1)$, we let

$$\begin{aligned} F_n := \left( \{ \varphi _n > t \} \cap \overline{\Omega } \right) \cup (F \setminus \overline{\Omega }). \end{aligned}$$

Clearly, $F_n \setminus \overline{\Omega } = F \setminus \overline{\Omega }$, which proves (5.9).

Also, Morse-Sard’s Theorem tells that, for a.e. $t \in (0, 1)$, the boundary of the level set $\{ \varphi _n > t \}$ is a smooth hypersurface. Hence $\partial F_n$ is smooth inside $\overline{\Omega }$, which gives (5.8).

We now claim that for a.e. $t \in (0, 1)$ fixed,

$$\begin{aligned} \lim _{n \rightarrow +\infty } |F_n \Delta F| = 0, \end{aligned}$$

(9.3)

and

$$\begin{aligned} \lim _{n \rightarrow +\infty } {{\mathrm{Per}}}_K(F_n; \Omega ) = {{\mathrm{Per}}}_K(F; \Omega ), \end{aligned}$$

(9.4)

up to a subsequence, that is (5.10) and (5.11), respectively.

We begin by checking (9.3). Let $\tau \in (0, 1)$ and notice that

$$\begin{aligned} \varphi _n - \chi _F&> \tau \quad&\text{ in } (\{ \varphi _n> \tau \} \setminus F) \cap \Omega \\ {\text{ and } }\quad \chi _F - \varphi _n&\ge 1 - \tau \quad&\text{ in } (F \setminus \{ \varphi _n > \tau \}) \cap \Omega . \end{aligned}$$

From this, we deduce that

$$\begin{aligned} \Vert \varphi _n - \chi _F \Vert _{L^1(\Omega )}&\ge \int _{(\{ \varphi _n> \tau \} \setminus F) \cap \Omega } (\varphi _n(x) - \chi _F(x)) \, dx + \int _{(F \setminus \{ \varphi _n> \tau \}) \cap \Omega } (\chi _F(x) - \varphi _n(x)) \, dx \\&\ge \tau |(\{ \varphi _n> \tau \} \setminus F) \cap \Omega | + (1 - \tau ) |(F \setminus \{ \varphi _n> \tau \}) \cap \Omega | \\&\ge \min \{ \tau , 1 - \tau \} \, |(\{ \varphi _n > \tau \} \Delta F) \cap \Omega |. \end{aligned}$$

Therefore, using this and (9.1),

$$\begin{aligned} \{ \varphi _n > \tau \} \longrightarrow F \quad \text{ in } L^1(\Omega ), \text{ for } \text{ a.e. } \tau \in (0, 1). \end{aligned}$$

(9.5)

Claim (9.3) follows as a particular case by taking $\tau = t$ in formula (9.5) above and recalling that $F_n \setminus \overline{\Omega } = F \setminus \overline{\Omega }$.

Next, we address the convergence of the perimeters stated in (9.4). Thanks to (9.5) and Lemma 4.1, we have

$$\begin{aligned} \mathcal {L}_K(F \cap \Omega , \Omega \setminus F) \le \liminf _{n \rightarrow +\infty } \mathcal {L}_K(\{ \varphi _n> \tau \} \cap \Omega , \Omega \setminus \{ \varphi _n > \tau \}) \quad \text{ for } \text{ a.e. } \tau \in (0, 1), \end{aligned}$$

or, equivalently,

$$\begin{aligned} \mathscr {K}_K(\chi _F; \Omega , \Omega ) \le \liminf _{n \rightarrow +\infty } \mathscr {K}_K(\chi _{\{ \varphi _n > \tau \}}; \Omega , \Omega ) \quad \text{ for } \text{ a.e. } \tau \in (0, 1). \end{aligned}$$

(9.6)

By applying, in sequence, (9.2), the generalized Coarea Formula of Lemma 4.2, Fatou’s Lemma and (9.6), we compute

$$\begin{aligned} \mathscr {K}_K(\chi _F; \Omega , \Omega )= & {} \lim _{n \rightarrow +\infty } \mathscr {K}_K(\varphi _n; \Omega , \Omega ) = \lim _{n \rightarrow +\infty } \int _{-\infty }^{+\infty } \mathscr {K}_K(\chi _{\{\varphi _n> \tau \}}; \Omega , \Omega ) \, d\tau \\\ge & {} \int _0^1 \liminf _{n \rightarrow +\infty } \mathscr {K}_K(\chi _{\{\varphi _n > \tau \}}; \Omega , \Omega ) \, d\tau \\\ge & {} \int _0^1 \mathscr {K}_K(\chi _F; \Omega , \Omega ) \, d\tau = \mathscr {K}_K(\chi _F; \Omega , \Omega ). \end{aligned}$$

By this and, again, (9.6) we conclude that

$$\begin{aligned} \liminf _{n \rightarrow +\infty } \mathscr {K}_K(\chi _{\{\varphi _n > \tau \}}; \Omega , \Omega ) = \mathscr {K}_K(\chi _F; \Omega , \Omega ) \quad \text{ for } \text{ a.e. } \tau \in (0, 1), \end{aligned}$$

and thence

$$\begin{aligned} \lim _{n \rightarrow +\infty } \mathcal {L}_K(F_n \cap \Omega , \Omega \setminus F_n) = \mathcal {L}_K(F \cap \Omega , \Omega \setminus F). \end{aligned}$$

(9.7)

On the other hand, we claim that

$$\begin{aligned}&\lim _{n \rightarrow +\infty } \mathcal {L}_K(F_n \setminus \Omega , \Omega \setminus F_n) = \mathcal {L}_K(F \setminus \Omega , \Omega \setminus F)\nonumber \\ \text{ and } \quad&\lim _{n \rightarrow +\infty } \mathcal {L}_K(F_n \cap \Omega , \mathbb {R}^d \setminus (F_n \cup \Omega )) = \mathcal {L}_K(F \cap \Omega , \mathbb {R}^d \setminus (F \cup \Omega )), \end{aligned}$$

(9.8)

up to subsequences. To check the validity of (9.8), we first notice that, by (9.3), $\chi _{F_n} \rightarrow \chi _F$ a.e. in $\mathbb {R}^d$ (up to extracting a subsequence), as $n\rightarrow +\infty$. Therefore, in view of Lemma 4.3 we may apply the Lebesgue’s Dominated Convergence Theorem to get

$$\begin{aligned} \lim _{n \rightarrow +\infty } \mathcal {L}_K(F_n \setminus \Omega , \Omega \setminus F_n)&= \lim _{n \rightarrow +\infty } \int _{\Omega } \chi _{\mathbb {R}^d \setminus F_n}(x) \left( \int _{\mathbb {R}^d \setminus \Omega } \chi _{F_n}(y) K(x, y) \, dy \right) dx \\&= \int _{\Omega } \chi _{\mathbb {R}^d \setminus F}(x) \left( \int _{\mathbb {R}^d \setminus \Omega } \chi _{F}(y) K(x, y) \, dy \right) dx \\&= \mathcal {L}_K(F \setminus \Omega , \Omega \setminus F), \end{aligned}$$

and similarly for the limit on the second line of (9.8). The combination of (9.7) and (9.8) yields the convergence of the K-perimeters claimed in (9.4).

The proof of Lemma 5.3 is thus finished. $\square$

Appendix 2: Optimality of the Width of the Strip Given in (1.17)

The goal of this appendix is to show that, for large values of the periodicity scale $\tau$, the interface of the planelike ground states for powerlike interactions, as in (1.9), oscillates, in general, by a quantity proportional to $\tau$ (i.e., the conclusion in (1.17) of Theorem 1.4 cannot be improved).

Of course, one needs to construct an ad-hoc example to check this optimality. The idea to construct this counterexample comes from similar phenomena in minimal surfaces and minimal foliations, in which the oscillation is produced by the fact that the metric is nonflat. For simplicity, we present here a two-dimensional explicit example, which goes as follows.

Given $\tau \in 4\mathbb {N}+1$ (to be taken large in the subsequent construction), we define

$$\begin{aligned}&Q:=\left\{ -\frac{\tau -1}{2},\dots ,0,\dots ,\frac{\tau -1}{2}\right\} ^2,\\&\widehat{Q}:=\left\{ -\frac{\tau -1}{4},\dots ,0,\dots ,\frac{\tau -1}{4}\right\} ^2 \end{aligned}$$

and

$$\begin{aligned} \widehat{\mathcal {Q}} :=\left\{ (i,j)\in \mathbb {Z}^2\times \mathbb {Z}^2 { \text{ s.t. } \text{ there } \text{ exists } }(i^{\prime },j^{\prime })\in \widehat{Q}\times \widehat{Q} { \text{ for } \text{ which } } i-i^{\prime }=j-j^{\prime }\in \tau \mathbb {Z}^2 \right\} . \end{aligned}$$

We set

$$\begin{aligned} J_{ij}:= {\left\{ \begin{array}{ll} \displaystyle \frac{\Lambda }{|i-j|^{2+s}} &{} { \text{ if } \, (i,j)\in \widehat{\mathcal {Q}} \text{ and } i \ne j}\\ \quad \, \, \, \, 0 &{} { \text{ if }\, i = j,} \\ \displaystyle \frac{1}{|i-j|^{2+s}} &{}{ \text{ otherwise, }} \end{array}\right. } \end{aligned}$$

for $\Lambda >1$ (to be chosen conveniently large in the sequel).

We claim that any planelike ground state with rationally independent slope $\omega \in \mathbb {R}^2$ (with $\omega \cdot n\ne 0$ for any $n \in \mathbb {Z}^2$) for the Hamiltonian associated to this case with vanishing magnetic field (i.e. $h_i:=0$ in (1.15)) possesses oscillations of order $\tau$, for large $\tau$.

For this, we argue by contradiction and suppose that (1.16) holds true with $M=M(\tau )$ sublinear in $\tau$, namely there exists a ground state $u=u_{\omega ,\tau }$ such that

$$\begin{aligned} \partial u \subset \left\{ i \in \mathbb {Z}^2 : \frac{\omega }{|\omega |} \cdot i \in [0, M(\tau )] \right\} , \end{aligned}$$

(10.1)

and

$$\begin{aligned} \lim _{\tau \rightarrow +\infty }\frac{ M(\tau )}{\tau }=0. \end{aligned}$$

(10.2)

Since $\omega$ is irrational, any straight line $r_\omega$ with direction normal to $\omega$ will get arbitrarily close to $\tau \mathbb {Z}^2$. In particular, up to a translation, we may assume that the origin lies in a $\frac{\tau }{64}$-neighborhood of $r_\omega$, and, from (10.1) and (10.2), we can write

$$\begin{aligned}&u_{i}=-1 \hbox { for any } i \hbox { for which } \frac{\omega }{|\omega |} \cdot i\ge \frac{\tau }{32}\nonumber \\ {\text{ and } } \quad&u_{i}=1 {\hbox { for any } i \hbox { for } \mathrm{which}} \, \frac{\omega }{|\omega |} \cdot i\le - \frac{\tau }{32}, \end{aligned}$$

(10.3)

as long as $\tau$ is large enough.

We now reach a contradiction with the minimality of u by constructing a suitable competitor v with less energy. To this aim, we define

$$\begin{aligned} v_i:= {\left\{ \begin{array}{ll} -1 &{}{ \text{ if } }i\in \widehat{Q},\\ u_i &{} { \text{ otherwise }}. \end{array}\right. } \end{aligned}$$

Since u is supposed to be minimal, we have that

$$\begin{aligned} 0 \,\ge H_{\widehat{Q}}(u)-H_{\widehat{Q}}(v)= & {} \sum _{(i,j)\in \mathbb {Z}^4\setminus (\mathbb {Z}^2\setminus \widehat{Q})^2 } J_{ij} (v_iv_j -u_iu_j)\nonumber \\= & {} \Lambda \sum _{i,j\in \widehat{Q}} \frac{1-u_i u_j}{|i-j|^{2+s}}-2 \sum \limits _{\begin{array}{c} {i\in {\widehat{Q}}} \\ {j \not \in {\widehat{Q}}} \end{array}} \frac{(1+u_i)\, u_j}{|i-j|^{2+s}}\nonumber \\\ge & {} 4\Lambda \sum \limits _{\begin{array}{c} {i,j\in \widehat{Q}}\\ {\{u_i=1\}}\\ {\{u_j=-1\}} \end{array}} \frac{1}{|i-j|^{2+s}}-4 \sum \limits _{\begin{array}{c} {i\in \widehat{Q}}\\ {j\not \in \widehat{Q} } \end{array}} \frac{1}{|i-j|^{2+s}}. \end{aligned}$$

(10.4)

Now, from (10.3), we know that the number of sites $i\in \widehat{Q}$ for which $u_i=1$ is at least of the order $c\tau ^2$, and similarly that the number of sites $j\in \widehat{Q}$ for which $u_j=-1$ is at least of the order $c\tau ^2$, with $c>0$ universal. Consequently, we have that

$$\begin{aligned} \sum \limits _{\begin{array}{c} {i,j\in {\widehat{Q}}}\\ {\{u_{i}=1\}} \\ {\{u_j=-1\}} \end{array}} \frac{1}{|i-j|^{2+s}} \ge \frac{c^{\prime }\,\tau ^4}{\tau ^{2+s}}=c^{\prime }\,\tau ^{2-s}, \end{aligned}$$

(10.5)

for some $c^{\prime }>0$. On the other hand, using the index $k:=j-i$,

$$\begin{aligned} \sum \limits _{\begin{array}{c} {i\in {\widehat{Q}}}\\ {j\not \in {\widehat{Q}}} \end{array}} \frac{1}{|i-j|^{2+s}}&\le C\, \sum \limits _{\begin{array}{c} {|i|_\infty \le \frac{\tau -1}{2}}\\ {|j|_\infty \ge \frac{\tau +1}{2} } \end{array}} \frac{1}{|i-j|_\infty ^{2+s}}\le C\, \sum \limits _{\begin{array}{c} {|i|_\infty \le \frac{\tau -1}{2}}\\ {|k|_\infty \ge \frac{\tau +1}{2} -|i|_\infty } \end{array}} \frac{1}{|k|_\infty ^{2+s}} \\&\le C^{\prime }\, \sum _{|i|_\infty \le \frac{\tau -1}{2}} \left( \frac{\tau +1}{2} -|i|_\infty \right) ^{-s} = C^{\prime \prime }\, \sum _{\ell =0}^{\frac{\tau -1}{2}} \left( \frac{\tau +1}{2} -\ell \right) ^{-s} \,\ell \\&\le C^{\prime \prime }\,\tau \, \sum _{\ell =0}^{\frac{\tau -1}{2}} \left( \frac{\tau +1}{2} -\ell \right) ^{-s} \le C^{\prime \prime \prime }\,\tau ^{2-s}, \end{aligned}$$

for some C, $C^{\prime }$, $C^{\prime \prime }$, $C^{\prime \prime \prime }>0$.

Thus, we insert this and (10.5) into (10.4) and we find that

$$\begin{aligned} 0 \ge 4\tau ^{2-s}\,(c^{\prime }\, \Lambda -C^{\prime \prime \prime }), \end{aligned}$$

which is a contradiction if $\Lambda$ is sufficiently large.

Conclusions

After a short review of the classical Ising model, we considered in this paper a spin system with long-range interactions. We gave rigorous proofs of three types of results:

• the construction of ground state solutions whose phase separation stays at a bounded distance from any given hyperplane,

• the construction of nonlocal minimal surfaces which stay at a bounded distance from any given hyperplane,

• the asymptotic link between ground states of long-range Ising models and nonlocal minimal surfaces.