Asymptotic expansion of a
nonlocal phase transition energy

In this paper, we initiate the study of the asymptotic behavior as $\varepsilon\searrow 0$ of the nonlocal phase coexistence model

where $n\geqslant 1$, $\Omega\subset\mathbb{R}^{n}$ is a bounded domain, $W$ is a double-well potential (see (1.2)), $\varepsilon>0$ is a small phase parameter, and $a_{\varepsilon}$, $b_{\varepsilon}>0$ take into account the scaling properties of the energy functional in terms of the fractional parameter $s\in(0,1)$ (see (1.3)). In particular, the effect of the potential energy, which favors the pure phases of the system, is balanced by the kinetic energy that takes into account the long-range particle interactions and is ferromagnetic.

The energy (1.1) was considered in [SV-gamma, SV-dens]. In the former, the authors prove that, under appropriate scaling, the energy $\Gamma$-converges to the classical perimeter functional when $s\in\left[\frac{1}{2},1\right)$ and to the nonlocal perimeter functional when $s\in\left(0,\frac{1}{2}\right)$.

In this work, we initiate the analysis of the higher-order asymptotic behavior as $\varepsilon\searrow 0$. This is important in the study of slow motion of interfaces in the corresponding evolutionary problem and showcases improved convergence rates for the minimum values. Different than the interior result in [SV-gamma], we also take into account the nonlocal Dirichlet boundary conditions by restricting the class of minimizers to those satisfying $u=g$ a.e. on $\Omega^{c}$ for a given function $g$ (see (1.16)). In this sense, the novelties provided by our work cover two distinct but intertwined aspects: on the one hand, we analyze the effect of the boundary data on the limit energy functional; on the other hand, we provide a higher-order expansion of the limit functional with respect to the perturbation parameter $\varepsilon$.

Let us now precisely describe our setting mathematically and state our main results.

We let $\Omega\subset\mathbb{R}^{n}$ be a bounded domain of class $C^{2}$. Roughly speaking, this domain is the “container” in which the phase transition takes place. Notice in (1.1) that the set

collects all the couples $(x,y)\in\mathbb{R}^{n}\times\mathbb{R}^{n}$ for which at least one entry lies in $\Omega$: that is, $\mathcal{E}_{\varepsilon}$ takes into account all the possible long-range interactions which involve particles inside the container. As is customary, $\Omega^{c}$ denotes the complement of $\Omega$ in $\mathbb{R}^{n}$.

To write the total energy of the system in a convenient notation, given a measurable function $u:\mathbb{R}^{n}\to\mathbb{R}$ and a measurable set $A\subset\mathbb{R}^{2n}$, we write

In particular, if $A=A_{1}\times A_{2}$ with $A_{1}$ and $A_{2}$ measurable subsets of $\mathbb{R}^{n}$, we write

We assume that $W\in C^{2}(\mathbb{R})$ is a double-well potential satisfying

Following [SV-gamma, SV-dens], the scaling in the fractional Allen–Cahn energy (1.1) is given by

As $s\nearrow 1$, the functional in (1.1) recovers the classical Allen–Cahn phase transition energy.

Let $g:\mathbb{R}^{n}\to[-1,1]$ be a given Lipschitz continuous function. We define

We establish a first-order asymptotic expansion of $m_{\varepsilon}$ for any fixed $s\in(0,1)$.

For $s\in(0,\frac{1}{2})$, we prove a first-order expansion in any spatial dimension. Morally speaking, we show that $m_{\varepsilon}/\varepsilon$ approximates the fractional perimeter of a nonlocal minimal surface in $\Omega$ with prescribed Dirichlet boundary data as $\varepsilon\searrow 0$. See the seminal paper [CRS] for more on nonlocal minimal surfaces. The precise result goes as follows.

Here above and in what follows, $\chi_{A}$ denotes the characteristic function associated to a measurable set $A\subset\mathbb{R}^{n}$.

Notably, for $s\in(0,\frac{1}{2})$ and in dimension $n=1$, we can find a specific scenario in which $m_{\varepsilon}$ does not have a meaningful second-order expansion and, in fact, intermediate fractional powers of $\varepsilon$ could be needed for a meaningful expansion (see Theorem 1.9 and Corollary 1.10 below).

The case $s\in\left[\frac{1}{2},1\right)$ is fundamentally different as the long-range interactions are lost in the limit as $\varepsilon\searrow 0$. In this paper, we provide the first-order asymptotic expansion of $m_{\varepsilon}$ in dimension $n=1$. Since the proof is already quite involved, we leave the general case $n\geqslant 2$ and the higher-order asymptotic expansion of $m_{\varepsilon}$ as future work.

We use the standard notation $P(E,\Omega)$ to denote the (classical) perimeter of a set $E$ in an open set $\Omega\subset\mathbb{R}$, see [Giusti].

For a function $u:\mathbb{R}\to\mathbb{R}$ such that $u\big{|}_{\Omega}=\chi_{E}-\chi_{E^{c}}$ for some set $E\subset\mathbb{R}$ with $P(E,\Omega)<+\infty$, the trace of $u$ along $\partial\Omega$, given by

is well-defined. As an abuse of notation in this setting, we often write $u(x)$ in place of $Tu(x)$, even when $x\in\partial\Omega$.

When $s\in\left[\frac{1}{2},1\right)$, we also take an extra assumption to control the behavior of the functions considered near the boundary. This is a technical assumption that we aim at relaxing in a forthcoming work, but, roughly speaking, it is used in our setting to prevent additional oscillations of minimizing heteroclinics.

This technical hypothesis goes as follows. Suppose that $\Omega=(\bar{x}_{1},\bar{x}_{2})\subset\mathbb{R}$. Let $\kappa\in\left[0,\frac{|\Omega|}{2}\right)$. Consider the class $Y_{\kappa}$ of measurable functions $u:\mathbb{R}\to\mathbb{R}$ such that, for $i=1,2$,

We point out that when $\kappa=0$, the set $Y_{0}$ consists of all the measurable functions $u:\mathbb{R}\to\mathbb{R}$.

Consider

In this setting, the first-order of the asymptotic expansion of the nonlocal phase coexistence functional goes as follows:

Roughly speaking, for any $\gamma\in(-1,1)$ and $s\in\left(\frac{1}{2},1\right)$, the penalization functional $\Psi$ mentioned in Theorem 1.2 is given by

where $X_{\gamma}$ is the class of functions $w\in H^{s}(\mathbb{R})$ such that $-1\leqslant w\leqslant\gamma$, $w(-\infty)=-1$, and $w=\gamma$ in $\mathbb{R}^{+}$, and similarly for $\Psi(+1,\gamma)$. That is, $\Psi(u(x),g(x))$ for $x\in\partial\Omega$ is the minimum energy connecting the interior value $u(x)\in\{\pm 1\}$ (in the trace sense) to the boundary value $\gamma=g(x)\in(-1,1)$.

Thus, in order to construct the penalization function $\Psi$, we need to construct the heteroclinic function connecting $\pm 1$ to $\gamma\in(-1,1)$. This construction is also somewhat delicate and does not follow trivially from the constructions of the heteroclinic function connecting $-1$ to $+1$ put forth in [CABSI, PAL]. Indeed, in our setting, cutting methods and monotonicity considerations are not available. Moreover, since $\gamma$ is not an equilibrium for the potential $W$, a bespoke analysis is needed.

The case $s=\frac{1}{2}$ is even more delicate, due to the presence of infinite energy contributions. To overcome this difficulty, the minimal layer solution when $s=\frac{1}{2}$ will be obtained by a limit procedure from the cases $s\in\left(\frac{1}{2},1\right)$.

We stress that the construction of this minimal heteroclinic connection is an important ingredient for the recovery sequence in the $\Gamma$-convergence expansion that we propose and introduces a novelty with respect to the available methods in the nonlocal setting. Indeed, while the interior estimates of the $\Gamma$-limits rely on the existing literature (see [SV-gamma]), the control of the boundary terms requires an ad-hoc analysis and an appropriate interpolation that involves this new heteroclinic connection.

We now describe in detail the mathematical setting related to this layer solution. Towards this end, for any $A\subseteq\mathbb{R}$, we define

and

where $B_{R}^{-}:=(-R,0)$.

Given $\gamma\in(-1,1)$, we define $X_{\gamma}$ to be the closure, with respect to the $H^{s}$-seminorm, of the set

Loosely speaking, $X_{\gamma}$ is the family of functions in $H^{s}(\mathbb{R})$ that connect $-1$ at $-\infty$ to $\gamma$ in $\mathbb{R}^{+}$.

With this notation, we can establish the existence of minimizers to the energy $\mathcal{G}_{s}$ in the class $X_{\gamma}$:

A similar result holds when $-1$ is replaced by $1$.

We faced several difficulties in proving the existence result in Theorem 1.3. First, if $\gamma>0$, then minimizers may instead prefer to connect $1$ to $\gamma$ in the sense that $w_{0}(-\infty)=1$. For instance, due to the boundary condition $w_{0}\equiv\gamma$ in $\mathbb{R}^{+}$, classical sliding methods as used in [PAL] are ineffectual in proving both monotonicity and uniqueness. To solve the issue, we impose the additional constraint $w\leqslant\gamma$ in the class $X_{\gamma}$. Then, since the energy on the right-hand side of (1.9) is finite for $s\in\left(\frac{1}{2},1\right)$, existence is proved using compactness. However, for $s=\frac{1}{2}$, the energy is infinite and must be defined through an appropriate rescaling. In the latter case, we give a novel approach to proving existence by sending $s\searrow\frac{1}{2}$. For this, we first prove uniform asymptotics in $s$ near $\frac{1}{2}$ for the heteroclinic orbits connecting $-1$ at $-\infty$ to $+1$ at $-\infty$. See Sections 6 and 7 for full details.

Theorems 1.1 and 1.2 are a consequence of a general approach for asymptotic developments in the classical setting of $\Gamma$-convergence which we now describe.

The notion of asymptotic development in the sense of $\Gamma$-convergence was introduced by Anzellotti and Baldo in [Baldo1], see also [Baldo2]. We will present the definitions and details surrounding asymptotic development in our setting. For reference, let us first recall the definition of $\Gamma$-convergence, see [Baldo1, Definition 1.1].

Consider the space $X$ of all the measurable functions $u:\mathbb{R}^{n}\to\mathbb{R}$ such that the restriction of $u$ to $\Omega$ belongs to $L^{1}(\Omega)$. We endow $X$, as well as its subspaces, with the metric of $L^{1}(\Omega)$:

Notice that the values outside $\Omega$ are neglected in this procedure.

We comprise the Dirichlet datum inside the functional by defining

and

Notice that $X_{g}$ is the functional space used in the construction of $m_{\varepsilon}$ in (1.4), so that

We also set $\mathcal{F}^{(0)}_{\varepsilon}:=\mathcal{F}_{\varepsilon}$ and ${\mathcal{U}}_{-1}:=X$.

Following [Baldo1, Baldo2] (see in particular [Baldo1, pages 109–110]), the asymptotic development of order $k\in\mathbb{N}$, written as

holds in the sense of $\Gamma$-convergence if

1. (1)

   $\mathcal{F}^{(0)}=\Gamma-\displaystyle\lim_{\varepsilon\searrow 0}\mathcal{F}_{\varepsilon}$ in ${\mathcal{U}}_{-1}$;

2. (2)

   for any $j\in\{0,\dots,k-1\}$, we have that $\mathcal{F}^{(j+1)}=\Gamma-\displaystyle\lim_{\varepsilon\searrow 0}\mathcal{F}_{\varepsilon}^{(j+1)}$ in ${\mathcal{U}}_{j}$, where

   			$\displaystyle{\mathcal{U}}_{j}:=\{u\in{\mathcal{U}}_{j-1}{\mbox{ s.t. }}\mathcal{F}^{(j)}(u)=m_{j}\}$
   	with		$\displaystyle m_{j}:=\inf_{{\mathcal{U}}_{j-1}}\mathcal{F}^{(j)}$
   	and		$\displaystyle\mathcal{F}^{(j+1)}_{\varepsilon}:=\frac{\mathcal{F}^{(j)}_{\varepsilon}-m_{j}}{\varepsilon}.$

One can show (see [Baldo1, pages 106 and 110]) that

and

Therefore, the asymptotic development provides a refined selection criteria for minimizers of $\mathcal{E}_{\varepsilon}$.

In this setting, we prove the following asymptotic behavior of $\mathcal{E}_{\varepsilon}$ in (1.1) in the sense of $\Gamma$-convergence.

For $s\in\left[\frac{1}{2},1\right)$, we recall (1.7) and define

Here, $X_{g}\cap Y_{\kappa}$ is used in the construction of $m_{\varepsilon}^{\kappa}$ in (1.8). In this setting, we prove the following first-order asymptotic of $\mathcal{F}_{\varepsilon}$.

In this framework and recalling (1.17), Theorems 1.1 and 1.2 are a consequence of Theorems 1.5 and 1.6, respectively.

For $s\in\left(0,\frac{1}{2}\right)$, notice in Theorems 1.1 and 1.5 that the nonlocal energies $\mathcal{F}_{\varepsilon}$ with exterior boundary conditions $u\equiv g$ in $\Omega^{c}$ give rise to a nonlocal energy with exterior boundary conditions. In contrast, for $s\in\left[\frac{1}{2},1\right)$, we see in Theorems 1.2 and 1.6 that the limiting energy localizes both in the interior and at the boundary, in the sense that the penalization energy only sees $g$ on $\partial\Omega$.

For $s\in(0,\frac{1}{2})$ and in a specific one-dimensional setting, we prove by direct calculation that $\mathcal{F}_{\varepsilon}$ does not have a meaningful asymptotic development of order $2$ or in fact any non-integer order $\mu+1>2-2s$. We present the precise result here. An analogous statement is expected to hold in a more general setting and is left for future work.

Since the inaugural works [Baldo1, Baldo2] there have been several papers devoted to asymptotic development in the local setting. See for instance [Braides] for additional details. See also [MR759767, MR2971613, MR3385247], where a second-order $\Gamma$-convergence expansion is produced for a classical, one-dimensional, Modica-Mortola energy functional, and [DalMaso, Leoni], for the higher-dimensional case.

To the best of our knowledge, we are the first to consider asymptotic development in the nonlocal setting. Our main obstacle to overcome is understanding the penalization function $\Psi$ and its role at the boundary for $s\in\left[\frac{1}{2},1\right)$.

The rest of the paper is organized as follows. First, in Section 2, we prove $\Gamma$-convergence to $\mathcal{F}^{(0)}$ as described in Remark 2.2. Then, we set notation for first-order $\Gamma$-convergence in Section 3. The proof of Theorem 1.5 and a discussion surrounding Theorem 1.9 for $s\in\left(0,\frac{1}{2}\right)$ is in Section 4. After that, we will assume for the remainder of the paper that $s\in\left[\frac{1}{2},1\right)$ and set notation in Section 5. Background and preliminaries on heteroclinic connections are provided in Section 6. Section 7 contains the proof of Theorem 1.3 and the construction of the penalization function $\Psi$. To prove Theorem 1.6 for $s\in\left[\frac{1}{2},1\right)$, we establish the $\liminf$-inequality in Section 8 (see Proposition 8.3) and the $\limsup$-inequality in Section 9 (see Proposition 9.10). Lastly, we collect some auxiliary energy estimates in Appendix A.