1. Introduction

Let $J := (0,\, 1)$ and let $u\colon \mathbb {R}^N \to \mathbb {R}$ be an $L^2$ function. Given the family of kernels $\{\rho _{\varepsilon} \}_{\varepsilon \in J}$, with $\rho _{\varepsilon} \colon \mathbb {R}^N \to [0,\,+\infty )$ measurable, we consider the energy functionals:

(1.1)\begin{equation} \mathscr{F}_{\varepsilon} [ u ] := \frac{1}{2}\int_{\mathbb{R}^N \times \mathbb{R}^N} \rho_{\varepsilon} (y-x) \frac{| u (y) - u (x) |^2}{| y - x |^2} \,\mathrm{d} y \,\mathrm{d} x. \end{equation}

We aim to characterize the class of kernels such that for every $u\in H^1(\mathbb {R}^N)$ the family $\{\mathscr {F}_{\varepsilon} [ u ]\}$ converges to (a variant of) $\| \nabla u \|^2_{L^2(\mathbb {R}^N)}$ as $\varepsilon \to 0$, see theorem 1.1.

Our study follows the line of research initiated in the renowned paper [Reference Bourgain, Brezis and Mironescu5]. The motivation advanced by the authors was the analysis of the Gagliardo seminorms:

\[ [u]^p_s := \int_{\mathbb{R}^N \times \mathbb{R}^N} \frac{| u (y) - u (x) |^{p}}{| y - x |^{N+sp}} \,\mathrm{d} y \,\mathrm{d} x, \quad \text{with}\ p\in(1,+\infty),s\in(0,1) \]

as $s\to 1$. They studied the asymptotics as $\varepsilon \to 0$ of double integrals with the same structure as the one in (1.1) for a family $\{\rho _{\varepsilon} \}\subset L^1(\mathbb {R}^N)$ of radial kernels and a general exponent $p\in (1,\,+\infty )$, and they proved that the Sobolev seminorm $\|\nabla u\|^p_{L^p(\mathbb {R}^N)}$ is retrieved in the limit. The case of the Gagliardo seminorms may be treated analogously upon taking some extra care of the tails of the fractional kernel (see, e.g. [Reference Leoni and Spector13, Sec. 1]).

The literature on nonlocal-to-local formulas has become extremely vast, and a detailed overview is beyond the scope of our contribution. Here, we restrict ourselves to the research that is most close in spirit to [Reference Bourgain, Brezis and Mironescu5]. The gap left open for the case $p=1$ was filled in [Reference Dávila9], where a characterization of functions of bounded variation was provided (see also [Reference Leoni and Spector13, Reference Ponce19]). The case of vector fields of bounded deformations was later addressed in [Reference Mengesha15] by considering a suitable symmetrization of the functionals in (1.1) (see also [Reference Mengesha and Du16] for the asymptotics of nonlocal elastic energies of peridynamic-type and [Reference Scott and Mengesha21] for a study of fractional Korn inequalities). The analysis of the asymptotic behaviour in the sense of $\Gamma$-convergence [Reference Dal Maso8] of the fractional perimeter functionals introduced in [Reference Caffarelli, Roquejoffre and Savin7] was undertaken in [Reference Ambrosio, De Philippis and Martinazzi2], and then extended in multiple directions by several contributions, e.g. [Reference Berendsen and Pagliari4, Reference De Luca, Kubin and Ponsiglione10, Reference Mazón, Rossi and Toledo14, Reference Pagliari17]. Finally, we point out that a general variational framework for the analysis of (static and dynamic) multiscale problems that feature nonlocal interactions has been very recently considered in the monograph [Reference Alicandro, Ansini, Braides, Piatnitski and Tribuzio1], again for kernels that, in our notation, are required to form a definitively bounded sequence in $L^1$.

A common trait of the works above is that they only concern sufficient conditions for the nonlocal-to-local formulas to hold. In the specific case of the functionals in (1.1) (see theorem 5.4 for a prototypical statement), this means that, given a measurable map $\rho _{\varepsilon} \colon \mathbb {R}^N \to [0,\,+\infty )$ for every $\varepsilon \in J$, a set of conditions on the family $\{\rho _{\varepsilon} \}_{\varepsilon \in J}$ is prescribed, so that the following can be deduced: there exist an infinitesimal sequence $\{\varepsilon _k\}\subset J$ and a positive Radon measure $\lambda$ on the unit sphere $\mathbb {S}^{N-1}$ that depends only on $\{\rho _{\varepsilon _k}\}_{\varepsilon _k\in J}$, such that for every $u\in H^1(\mathbb {R}^N)$:

(1.2)\begin{equation} \lim_{k\to+\infty} \mathscr{F}_{\varepsilon_k}[u] = \int_{\mathbb{R}^N} \int_{\mathbb{S}^{N-1}} | \nabla u(x) \cdot \sigma |^2 \,\mathrm{d} \lambda(\sigma) \,\mathrm{d} x. \end{equation}

We refer to such equality as the Bourgain–Brezis–Mironescu formula, in short BBM formula. The novelty of our contribution is that we devise conditions that are both necessary and sufficient for (1.2) to hold (see also § 5.3 for some remarks about energies with non-quadratic growth). Precisely, we establish the following.

Roughly speaking, condition (i) prescribes that for $\varepsilon \in J$ small enough each kernel $\rho _{\varepsilon}$ must have finite mass in any large ball around the origin, and that, at the same time, the contributions accounting for long-range interactions must be asymptotically negligible. Indeed, as we show in § 5.1, (1.3) is equivalent to the following uniform decay condition: there exists $M\geq 0$ such that for every $R>0$:

\[ \limsup_{k\to +\infty}\int_{\mathbb{R}^N}\frac{\rho_{\varepsilon_k}(z)}{R^2+|z|^2}\,\mathrm{d} z \leq \frac{M}{R^2}. \]

When $R=1$, the previous inequality entails that for $k$ large enough $\rho _{\varepsilon _k}\in L^1_{\rm loc}(\mathbb {R}^N)$, so that, in particular, position (1.4) actually defines a Radon measure on $\mathbb {R}^N$. A useful way to regard the measures $\nu _k$ in (1.4) is to think of them as quantities encoding medium-range interactions, although this is not immediately evident from the definition. From this point of view, condition (ii) shows us that, in the limit, such interactions must vanish outside of the origin. We will elaborate further on this point in this introduction.

It turns out that conditions (i) and (ii) are also sufficient for the BBM formula to hold, so that, in light of theorem 1.1, they are sharp. To establish the sufficiency, we need the following compactness result, which is interesting on its own:

Theorem 1.2 (Asymptotic behaviour of nonlocal energies)

For every $\varepsilon \in J$, let $\rho _{\varepsilon} \colon \mathbb {R}^N \to [0,\,+\infty )$ be measurable and let $\mathscr {F}_{\varepsilon}$ be as in (1.1).

Suppose that there exists $M\geq 0$ with the property that for every $R>0$:

(1.5)\begin{equation} \limsup_{\varepsilon\to 0} \left[ \int_{B(0,R)} \rho_{\varepsilon}(z)\,\mathrm{d} z + R^2 \int_{B(0,R)^c} \frac{\rho_{\varepsilon}( z)}{| z |^2} \,\mathrm{d} z \right] \leq M. \end{equation}

Then, there exist an infinitesimal sequence $\{\varepsilon _k\}\subset J$ and two finite positive Radon measures $\mu$ and $\nu$, respectively on $\mathbb {S}^{N-1}$ and $\mathbb {R}^N$, that depend only on $\{\rho _{\varepsilon _k}\}$, and such that for every $u\in H^1(\mathbb {R}^N)$ there holds:

(1.6)\begin{align} \lim_{k\to+\infty} \mathscr{F}_{\varepsilon_k}[u] & = \frac{1}{2}\int_{\mathbb{R}^N} \left[ \vphantom{\frac{\rho_{\varepsilon}( z)}{| z |^2}} \int_{\mathbb{S}^{N-1}} | \nabla u(x) \cdot \sigma |^2 \,\mathrm{d} \mu(\sigma)\right.\notag\\ & \quad + \left. \int_{\mathbb{R}^N\setminus\{0\}} \frac{| u (x+z) - u (x) |^2}{| z |^2} \,\mathrm{d} \nu(z) \right] \mathrm{d} x. \end{align}

Moreover, the right-hand side of (1.6) is finite for every $u\in H^1(\mathbb {R}^N)$.

Theorem 1.2 shows that, while the integrability and decay conditions in (i) are sufficient to establish the convergence of the functionals in (1.1), in the absence of condition (ii) we cannot exclude the persistence of nonlocal terms in the limit. Indeed, the measure $\nu$ is retrieved as the limit (in the sense of weak-$\ast$ convergence) of the medium-range interactions encoded by (1.4). The measure $\mu$ captures instead the concentration of the sequence $\{\rho _{\varepsilon _k}\}$ around the origin, and it characterizes the (possibly zero) local term in the limiting energy. Loosely speaking, for every Borel subset $E\subseteq \mathbb {S}^{N-1}$, $\mu$ is given by

\[ \mu(E):= \lim_{\delta\to 0}\int_{C_\delta(E)}\rho_{\varepsilon_\delta}(z) \,\mathrm{d} z \]

where $C_\delta (E)$ is the intersection of the cone spanned by $E$ with $B(0,\,\delta )$, $\{\varepsilon _\delta \}$ is a suitable subfamily, and the limit is taken in the sense of the weak-$\ast$ convergence of measures. We refer to steps 3 and 4 in the proof of proposition 3.2 for the precise definition. In particular, when the kernels $\rho _{\varepsilon _k}$ are radial (cf. [Reference Bourgain, Brezis and Mironescu5]), then $\mu =c\mathscr {H}^{N-1}\lfloor {\mathbb {S}^{N-1}}$ for a constant $c\geq 0$.

We conclude our analysis by showing that, when (ii) is imposed as well, the limiting nonlocal effects vanish.

Corollary 1.3 (Sharp sufficient conditions for the BBM formula)

Let us suppose that same hypotheses of theorem 1.2 hold, and let us suppose also that the family $\{\nu _{\varepsilon} \}_{\varepsilon \in J}$ of Radon measures on $\mathbb {R}^N$ defined by

(1.7)\begin{equation} \langle \nu_{\varepsilon} , f \rangle := \int_{\mathbb{R}^N} \rho_{\varepsilon}(z) f(z) \,\mathrm{d} z \quad\text{for all}\ f\in C_c(\mathbb{R}^N), \end{equation}

locally weakly-$\ast$ converges in the sense of Radon measures to $\alpha \delta _0$, where $\alpha \geq 0$ is a positive constant, and $\delta _0$ is the Dirac delta in $0$. Then, there exist an infinitesimal sequence $\{\varepsilon _k\}\subset J$ and a finite positive Radon measure $\mu$ on $\mathbb {S}^{N-1}$ such that the BBM formula holds, that is,

(1.8)\begin{equation} \lim_{k\to+\infty} \mathscr{F}_{\varepsilon_k}[u] = \frac{1}{2} \int_{\mathbb{R}^N}\int_{\mathbb{S}^{N-1}} | \nabla u(x) \cdot \sigma |^2 \,\mathrm{d} \mu(\sigma) \,\mathrm{d} x. \end{equation}

We refer to remark 4.3 for an alternative formulation of the right-hand side of (1.8) in terms of the action of a quadratic form.

Our approach grounds on the use of the Fourier transform, which allows recasting the family of nonlocal functionals in (1.1) into double integrals of the form:

(1.9)\begin{equation} \int_{\mathbb{R}^N} | \psi (\xi) |^2 \int_{\mathbb{R}^N} \rho_{\varepsilon}( z)\frac{1 - \cos( z\cdot \xi )}{| z |^2} \,\mathrm{d} z \,\mathrm{d} \xi, \end{equation}

with $\psi$ in a suitable weighted $L^2$ space (see (2.1) and (2.3)). The technical preliminaries about the Fourier transform and those on Radon measures to be used later in this work are collected in § 2. In particular, the functionals in (1.9) and equivalent formulations of the BBM formula in Fourier variables are retrieved in lemma 2.1.

From § 3, we turn to the proof of our results. First, we establish theorem 1.2 by observing that the condition in (1.5) grants not only that the integrals with respect to $z$ in (1.9), as a function of $\xi$, grow at most as $1+|\xi |^2$ (see lemma 3.1), but also that they converge pointwise to the Fourier transform of the integrals within the square brackets in (1.6) (see proposition 3.2). The dominated convergence theorem then applies, and (1.6) is retrieved.

The pointwise convergence of the nonlocal energies provided by proposition 3.2 plays a central role in our analysis. It is obtained by studying separately the behaviours of the family $\{\rho _{\varepsilon} \}$ at three distinct interaction ranges, respectively short, medium and long, that we encode by means of an additional parameter $\delta \in J$. Short-range interactions arise from the contributions of shrinking balls of radius $\delta$ centred in the origin, and, as $\delta \to 0$, they asymptotically approach the gradient term in (1.6). Medium-range interactions originate from the contributions to the energy stored in annuli that lie at a distance $\delta$ from the origin. In the limit, their presence leads to the nonlocal term in (1.6), that is, the integral with respect to the measure $\nu$. Finally, long-range interactions occur outside of balls of radius $\delta ^{-1}$ centred in the origin, and their contributions are negligible when $\delta \to 0$.

The proofs of our two other results are provided in § 4. With theorem 1.2 on hand, corollary 1.3, that is, the sufficiency of conditions (i) and (ii) in theorem 1.1 for the BBM formula, follows quickly: it is enough to observe that (ii) forces the integral with respect to $\nu$ in (1.6) to vanish. In this sense, (ii) may be regarded as a locality condition, since it requires that in the limit the kernels concentrate at the origin. Conditions of this sort appear to be natural as far as sufficient criteria for the convergence of the nonlocal energies to variants of the Dirichlet norm are sought after (cf., e.g. (5.4) in theorem 5.4 or [Reference Alicandro, Ansini, Braides, Piatnitski and Tribuzio1, Thm. 3.1]). The key novelty of our contribution is that we prove item (ii) in theorem 1.1 to be the weakest locality requirement for the BBM formula (1.2) to hold.

Proving theorem 1.1, that is, the necessity of (i) and (ii) for the validity of the BBM formula, is a more delicate issue. The key step is established in proposition 4.1, where, by a suitable scaling of the functions in (1.9) (see remark 4.2), it is proved that (5.5) implies (i). The weak-$\ast$ convergence of the sequence $\{\nu _k\}$ in (ii) to a multiple of the Dirac delta in $0$ follows then from a homogeneity argument. We conclude our contribution in § 5 by clarifying how it compares with the existing literature and by pointing out possible future research directions.

As we briefly outlined above, there have been intense research efforts in the asymptotic analysis of nonlocal energies of the form (1.1). It is to be noted that such functionals also arise in applications, a case of interest being represented, for instance, by nonlocal models in micromagnetics. Indeed, as pointed out in [Reference Rogers20], if the classical symmetric exchange energy given by the Dirichlet integral of the magnetization is replaced by a nonlocal Heisenberg functional of the form (1.1), then a model closer to atomistic theories is obtained, and, in addition, the class of admissible magnetizations may be enlarged to include discontinuous and even ‘measure-valued’ fields. This observation is crucial in nonconvex problems such as those of ferromagnetism, in which the highly oscillatory ‘domain structures’ observed in ferromagnetic materials cannot be captured by magnetizations with Sobolev regularity. In such nonlocal micromagnetics models, knowing what classes of kernels $\rho _{\varepsilon}$ lead to an approximation of the classical Dirichlet energies amounts to a selection criterion to establish whether nonlocal descriptions can be replaced by local ones or, instead, such approximations are not mathematically correct. We refer to [Reference Di Fratta and Slastikov11] for further discussion on this topic.