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A Bourgain–Brezis–Mironescu formula accounting for nonlocal antisymmetric exchange interactions

Elisa Davoli Giovanni Di Fratta  and  Rossella Giorgio
Abstract.

The present study concerns the nonlocal-to-local convergence of a family of exchange energy functionals in the limit of very short-range interactions. The analysis accounts for both symmetric and antisymmetric exchange. Our result is twofold. First, we extend the Bourgain-Brezis-Mironescu formula to encompass the scenario where antisymmetric contributions are encoded into the energy. Second, we prove that, under physically relevant assumptions on the families of exchange kernels, the family of nonlocal functionals Gamma-converges to their local counterparts. As a byproduct of our analysis, we obtain a rigorous justification of Dzyaloshinskii–Moriya interactions in chiral magnets under the form commonly adopted in the variational theory of micromagnetism when modeling antisymmetric exchange interactions.

Key words and phrases:
Nonlocal energies, Bourgain-Brezis-Mironescu formula, Γ\Gamma-Convergence, antisymmetric exchange interactions, Micromagnetics, Dzyaloshinskii–Moriya interaction (DMI), Magnetic skyrmions.
1991 Mathematics Subject Classification:
46E35; 49J45; 49S05

1. Introduction and motivation

The present paper investigates the short-range interaction limit of a family of nonlocal exchange energies of the form

ε(m):=ε(m)+ε(m),\mathcal{E}_{\varepsilon}(m):=\mathcal{F}_{\varepsilon}(m)+\mathcal{H}_{\varepsilon}(m), (1.1)

where the energy functionals ε\mathcal{F}_{\varepsilon} and ε\mathcal{H}_{\varepsilon} are given by

ε(m)\displaystyle\mathcal{F}_{\varepsilon}(m) :=Ω×Ωρε(xy)|m(x)m(y)|2|xy|2dxdy,\displaystyle:=\iint_{\Omega\times\Omega}\rho_{\varepsilon}(x-y)\frac{|m(x)-m(y)|^{2}}{|x-y|^{2}}\mathrm{d}x\hskip 1.69998pt\mathrm{d}y, (1.2)
ε(m)\displaystyle\mathcal{H}_{\varepsilon}(m) :=Ω×Ωνε(xy)(m(x)×m(y))|xy|dxdy,\displaystyle:=\iint_{\Omega\times\Omega}\nu_{\varepsilon}(x-y)\cdot\frac{(m(x)\times m(y))}{|x-y|}\hskip 1.69998pt\mathrm{d}x\hskip 1.69998pt\mathrm{d}y, (1.3)

and are both defined on a suitable (metric) subspace of L2(Ω;𝕊2)L^{2}(\Omega;\mathbb{S}^{2}). Here, L2(Ω;𝕊2)L^{2}(\Omega;\mathbb{S}^{2}) denotes the space of vector-valued maps m:Ω𝕊2m:\Omega\to\mathbb{S}^{2}, where Ω\Omega is a bounded Lipschitz domain of 3\mathbb{R}^{3} and 𝕊2\mathbb{S}^{2} the unit sphere of 3\mathbb{R}^{3}.

The energy functional ε\mathcal{E}_{\varepsilon} is the result of two different types of interactions. The term ε\mathcal{F}_{\varepsilon} in (1.2) accounts for the so-called symmetric exchange interactions, whereas the energy term ε\mathcal{H}_{\varepsilon} in (1.3) accounts for antisymmetric exchange interactions. The scalar kernel ρε:3+\rho_{\varepsilon}:\mathbb{R}^{3}\to\mathbb{R}_{+} and the vector-valued kernel νε:33\nu_{\varepsilon}:\mathbb{R}^{3}\to\mathbb{R}^{3} model the strength and positional configuration of the exchange interactions at spatial scale ε>0\varepsilon>0. They will be referred to as symmetric and antisymmetric exchange kernels, respectively.

The main aim of this paper is to show that, under physically relevant assumptions on the families of exchange kernels (ρε)ε\mathopen{}\mathclose{{\left(\rho_{\varepsilon}}}\right)_{\varepsilon} and (νε)ε(\nu_{\varepsilon})_{\varepsilon}, the family ε\mathcal{E}_{\varepsilon} converges, in the sense of Γ\Gamma-convergence in L2(Ω;𝕊2)L^{2}(\Omega;\mathbb{S}^{2}), and up to constant factors, to the local energy functional

(m):=(m)+(m):=12Ω|m(x)|2+i=13Ωm(x)(di×im(x))dx,\mathcal{E}(m):=\mathcal{F}(m)+\mathcal{H}(m):=\frac{1}{2}\int_{\Omega}|\nabla m(x)|^{2}+\sum_{i=1}^{3}\int_{\Omega}m(x)\cdot(d_{i}\times\partial_{i}m(x))\mathrm{d}x, (1.4)

provided that mH1(Ω;𝕊2)m\in H^{1}(\Omega;\mathbb{S}^{2}). Here, the quantities di3d_{i}\in\mathbb{R}^{3} denote constant vectors referred to as Dzyaloshinskii vectors in the Physics literature; their expression strongly depends on the limiting behavior of the family (νε)ε(\nu_{\varepsilon})_{\varepsilon}.

Rigorous statements will be given in Section 2, where we formulate explicit assumptions on the exchange kernels.

1.1. Outline

The paper is organized as follows. In the rest of this Section, we present a brief overview of the physical framework that motivated our investigation and elucidate the significance of our findings in relation to prior research on the subject. In Section 2, we present the precise formulation of the problem and provide a detailed account of the contributions of our work. Proofs of our main results are given in Sections 3 and  4. Specifically, Section 3 focuses on the pointwise convergence of the antisymmetric exchange interactions (1.3), while Section 4 establishes the Γ\Gamma-convergence of the nonlocal family (ε)\mathopen{}\mathclose{{\left(\mathcal{E}_{\varepsilon}}}\right) in (1.1) to the local energy functional \mathcal{E} in (1.4).

1.2. Physics context: symmetric and antisymmetric exchange interactions in micromagnetics

Reliable theoretical models for studying magnetic phenomena must depend on the relevant length scales. At the mesoscopic scale, there is a well-established and effective variational theory of micromagnetism, whose roots may be found in the works of Landau–Lifshitz [18] and Brown [7, 6] on fine ferromagnetic particles. In this theory, for a rigid ferromagnetic particle occupying a region Ω3\Omega\subseteq\mathbb{R}^{3}, the order parameter is the magnetization field M:Ω3M:\Omega\rightarrow\mathbb{}\mathbb{R}^{3}. The modulus of MM, Ms:=|M|M_{s}:=|M|, is called spontaneous magnetization and is a function of the temperature that vanishes above the so-called Curie point TcT_{c}: a critical value strongly depends on the specific crystal structure of the ferromagnet. When the specimen is at a fixed temperature well below TcT_{c}, the function MsM_{s} can be assumed constant in Ω\Omega, and the magnetization can be conveniently written as M:=MsmM:=M_{s}m, where m:Ω𝕊2m:\Omega\rightarrow\mathbb{S}^{2} is a vector field with values in the unit sphere of 3\mathbb{R}^{3} (cf. [7, 15]).

Despite |M||M| being constant in Ω\Omega, this is generally not the case for the direction of MM, and according to the variational theory of micromagnetism, the observable magnetization patterns are the local minimizers of the micromagnetic energy functional, which, after normalization, reads as111In writing (1.5) is non-convex, non-local, and contains multiple length scales, we neglected the magnetocrystalline anisotropy and Zeeman energy, but only to shorten the notation. Indeed, although these contributions are of fundamental importance in ferromagnetism [7, 6], from the variational point of view, they behave like continuous perturbations, their analysis is usually straightforward, and in our specific context, they play no role.

𝒢(m):=(m)+𝒲(m):=12Ω|m|2+123|h𝖽[mχΩ]|2.\mathcal{G}(m):=\mathcal{F}(m)+\mathcal{W}(m):=\frac{1}{2}\int_{\Omega}|\nabla m|^{2}+\frac{1}{2}\int_{\mathbb{R}^{3}}|h_{\mathsf{d}}[m\chi_{\Omega}]|^{2}. (1.5)

Here, mH1(Ω;𝕊2)m\in H^{1}(\Omega;\mathbb{S}^{2}), and mχΩm\chi_{\Omega} is the extension by zero of mm to 3\mathbb{R}^{3}. The Dirichlet energy \mathcal{F}, i.e., the first term in (1.5), penalizes nonuniformities in the magnetization orientation, whereas the magnetostatic self-energy 𝒲\mathcal{W}, i.e., the second term in (1.5), is the energy associated with the demagnetizing field hdh_{\text{d}} generated by mχΩm\chi_{\Omega}, which describes the long-range dipole interaction of the magnetic moments: for vL2(3,3)v\in L^{2}\mathopen{}\mathclose{{\left(\mathbb{R}^{3},\mathbb{R}^{3}}}\right), the demagnetizing field h𝖽[v]L2(3,3)-h_{\mathsf{d}}[v]\in L^{2}\mathopen{}\mathclose{{\left(\mathbb{R}^{3},\mathbb{R}^{3}}}\right) can be characterized as the L2L^{2}-projection of vv on the space of gradients H˙1(3):={v:v𝒟(3),vL2(3,3)}\nabla\dot{H}^{1}\mathopen{}\mathclose{{\left(\mathbb{R}^{3}}}\right):=\mathopen{}\mathclose{{\left\{\nabla v:v\in\mathcal{D}^{\prime}(\mathbb{R}^{3}),\nabla v\in L^{2}(\mathbb{R}^{3},\mathbb{R}^{3})}}\right\}, see [10, 25] for details.

Much of the pattern observed in ferromagnetic materials is explained by the competition between the two contributions in (1.5); in particular, the formation of almost uniform magnetization regions (magnetic domains) separated by thin transition layers (domain walls), as predicted by the Weiss theory of ferromagnetism (cf. [7, 15]).

However, recent advancements in nanotechnology have led to the discovery of magnetic skyrmions: chiral spin textures that carry a nontrivial topological charge. Unlike conventional magnetic domains, magnetic skyrmions exhibit unusual swirling textures, and they arise in ferromagnetic materials with low crystallographic symmetry. The primary mechanism behind their formation and stability is weak antisymmetric exchange interactions, also known as Dzyaloshinskii–Moriya interactions (DMI), which result from the combination of spin-orbit and superexchange interactions [12, 20].

In the continuum theory of micromagnetism, DMI is accounted through the so-called chirality tensor m×m:=(im×m)i\nabla m\times m:=(\partial_{i}m\times m)_{i}, whose components are the Lifshitz invariants of mm (see, e.g., [14, Supplementary information]). The bulk DMI energy density corresponds to the trace of the chirality tensor: in the presence of the bulk DMI, the micromagnetic energy functional (1.5) has to be remodeled under the form

𝒢(m):=(m)+bulk(m)+𝒲(m):=12Ω|m|2+γΩcurlmm+123|h𝖽[mχΩ]|2.\mathcal{G}(m):=\mathcal{F}(m)+\mathcal{H}_{\operatorname{bulk}}(m)+\mathcal{W}(m):=\frac{1}{2}\int_{\Omega}|\nabla m|^{2}+\gamma\int_{\Omega}\mathrm{curl}m\cdot m+\frac{1}{2}\int_{\mathbb{R}^{3}}|h_{\mathsf{d}}[m\chi_{\Omega}]|^{2}\underset{}{}. (1.6)

The normalized constant γ\gamma\in\mathbb{R} is the bulk DMI constant, and its sign affects the chirality of the ferromagnetic system [2, 21] (see [11, Sec. 4] for further forms of DMI). Note that the bulk DMI bulk\mathcal{H}_{\operatorname{bulk}} in (1.6) is what the energy term \mathcal{H} in (1.4) reduces to when the Dzyaloshinskii vectors have the form di:=γeid_{i}:=\gamma e_{i}, with eie_{i} being the ii-th element of the standard basis of 3\mathbb{R}^{3} (observe that curlm=i=13ei×im\operatorname{curl}m=\sum_{i=1}^{3}e_{i}\times\partial_{i}m).

1.3. The L2L^{2}-theory of symmetric and antisymmetric exchange interactions. The Heisenberg setting

Ferromagnetism occurs in materials where the spins tend to align with each other, thereby generating an observable magnetic field outside of the media. Symmetric exchange interactions are the primary mechanism behind this effect, and in the isotropic Heisenberg model their behavior is described by the Hamiltonian (cf. [17])

Λ=S2(i,j)Λ×ΛJijmimj.\mathcal{F}_{\Lambda}=-S^{2}\sum_{(i,j)\in\Lambda\times\Lambda}J_{ij}m_{i}\cdot m_{j}. (1.7)

Here, Jij=JjiJ_{ij}=J_{ji} is the symmetric exchange constant between the spins mi,mj𝕊2m_{i},m_{j}\in\mathbb{S}^{2}, occupying the ii-th and jj-th site of the crystal lattice Λ\Lambda, and SS is the magnitude of the spin. The exchange constant JijJ_{ij} is a positive quantity in ferromagnetism; it depends on the ferromagnet’s crystal structure and weighs the intensity of the interaction among different spins.

When antisymmetric interactions cannot be neglected because, e.g., the ferromagnetic crystal lacks inversion symmetry, the DMI induces a spin canting of the magnetic moments, and the Hamiltonian (1.7) has to be remodeled under the form (cf. [17])

Λ:=Λ+Λ:=S2(i,j)Λ×ΛJijmimj+(i,j)Λ×Λdij(mi×mj).\mathcal{E}_{\Lambda}:=\mathcal{F}_{\Lambda}+\mathcal{H}_{\Lambda}:=-S^{2}\sum_{(i,j)\in\Lambda\times\Lambda}J_{ij}m_{i}\cdot m_{j}+\sum_{(i,j)\in\Lambda\times\Lambda}d_{ij}\cdot(m_{i}\times m_{j}). (1.8)

The Dzyaloshinskii vector dij=djid_{ij}=-d_{ji} is an axial vector that depends, other than from the relative distance between the spins mim_{i} and mjm_{j}, on the symmetry of the crystal lattice; its precise form has to be determined following Moriya’s rules [20]. We stress that while the term Λ\mathcal{F}_{\Lambda} is symmetric, in the sense that magnetic moments with right-handed ()\mathopen{}\mathclose{{\left({\mathopen{}\mathclose{{\left.\nwarrow\hskip 1.69998pt\nearrow}}\right)}}}\right. or left-handed ()\mathopen{}\mathclose{{\left({\mathopen{}\mathclose{{\left.\nearrow\hskip 1.69998pt\nwarrow}}\right)}}}\right. alignment give the same contributions, the discrete antisymmetric exchange energy Λ\mathcal{H}_{\Lambda} distinguishes between those two states via the local chirality imposed by the Dzyaloshinskii vectors dijd_{ij}.

In the limit of a continuous distribution of lattice sites occupying a region Ω3\Omega\subseteq\mathbb{R}^{3}, the exchange energy (1.8), up to a constant term, can be expressed as

J,d(m):=J(m)+d(m)\displaystyle\mathcal{E}_{J,d}(m):=\mathcal{F}_{J}(m)+\mathcal{H}_{d}(m) :=12Ω×ΩJ(xy)|m(x)m(y)|2dxdy\displaystyle:=\frac{1}{2}\int_{\Omega\times\Omega}J(x-y)|m(x)-m(y)|^{2}\mathrm{d}x\,\mathrm{d}y
+Ω×Ωd(xy)(m(y)×m(x))dxdy,\displaystyle\qquad\qquad\qquad+\int_{\Omega\times\Omega}d(x-y)\cdot(m(y)\times m(x))\mathrm{d}x\,\mathrm{d}y, (1.9)

with m:Ω𝕊2m:\Omega\rightarrow\mathbb{S}^{2} the normalized magnetization density, J0J\geqslant 0 a symmetric exchange kernel with even symmetry, i.e., such that J(z)=J(z)J(z)=J(-z) for every z3z\in\mathbb{R}^{3},  and dd an antisymmetric exchange kernel with odd symmetry, i.e., such that d(z)=d(z)d(-z)=-d(z) for every z3z\in\mathbb{R}^{3}.

The energy functional ε\mathcal{E}_{\varepsilon} in (1.1) is a faithful analog of the energy J,d\mathcal{E}_{J,d} in (1.9), and we found it more convenient to work with ε\mathcal{E}_{\varepsilon} because of scaling reasons. Specifically, we set J(z):=ρ(z)/|z|2J(z):=\rho(z)/|z|^{2} and d(z):=νε(z)/|z|d(z):=\nu_{\varepsilon}(z)/|z|. Overall, the exchange terms ε\mathcal{F}_{\varepsilon}, ε\mathcal{H}_{\varepsilon}, in (1.2) and (1.3), are the continuous counterparts, respectively, of the symmetric Heisenberg Hamiltonian associated with a many-electron system and of the antisymmetric exchange interactions due to the spin-orbit coupling between neighboring magnetic spins.

1.4. State of the art and contributions of the present work

At first glance, the relationship between the nonlocal energies ε\mathcal{F}_{\varepsilon}, ε\mathcal{H}_{\varepsilon}, in (1.2), (1.3), and the terms \mathcal{F} and \mathcal{H} in (1.4) is not evident, but it can be formally revealed through a first-order asymptotic expansion of the magnetization m(x)m(x) in a neighborhood of xΩx\in\Omega, i.e., by setting m(y)=m(x)+Dm(x)(yx)+𝒪(|yx|)m(y)=m(x)+\mathrm{D}m(x)(y-x)+\mathcal{O}(|y-x|). The asymptotic analysis becomes more reliable the more one can neglect variations of mm around xΩx\in\Omega, i.e., the more the kernels ρε\rho_{\varepsilon} and νε\nu_{\varepsilon} concentrate their mass around the origin, i.e., the more the exchange interactions act on a very short range only. The question then is whether and in which sense the exchange energy \mathcal{E} in (1.1) is a short-range approximation of the Heisenberg L2L^{2}-description in (1.9). In this paper, we give an affirmative answer to this question under physically relevant hypotheses on the kernels.

When only symmetric exchange interactions are considered, the (affirmative) answer is already known because of the Γ\Gamma-convergence result established in [24]. Surprisingly, the motivation for the results in [24] came from very different reasons: a question left open in the seminal paper [3] where a new characterization of Sobolev spaces is presented, and the Bourgain–Brezis–Mironescu (BBM) formula made its first appearance. In [3], pointwise convergence of ε\mathcal{F}_{\varepsilon} to \mathcal{F} is established when (ρε)ε+(\rho_{\varepsilon})_{\varepsilon\in\mathbb{R}_{+}} is a family of radial mollifiers concentrating their mass at the origin as ε0\varepsilon\rightarrow 0; also, a technical lemma provides upper and lower bounds on the variational convergence of ε\mathcal{F}_{\varepsilon}, but not sufficient to deduce Γ\Gamma-convergence, finally obtained in [24]. In summary, in the absence of DMI, the classical symmetric exchange energy can be considered as the very short-rage limit of the family of nonlocal energies (ε)(\mathcal{F}_{\varepsilon}), and the main aim of this paper is to extend the result to the case in which also antisymmetric interactions are present and under legitimate hypotheses on the antisymmetric exchange kernels νε\nu_{\varepsilon}.

Over the years, several papers have presented new BBM-type formulas that expanded the original results in [3] into various directions [5, 24, 1, 9, 19, 16] (see also [26, 22] for some applications to the magnetic Schrödinger operator). Also, formal asymptotics predicts that the choice of appropriate exchange kernels is crucial for ε\mathcal{E}_{\varepsilon} to reduce to \mathcal{E}, and we refer the reader to [8, 13] for some results about the class of admissible kernels.

In contrast to the symmetric energy functional ε\mathcal{F}_{\varepsilon}, the variational convergence of ε\mathcal{H}_{\varepsilon} in the asymptotic regime of very short-range interactions had not been investigated so far. To the best of the authors’ knowledge, no variational analysis involving ε\mathcal{H}_{\varepsilon} has been carried out thus far; this includes basic questions like the existence of 𝕊2\mathbb{S}^{2}-valued minimizers of ε\mathcal{E}_{\varepsilon} or the qualitative behaviors of minimizing sequences; these are significant issues because of their direct bearing on the emergence of magnetic skyrmions, and will be the subject of a forthcoming paper.

Notation.

In what follows, we will adopt standard notation for the spaces of Lebesgue and Sobolev functions, as well as for the space of functions with bounded variation (BVBV). Balls with center xx and radius rr will be denoted by Br(x)B_{r}(x) and characteristic functions of sets AA by χA\chi_{A}, with the convention that χA(x)=1\chi_{A}(x)=1 if xAx\in A and χA(x)=0\chi_{A}(x)=0 otherwise. Whenever not explicitly mentioned otherwise, C>0C>0 will alway denote a generic constant, only dependent on the data of the problem, and whose value might, in principle, change from line to line.

2. Statement of main results

For a given ε>0\varepsilon>0, we denote by Dε(Ω;𝕊2)D_{\varepsilon}(\Omega;\mathbb{S}^{2}) the subspace of L2(Ω;𝕊2)L^{2}(\Omega;\mathbb{S}^{2}) where ε\mathcal{E}_{\varepsilon} is finite:

Dε(Ω;𝕊2)={mL2(Ω;𝕊2):ε(m)<+}.D_{\varepsilon}(\Omega;\mathbb{S}^{2})=\{m\in L^{2}(\Omega;\mathbb{S}^{2}):\mathcal{E}_{\varepsilon}(m)<+\infty\}. (2.1)

Also, since many of our results still hold when mm is not constrained to take values on 𝕊2\mathbb{S}^{2}, and even if Ω\Omega is unbounded, it is convenient for us to denote by Dε(Ω;3)D_{\varepsilon}(\Omega;\mathbb{R}^{3}) the analog unconstrained subspace of L2(Ω;3)L^{2}(\Omega;\mathbb{R}^{3}). Of course, the space Dε(Ω;𝕊2)D_{\varepsilon}(\Omega;\mathbb{S}^{2}) on which the energy ε\mathcal{E}_{\varepsilon} is finite depends on the particular choice of the kernels ρε\rho_{\varepsilon} and νε\nu_{\varepsilon}. In what follows, we will make the following assumptions.

The symmetric exchange kernels ρε\rho_{\varepsilon}

Driven by formal asymptotics, we consider a family of kernels (ρε)ε(\rho_{\varepsilon})_{\varepsilon} satisfying the following hypotheses already adopted in [24] —which are more general than the ones initially proposed in [3] (see also [8] for an extensive discussion).

  1. (G1)

    For every ε>0\varepsilon>0, ρε0\rho_{\varepsilon}\geqslant 0 in 3\mathbb{R}^{3} and ρεL1(3)BVloc(3)\rho_{\varepsilon}\in L^{1}(\mathbb{R}^{3})\color[rgb]{0,0,0}\cap BV_{\rm loc}(\mathbb{R}^{3})\color[rgb]{0,0,0} with ρεL1(3)=1\|\rho_{\varepsilon}\|_{L^{1}(\mathbb{R}^{3})}=1.

  2. (G2)

    For every δ>0\delta>0

    limε03Bδ(0)ρε(y)dy=0.\lim_{\varepsilon\to 0}\int_{\mathbb{R}^{3}\setminus B_{\delta}(0)}\rho_{\varepsilon}(y)\hskip 1.69998pt\mathrm{d}y=0. (2.2)
  3. (G3)

    There exist linearly independent directions v1,v2,v3𝕊2v_{1},v_{2},v_{3}\in\mathbb{S}^{2} and δ>0\delta>0 such that Cδ(vi)Cδ(vj)=C_{\delta}(v_{i})\cap C_{\delta}(v_{j})=\emptyset for iji\neq j, and for any i=1,2,3i=1,2,3

    lim supε0Cδ(vi)ρε(y)dy>0.\underset{\varepsilon\to 0}{\limsup}\int_{C_{\delta}(v_{i})}\rho_{\varepsilon}(y)\mathrm{d}y>0. (2.3)

    Here, for v𝕊2v\in\mathbb{S}^{2} and δ>0\delta>0, Cδ(v):=+Eδ(v)C_{\delta}(v):=\mathbb{R}_{+}E_{\delta}(v) denotes the cone centered at the origin, whose projection on 𝕊2\mathbb{S}^{2} is given by Eδ(v):={w𝕊2:wv>(1δ)}E_{\delta}(v):=\mathopen{}\mathclose{{\left\{w\in\mathbb{S}^{2}:w\cdot v>(1-\delta)}}\right\}.

  4. (G4)

    There exists a constant 0<κ10<\kappa\leqslant 1 such that, denoting by ρεrad\rho_{\varepsilon}^{\rm rad} the radial functions ρεrad:++\rho_{\varepsilon}^{\rm rad}:\mathbb{R}_{+}\to\mathbb{R}_{+}, defined for every ε>0\varepsilon>0 as

    ρεrad(t):=essinf{ρε(x):x3 with |x|=t} for every t+,\rho_{\varepsilon}^{\rm rad}(t):={\rm ess}\inf\{\rho_{\varepsilon}(x):\,x\in\mathbb{R}^{3}\text{ with }|x|=t\}\quad\text{ for every }t\in\mathbb{R}_{+}, (2.4)

    there holds

    infε>03ρεrad(|x|)dxκ.\inf_{\varepsilon>0}\int_{\mathbb{R}^{3}}\rho_{\varepsilon}^{\rm rad}(|x|)\mathrm{d}x\geqslant\kappa. (2.5)
Remark 2.1 (On Hypotheses (G1), (G3), and (G4)).

A few words about Hypotheses (G1),(G3) and (G4) are in order. Hypothesis (G3) is a weaker condition than the radiality assumed in [3]. Roughly speaking, it assures that in the limit of very short-range interactions (ε0\varepsilon\to 0), the family (ρε)ε(\rho_{\varepsilon})_{\varepsilon} has nontrivial support at least around three linearly independent directions. Condition (G3) has been introduced in [24] in order to prove a characterization of W1,p(Ω)W^{1,p}(\Omega), 1<p<1<p<\infty. Our work uses it to prove a regularity result for our nonlocal functional (1.1) (see Theorem 2.2). In view of (G1), Hypothesis (G4) is automatically satisfied when the kernels (ρε)ε(\rho_{\varepsilon})_{\varepsilon} are radial. If this is not the case, Hypothesis (G4) provides a non-degeneracy condition for suitable radial lower envelopes of the kernels. In particular, in the prototypical case in which the family (ρε)ε(\rho_{\varepsilon})_{\varepsilon} is given by

ρε(x):=1ε3ρ(xε),\rho_{\varepsilon}(x):=\frac{1}{\varepsilon^{3}}\rho\mathopen{}\mathclose{{\left(\frac{x}{\varepsilon}}}\right),

for every ε>0\varepsilon>0 and x3x\in\mathbb{R}^{3}, where ρCc(3)\rho\in C^{\infty}_{c}(\mathbb{R}^{3}) is such that 0ρ10\leqslant\rho\leqslant 1 in 3\mathbb{R}^{3} and ρL1(3)=1\|\rho\|_{L^{1}(\mathbb{R}^{3})}=1, Hypothesis (G4) is directly satisfied as long as ρ\rho is strictly positive at the origin. In fact, assume that ρ(0)=2δ>0\rho(0)=2\delta>0. By the regularity of ρ\rho, there exists a ball of radius r>0r>0 such that ρ(x)δ\rho(x)\geqslant\delta for every xBr(0)¯x\in\overline{B_{r}(0)}. In particular, ρεrad(|x|)δε3\rho_{\varepsilon}^{\rm rad}(|x|)\geqslant\frac{\delta}{\varepsilon^{3}} for every ε>0\varepsilon>0 and xBrε(0)¯x\in\overline{B_{r\varepsilon}(0)}, so that (2.5) holds with κ=δ|Br(0)|\kappa=\delta|B_{r}(0)|.

Finally, the regularity of the kernels in (G1) can be weakened to just L1(3)L^{1}(\mathbb{R}^{3}) in the case in which the kernels are radial. The further BVlocBV_{\rm loc}- regularity is only needed to provide a meaning in the sense of traces to the restriction of the kernels ρε\rho_{\varepsilon} to spheres of radius rr, so that the definition in (2.4) is well-posed.

The antisymmetric exchange kernels νε\nu_{\varepsilon}

Motivated by formal asymptotics, we consider a family of vector-valued kernels (νε)ε(\nu_{\varepsilon})_{\varepsilon} satisfying the following assumptions.

  1. (H1)

    For every ε>0\varepsilon>0, νε\nu_{\varepsilon} is odd, i.e., νε(y)=νε(y)\nu_{\varepsilon}(-y)=-\nu_{\varepsilon}(y) for each y3y\in\mathbb{R}^{3}, and νεL1(3;3)\nu_{\varepsilon}\in L^{1}(\mathbb{R}^{3};\mathbb{R}^{3}) with νεL1(3)=1\|\nu_{\varepsilon}\|_{L^{1}(\mathbb{R}^{3})}=1.

  2. (H2)

    For every δ>0\delta>0 there holds

    limε03Bδ(0)|νε(y)|dy=0.\lim_{\varepsilon\to 0}\int_{\mathbb{R}^{3}\setminus B_{\delta}(0)}|\nu_{\varepsilon}(y)|\hskip 1.69998pt\mathrm{d}y=0. (2.6)
  3. (H3)

    For i=1,2,3i=1,2,3, the following limit exists

    limε03νε(y)yi|y|dy=:di3.\lim_{\varepsilon\to 0}\int_{\mathbb{R}^{3}}\nu_{\varepsilon}(y)\frac{y_{i}}{|y|}\hskip 1.69998pt\mathrm{d}y=:d_{i}\in\mathbb{R}^{3}. (2.7)

    Motivated by their physical significance, we will refer to the vectors did_{i} as the Dzyaloshinskii vectors. Note that |di|1|d_{i}|\leqslant 1 for every i=1,2,3i=1,2,3.

Later on, we will need the following condition (A1), which connects the integrability of the two families of kernels (ρε)ε(\rho_{\varepsilon})_{\varepsilon} and (νε)ε(\nu_{\varepsilon})_{\varepsilon}.

  1. (A1)

    There exist a constant C>0C>0 such that

    supε>0νερεL(3)C.\sup_{\varepsilon>0}\hskip 1.69998pt\mathopen{}\mathclose{{\left\lVert\hskip 1.69998pt\frac{\nu_{\varepsilon}}{\rho_{\varepsilon}}\hskip 1.69998pt}}\right\rVert_{L^{\infty}(\mathbb{R}^{3})}\leqslant C. (2.8)

Throughout the work, we specify in which cases assumption (A1) is needed.

Remark 2.2 (On the odd symmetry of νε\nu_{\varepsilon}).

We emphasize that the requirement of νε\nu_{\varepsilon} being odd causes no loss of generality. Indeed, one can always decompose a vector-valued kernel into its even and odd parts, and an immediate symmetry argument shows that the presence of an even component in νε\nu_{\varepsilon} would give no contribution to ε\mathcal{H}_{\varepsilon} due to the antisymmetric nature of the functional ε\mathcal{H}_{\varepsilon} and its linearity with respect to the kernel. Also, our results still hold when one relaxes (H3), requiring that (2.7) holds up to the extraction of a subsequence, but we do not insist on these refinements.

The main result of this paper consists of a compactness and Γ\Gamma-convergence analysis for our nonlocal exchange functionals in (1.1).

Theorem 2.1.

(Compactness and Γ\Gamma-convergence) Let Ω3\Omega\subseteq\mathbb{R}^{3} be a bounded Lipschitz domain. Assume (G1)(G4), (H1)(H3), and (A1). The following assertions hold.

  1. (i)

    (Compactness) If (mε)εL2(Ω;𝕊2)(m_{\varepsilon})_{\varepsilon}\subset L^{2}(\Omega;\mathbb{S}^{2}) is such that

    lim infε0ε(mε)<+,\liminf_{\varepsilon\rightarrow 0}\hskip 1.69998pt\mathcal{E}_{\varepsilon}(m_{\varepsilon})<+\infty, (2.9)

    then, there exists mH1(Ω;𝕊2)m\in H^{1}(\Omega;\mathbb{S}^{2}) such that, possibly up to a non-relabeled subsequence, mεmm_{\varepsilon}\to m strongly in L2(Ω;𝕊2)L^{2}(\Omega;\mathbb{S}^{2}).

  2. (ii)

    (Γ\Gamma-convergence) There exists a finite Radon measure μ(𝕊2)\mu\in\mathcal{M}(\mathbb{S}^{2}), with μ(𝕊2)=1\mu\mathopen{}\mathclose{{\left(\mathbb{S}^{2}}}\right)=1, such that, possibly up to a non-relabeled subsequence,

    Γ-limε0ε=μ,strongly in L2(Ω;𝕊2),\Gamma\text{-}\lim_{\varepsilon\to 0}\mathcal{E}_{\varepsilon}=\mathcal{E}_{\mu},\quad\text{strongly in }L^{2}\mathopen{}\mathclose{{\left(\Omega;\mathbb{S}^{2}}}\right), (2.10)

    with μ(m)=+\mathcal{E}_{\mu}(m)=+\infty if mL2(Ω;𝕊2)\H1(Ω;𝕊2)m\in L^{2}(\Omega;\mathbb{S}^{2})\backslash H^{1}(\Omega;\mathbb{S}^{2}) and

    μ(m)=Ω(𝕊2|σm(x)|2dμ(σ))dx+i=13Ωm(x)(di×im(x))dx\mathcal{E}_{\mu}(m)=\int_{\Omega}\mathopen{}\mathclose{{\left(\int_{\mathbb{S}^{2}}|\partial_{\sigma}m(x)|^{2}\mathrm{d}\mu(\sigma)}}\right)\mathrm{d}x+\sum_{i=1}^{3}\int_{\Omega}m(x)\cdot(d_{i}\times\partial_{i}m(x))\hskip 1.69998pt\mathrm{d}x (2.11)

    if mH1(Ω;𝕊2)m\in H^{1}(\Omega;\mathbb{S}^{2}).

Remark 2.3 (The micromagnetic case).

We note that choosing kernels (νε)ε(\nu_{\varepsilon})_{\varepsilon} such that di=γeid_{i}=\gamma e_{i} for i=1,2,3i=1,2,3 in (H3), and kernels (ρε)ε\mathopen{}\mathclose{{\left(\rho_{\varepsilon}}}\right)_{\varepsilon} radial, from Theorem 2.1 we obtain that with respect to the strong topology of L2L^{2}, there holds Γ-limε0ε=\Gamma\text{-}\lim_{\varepsilon\to 0}\mathcal{E}_{\varepsilon}=\mathcal{E} with

(m)=13Ω|m(x)|2dx+γΩcurlm(x)m(x)dx\mathcal{E}(m)=\frac{1}{3}\int_{\Omega}|\nabla m(x)|^{2}\mathrm{d}x+\gamma\int_{\Omega}\mathrm{curl}m(x)\cdot m(x)\hskip 1.69998pt\mathrm{d}x (2.12)

if mH1(Ω;𝕊2)m\in H^{1}(\Omega;\mathbb{S}^{2}), and (m)=+\mathcal{E}(m)=+\infty otherwise (if mL2(Ω;𝕊2)\H1(Ω;𝕊2)m\in L^{2}(\Omega;\mathbb{S}^{2})\backslash H^{1}(\Omega;\mathbb{S}^{2})). Up to a constant factor, this is nothing but the micromagnetic energy functional (1.6) thoroughly studied in recent years as a theoretical foundation for the analysis of magnetic skyrmions emerging from bulk DMI.

Remark 2.4 (Examples of kernels ρε\rho_{\varepsilon} and νε\nu_{\varepsilon}).

The conditions imposed on the families of kernels (ρε)ε(\rho_{\varepsilon})_{\varepsilon} and (νε)ε(\nu_{\varepsilon})_{\varepsilon} are not meant to be sharp but rather to cover most situations of interest in applications, in particular, for the analysis of magnetic skyrmions in chiral ferromagnetic materials. An explicit example of kernels satisfying the hypotheses above are presented below. Let ν(y):=4|𝕊2|yχB1(0)(y)\nu(y):=\frac{4}{|\mathbb{S}^{2}|}y\chi_{B_{1}(0)}(y) and ρ(y):=|ν(y)|\rho(y):=|\nu(y)| for every y3y\in\mathbb{R}^{3}, where we have denoted by χB1(0)(y)\chi_{B_{1}(0)}(y) the characteristic function of the unit ball in 3\mathbb{R}^{3}. Note that ν\nu is odd, νL1(3)=ρL1(3)=1\|\nu\|_{L^{1}(\mathbb{R}^{3})}=\|\rho\|_{L^{1}(\mathbb{R}^{3})}=1, and ρ\rho is radial. Then, setting

νε(y):=1ε3ν(yε)and ρε(y):=1ε3ρ(yε)\nu_{\varepsilon}(y):=\frac{1}{\varepsilon^{3}}\nu\mathopen{}\mathclose{{\left(\frac{y}{\varepsilon}}}\right)\quad\text{and }\rho_{\varepsilon}(y):=\frac{1}{\varepsilon^{3}}\rho\mathopen{}\mathclose{{\left(\frac{y}{\varepsilon}}}\right)

for every ε>0\varepsilon>0 and y3y\in\mathbb{R}^{3}, Hypotheses (G1)(G4), as well as (H1)(H2) and (A1) are directly fulfilled. Additionally, (H3) holds with di=13eid_{i}=\frac{1}{3}e_{i}, i=1,2,3i=1,2,3.

Remark 2.5 (On the measure μ\mu).

Note that, on the one hand, as for the purely symmetric case, cf. [24, Lemma 8], the measure μ\mu in Theorem 2.1 (ii) is, in principle, dependent on the choice of the extracted subsequence. For some specific choices of the kernels (ρε)ε(\rho_{\varepsilon})_{\varepsilon}, on the other hand, for example when they are radial, the limiting functional is uniquely identified independently of the extracted subsequence, see also Remark 2.9 below, so that the Gamma-convergence result actually holds for the entire sequence (ε)ε(\mathcal{E}_{\varepsilon})_{\varepsilon}.

The proof of Theorem 2.1 is provided in Section 4. The argument relies upon the following results, which are of interest in their own rights. The first assures that the nonlocal functional (1.1) is well-defined on H1(Ω;𝕊2)H^{1}(\Omega;\mathbb{S}^{2}).

Theorem 2.2.

Assume (G1) and (H1). For every mH1(Ω;3)m\in H^{1}(\Omega;\mathbb{R}^{3}) there holds

supε>0ε(m)CmH1(Ω)2,\sup_{\varepsilon>0}\mathcal{E}_{\varepsilon}(m)\leqslant C\hskip 1.69998pt\|m\|^{2}_{H^{1}(\Omega)}, (2.13)

for some constant C>0C>0 depending only on Ω\Omega.

Viceversa, assume (G1)(G3), as well as (H1) and (A1). If mL2(Ω;3)m\in L^{2}(\Omega;\mathbb{R}^{3}) is such that

supε>0ε(m)<+,\sup_{\varepsilon>0}\hskip 1.69998pt\mathcal{E}_{\varepsilon}(m)<+\infty, (2.14)

then mH1(Ω;3)m\in H^{1}(\Omega;\mathbb{R}^{3}).

Remark 2.6 (The domain of the nonlocal functionals).

Estimate (2.13) gives, in particular (cf. (2.1)), that H1(Ω;3)Dε(Ω;3)H^{1}(\Omega;\mathbb{R}^{3})\subseteq D_{\varepsilon}(\Omega;\mathbb{R}^{3}) and H1(Ω;𝕊2)Dε(Ω;𝕊2)H^{1}(\Omega;\mathbb{S}^{2})\subseteq D_{\varepsilon}(\Omega;\mathbb{S}^{2}). Also, it will be evident from the proof that H1(Ω;3)Dε(Ω;3)H^{1}(\Omega;\mathbb{R}^{3})\subseteq D_{\varepsilon}(\Omega;\mathbb{R}^{3}) even when Ω\Omega is unbounded. In particular, H1(3;3)Dε(3;3)H^{1}(\mathbb{R}^{3};\mathbb{R}^{3})\subseteq D_{\varepsilon}(\mathbb{R}^{3};\mathbb{R}^{3}).

Remark 2.7 (On the assumptions of Theorem 2.2).

Estimate (2.13) guarantees that if (G1) and (H1) hold and mH1(Ω;3)m\in H^{1}(\Omega;\mathbb{R}^{3}), then supε>0ε(m)<+\sup_{\varepsilon>0}\mathcal{E}_{\varepsilon}(m)<+\infty. The converse statement holds under the additional assumptions (G2), (G3), (H1), and (A1). Assumptions (G2)(G3) are the main requirements in [24, Theorem 5]. Roughly speaking, (G3) prevents degenerate mass behavior of the family (ρε)ε(\rho_{\varepsilon})_{\varepsilon}, so that the partial derivatives of mm can be controlled over all directions. The further assumption (A1) expresses an integrability relation between the two families of kernels (ρε)ε(\rho_{\varepsilon})_{\varepsilon} and (νε)ε(\nu_{\varepsilon})_{\varepsilon} which, nevertheless, naturally arise when modeling antisymmetric exchange interactions in the variational theory of micromagnetism.

Another crucial ingredient for the proof of Theorem 2.1 is a uniform convergence result for the antisymmetric exchange energies ε\mathcal{H}_{\varepsilon}, contained in the following Theorem 2.3.

Theorem 2.3.

(Pointwise and uniform convergence of the antisymmetric exchange term) Assume (H1)(H3) and let Ω\Omega be bounded. For every mH1(Ω;3)m\in H^{1}(\Omega;\mathbb{R}^{3}) there holds

limε0ε(m)=(m),\lim_{\varepsilon\to 0}\mathcal{H}_{\varepsilon}(m)=\mathcal{H}(m), (2.15)

where, we recall (see (1.2) and (1.4))

ε(m)=Ω×Ωνε(xy)|xy|(m(x)×m(y))dxdy,(m)=i=13Ωm(x)(di×im(x))dx,\mathcal{H}_{\varepsilon}(m)=\iint_{\Omega\times\Omega}\frac{\nu_{\varepsilon}(x-y)}{|x-y|}\cdot(m(x)\times m(y))\hskip 1.69998pt\mathrm{d}x\hskip 1.69998pt\mathrm{d}y\hskip 1.69998pt,\quad\mathcal{H}(m)=\sum_{i=1}^{3}\int_{\Omega}m(x)\cdot(d_{i}\times\partial_{i}m(x))\hskip 1.69998pt\mathrm{d}x,

and the di3d_{i}\in\mathbb{R}^{3} are the Dzyaloshinskii vectors defined in (H3).

Moreover, if (mε)ε>0(m_{\varepsilon})_{\varepsilon>0} is a family in C2(Ω¯;3)C^{2}(\bar{\Omega};\mathbb{R}^{3}) such that mεmm_{\varepsilon}\rightarrow m in C2(Ω¯;3)C^{2}(\bar{\Omega};\mathbb{R}^{3}), then

limε0ε(mε)=(m).\lim_{\varepsilon\to 0}\mathcal{H}_{\varepsilon}(m_{\varepsilon})=\mathcal{H}(m). (2.16)
Remark 2.8 (The case of unbounded Ω\Omega).

It will be evident from the proof that (2.15) holds even when Ω\Omega is unbounded. In particular, when Ω=3\Omega=\mathbb{R}^{3}. Also, for simplicity, we required (H3) to ensure the convergence of the whole sequence in (2.7). It will be clear from the proof that this requirement can be weakened by only requiring that the convergence in (H3) holds up to the extraction of a subsequence.

As a direct consequence of Theorem 2.3, and [24, Theorem 1], we infer the following result about the pointwise asymptotic behavior of the total nonlocal energy (1.1).

Corollary 2.1.

(Pointwise convergence of the total energy) Assume (G1)(G3), as well as (H1)(H3) and (A1). Then, there exists a finite Radon measure μ(𝕊2)\mu\in\mathcal{M}(\mathbb{S}^{2}), with μ(𝕊2)=1\mu\mathopen{}\mathclose{{\left(\mathbb{S}^{2}}}\right)=1, such that, possibly up to a non-relabeled subsequence, for every mH1(Ω;3)m\in H^{1}(\Omega;\mathbb{R}^{3}), there holds

limε0ε(m)=Ω(𝕊2|σm|2dμ(σ))dx+i=13Ωm(x)(di×im(x))dx,\lim_{\varepsilon\rightarrow 0}\mathcal{E}_{\varepsilon}(m)=\int_{\Omega}\mathopen{}\mathclose{{\left(\int_{\mathbb{S}^{2}}|\partial_{\sigma}m|^{2}\mathrm{d}\mu(\sigma)}}\right)\mathrm{d}x+\sum_{i=1}^{3}\int_{\Omega}m(x)\cdot(d_{i}\times\partial_{i}m(x))\hskip 1.69998pt\mathrm{d}x, (2.17)

with the di3d_{i}\in\mathbb{R}^{3} being the Dzyaloshinskii vectors introduced in (H3).

Moreover, if the family (ρε)ε(\rho_{\varepsilon})_{\varepsilon} consists of radial kernels, then for every mH1(Ω;3)m\in H^{1}(\Omega;\mathbb{R}^{3}) we have

limε0[ε(m)+ε(m)]=13Ω|m|2dx+i=13Ωm(x)(di×im(x))dx.\lim_{\varepsilon\rightarrow 0}\mathopen{}\mathclose{{\left[\mathcal{F}_{\varepsilon}(m)+\mathcal{H}_{\varepsilon}(m)}}\right]=\frac{1}{3}\int_{\Omega}|\nabla m|^{2}\hskip 1.69998pt\mathrm{d}x+\sum_{i=1}^{3}\int_{\Omega}m(x)\cdot(d_{i}\times\partial_{i}m(x))\hskip 1.69998pt\mathrm{d}x.
Remark 2.9 (Structure of the limiting symmetric term).

Since we are in the quadratic setting, one has |σm|2=k(σσ)mkmk|\partial_{\sigma}m|^{2}=\sum_{k}(\sigma\otimes\sigma)\nabla m_{k}\cdot\nabla m_{k} and, therefore, one can rewrite the first term in (2.17) under the form

limε0ε(m)=kΩAmkmkdx\lim_{\varepsilon\rightarrow 0}\mathcal{F}_{\varepsilon}(m)=\sum_{k}\int_{\Omega}A\nabla m_{k}\cdot\nabla m_{k}\mathrm{d}x (2.18)

where AA is the anisotropic matrix given by A:=𝕊2(σσ)dμ(σ)A:=\int_{\mathbb{S}^{2}}(\sigma\otimes\sigma)\mathrm{d}\mu(\sigma). If the kernels are radial, then μ\mu is isotropic, in the sense that the resulting matrix AA is given by (1/3)I(1/3)I with II being the 3×33\times 3 identity matrix.

3. Pointwise and uniform behavior of the energy (proofs of Theorem 2.2 and Theorem 2.3)

Proof of Theorem 2.2.

We split the proof in two steps. In Step 1, we derive the estimate (2.13), while in Step 2, we prove that if (2.14) holds, then mH1(Ω;3)m\in H^{1}(\Omega;\mathbb{R}^{3}).

Step 1. Recall that, by definition, ε(m)=ε(m)+ε(m)\mathcal{E}_{\varepsilon}(m)=\mathcal{F}_{\varepsilon}(m)+\mathcal{H}_{\varepsilon}(m). For ease of computation, we treat the two energy terms separately. For mH1(Ω;3)m\in H^{1}(\Omega;\mathbb{R}^{3}), we denote by m~H1(3;3)\tilde{m}\in H^{1}(\mathbb{R}^{3};\mathbb{R}^{3}) the extension of mm to the whole of 3\mathbb{R}^{3} given by the classical extension operator on H1(Ω;3)H^{1}(\Omega;\mathbb{R}^{3}).

For the symmetric exchange term ε\mathcal{F}_{\varepsilon} we have

ε(m)\displaystyle\mathcal{F}_{\varepsilon}(m) 3×3ρε(xy)|m~(x)m~(y)|2|xy|2dxdy\displaystyle\leqslant\iint_{\mathbb{R}^{3}\times\mathbb{R}^{3}}\rho_{\varepsilon}(x-y)\frac{|\tilde{m}(x)-\tilde{m}(y)|^{2}}{|x-y|^{2}}\hskip 1.69998pt\mathrm{d}x\hskip 1.69998pt\mathrm{d}y
=3ρε(h)(3|m~(xh)m~(x)|2|h|2dx)dh\displaystyle=\int_{\mathbb{R}^{3}}\rho_{\varepsilon}(h)\mathopen{}\mathclose{{\left(\int_{\mathbb{R}^{3}}\frac{|\tilde{m}(x-h)-\tilde{m}(x)|^{2}}{|h|^{2}}\hskip 1.69998pt\mathrm{d}x}}\right)\mathrm{d}h (3.1)
=3ρε(h)τhm~m~L2(3)2|h|2dh\displaystyle=\int_{\mathbb{R}^{3}}\rho_{\varepsilon}(h)\hskip 1.69998pt\frac{\|\tau_{-h}\tilde{m}-\tilde{m}\|^{2}_{L^{2}(\mathbb{R}^{3})}}{|h|^{2}}\hskip 1.69998pt\mathrm{d}h
ρεL1(3)m~L2(3)2\displaystyle\leqslant\|\rho_{\varepsilon}\|_{L^{1}(\mathbb{R}^{3})}\hskip 1.69998pt\|\nabla\tilde{m}\|^{2}_{L^{2}(\mathbb{R}^{3})}
CmH1(Ω)2,\displaystyle\leqslant\hskip 1.69998ptC\hskip 1.69998pt\|m\|^{2}_{H^{1}(\Omega)}, (3.2)

for some constant C>0C>0 depending only on Ω\Omega through the linear extension operator. The equality (3.1) is the result of the change of variables yxhy\mapsto x-h for fixed x3x\in\mathbb{R}^{3}, whereas to get (3.2), we used (G1) and standard properties of Sobolev spaces (cf., e.g., [3, Thm. 1] or [4, Prop. 9.3]).

Similarly, for the antisymmetric exchange term, we obtain

|ε(m)|\displaystyle|\mathcal{H}_{\varepsilon}(m)| Ω×Ω|νε(xy)||xy||m(x)×[m(y)m(x)]|dxdy\displaystyle\leqslant\iint_{\Omega\times\Omega}\frac{\mathopen{}\mathclose{{\left|\nu_{\varepsilon}(x-y)}}\right|}{|x-y|}\cdot|m(x)\times[m(y)-m(x)]|\hskip 1.69998pt\mathrm{d}x\hskip 1.69998pt\mathrm{d}y\hskip 1.69998pt
3×3|νε(xy)||xy||m~(x)||m~(y)m~(x)|dxdy\displaystyle\leqslant\iint_{\mathbb{R}^{3}\times\mathbb{R}^{3}}\frac{|\nu_{\varepsilon}(x-y)|}{|x-y|}|\tilde{m}(x)||\tilde{m}(y)-\tilde{m}(x)|\hskip 1.69998pt\mathrm{d}x\hskip 1.69998pt\mathrm{d}y
=3|νε(h)|(3|m~(x)||m~(x+h)m~(x)||h|dx)dh,\displaystyle=\int_{\mathbb{R}^{3}}|\nu_{\varepsilon}(h)|\mathopen{}\mathclose{{\left(\int_{\mathbb{R}^{3}}|\tilde{m}(x)|\frac{|\tilde{m}(x+h)-\tilde{m}(x)|}{|h|}\hskip 1.69998pt\mathrm{d}x}}\right)\mathrm{d}h, (3.3)

where in (3.3) we performed the change of variables yx+hy\mapsto x+h for fixed x3x\in\mathbb{R}^{3} and then used the odd symmetry of the kernel νε\nu_{\varepsilon}. Applying Hölder’s inequality as well as classical properties of Sobolev spaces (see [4, Prop. 9.3]), from (3.3) and (H1) we infer that

|ε(m)|\displaystyle|\mathcal{H}_{\varepsilon}(m)| 3|νε(h)|m~L2(3)τhm~m~L2(3)|h|dh\displaystyle\leqslant\int_{\mathbb{R}^{3}}|\nu_{\varepsilon}(h)|\|\tilde{m}\|_{L^{2}(\mathbb{R}^{3})}\frac{\|\tau_{h}\tilde{m}-\tilde{m}\|_{L^{2}(\mathbb{R}^{3})}}{|h|}\hskip 1.69998pt\mathrm{d}h
νεL1(3)m~L2(3)m~L2(3)\displaystyle\leqslant\|\nu_{\varepsilon}\|_{L^{1}(\mathbb{R}^{3})}\|\tilde{m}\|_{L^{2}(\mathbb{R}^{3})}\|\nabla\tilde{m}\|_{L^{2}(\mathbb{R}^{3})}
CmH1(Ω)2,\displaystyle\leqslant C\hskip 1.69998pt\|m\|^{2}_{H^{1}(\Omega)},

for some constant C>0C>0 depending only on Ω\Omega. This completes the proof of (2.13).

Step 2. We claim that it is sufficient to show that if supε>0ε(m)<+\sup_{\varepsilon>0}\hskip 1.69998pt\mathcal{E}_{\varepsilon}(m)<+\infty, then supε>0ε(m)<+\sup_{\varepsilon>0}\hskip 1.69998pt\mathcal{F}_{\varepsilon}(m)<+\infty. Indeed, as soon as we show that supε>0ε(m)<+\sup_{\varepsilon>0}\hskip 1.69998pt\mathcal{F}_{\varepsilon}(m)<+\infty, we can then invoke [24, Theorem 5] which shows that if supε>0ε(m)<+\sup_{\varepsilon>0}\hskip 1.69998pt\mathcal{F}_{\varepsilon}(m)<+\infty and the family (ρε)ε\mathopen{}\mathclose{{\left(\rho_{\varepsilon}}}\right)_{\varepsilon} satisfies (G1)(G3), then mH1(Ω;3)m\in H^{1}(\Omega;\mathbb{R}^{3}) and there exists α>0\alpha>0 (depending only on the family of kernels (ρε)ε\mathopen{}\mathclose{{\left(\rho_{\varepsilon}}}\right)_{\varepsilon}) such that

α2Ω|m|2dxlim supε0Ω×Ωρε(xy)|m(x)m(y)|2|xy|2dxdy.\alpha^{2}\int_{\Omega}|\nabla m|^{2}\mathrm{d}x\leqslant\limsup_{\varepsilon\to 0}\iint_{\Omega\times\Omega}\rho_{\varepsilon}(x-y)\frac{|m(x)-m(y)|^{2}}{|x-y|^{2}}\hskip 1.69998pt\mathrm{d}x\hskip 1.69998pt\mathrm{d}y. (3.4)

Since ε(m)\mathcal{H}_{\varepsilon}(m) is a quantity without a definite sign, we use Young’s inequality for products to estimate the term ε(m)\mathcal{H}_{\varepsilon}(m) from below. We get that for every δ>0\delta>0 there holds

ε(m)\displaystyle\mathcal{H}_{\varepsilon}(m) =Ω×Ωm(x)m(y)|xy|(m(y)×νε(xy))dxdy\displaystyle=\iint_{\Omega\times\Omega}\frac{m(x)-m(y)}{|x-y|}\cdot\mathopen{}\mathclose{{\left(m(y)\times\nu_{\varepsilon}(x-y)}}\right)\hskip 1.69998pt\mathrm{d}x\hskip 1.69998pt\mathrm{d}y
=Ω×Ω(ρε(xy)m(x)m(y)|xy|)(m(y)×νε(xy)ρε(xy))dxdy\displaystyle=\iint_{\Omega\times\Omega}\mathopen{}\mathclose{{\left(\sqrt{\rho_{\varepsilon}(x-y)}\frac{m(x)-m(y)}{|x-y|}}}\right)\cdot\mathopen{}\mathclose{{\left(m(y)\times\frac{\nu_{\varepsilon}(x-y)}{\sqrt{\rho_{\varepsilon}(x-y)}}}}\right)\hskip 1.69998pt\mathrm{d}x\hskip 1.69998pt\mathrm{d}y
δ2ε(m)14δ2Ω×Ω|m(y)×νε(xy)|2ρε(xy)dxdy.\displaystyle\geqslant-\delta^{2}\mathcal{F}_{\varepsilon}(m)-\frac{1}{4\delta^{2}}\iint_{\Omega\times\Omega}\frac{|m(y)\times\nu_{\varepsilon}(x-y)|^{2}}{\rho_{\varepsilon}(x-y)}\mathrm{d}x\hskip 1.69998pt\mathrm{d}y. (3.5)

Adding ε(m)\mathcal{F}_{\varepsilon}(m) to both sides of the previous estimate, we get that for every δ>0\delta>0 there holds

(1δ2)ε(m)14δ2Ω×Ω|m(y)×νε(xy)|2ρε(xy)dxdyε(m).(1-\delta^{2})\mathcal{F}_{\varepsilon}(m)-\frac{1}{4\delta^{2}}\iint_{\Omega\times\Omega}\frac{|m(y)\times\nu_{\varepsilon}(x-y)|^{2}}{\rho_{\varepsilon}(x-y)}\mathrm{d}x\hskip 1.69998pt\mathrm{d}y\leqslant\mathcal{E}_{\varepsilon}(m). (3.6)

Now, by (A1), and taking into account that νεL1(3)=1\|\nu_{\varepsilon}\|_{L^{1}(\mathbb{R}^{3})}=1, we infer

Ω×Ω|m(y)×νε(xy)|2ρε(xy)dxdy\displaystyle\iint_{\Omega\times\Omega}\frac{|m(y)\times\nu_{\varepsilon}(x-y)|^{2}}{\rho_{\varepsilon}(x-y)}\mathrm{d}x\hskip 1.69998pt\mathrm{d}y Ω|m(y)|2(3|νε(z)|2ρε(z)dz)dy\displaystyle\leqslant\int_{\Omega}|m(y)|^{2}\mathopen{}\mathclose{{\left(\int_{\mathbb{R}^{3}}\frac{|\nu_{\varepsilon}(z)|^{2}}{\rho_{\varepsilon}(z)}\mathrm{d}z}}\right)\mathrm{d}y
14(Ω|m(y)|2dy)νεL1(3)νερεL(3)\displaystyle\leqslant\frac{1}{4}\mathopen{}\mathclose{{\left(\int_{\Omega}|m(y)|^{2}\mathrm{d}y}}\right)\lVert\nu_{\varepsilon}\rVert_{L^{1}(\mathbb{R}^{3})}\hskip 1.69998pt\mathopen{}\mathclose{{\left\lVert\frac{\nu_{\varepsilon}}{\rho_{\varepsilon}}}}\right\rVert_{L^{\infty}(\mathbb{R}^{3})}
CmL2(Ω)2,\displaystyle\leqslant C\hskip 1.69998pt\|m\|_{L^{2}(\Omega)}^{2}, (3.7)

with C>0C>0 proportional to the constant from (A1). Hence, from (3.6) and (3.7) we deduce

(1δ2)ε(m)ε(m)+CmL2(Ω)2(1-\delta^{2})\mathcal{F}_{\varepsilon}(m)\leqslant\mathcal{E}_{\varepsilon}(m)+C\|m\|_{L^{2}(\Omega)}^{2} (3.8)

By taking δ2:=1/2\delta^{2}:=1/2 we conclude that if supε>0ε(m)<+\sup_{\varepsilon>0}\hskip 1.69998pt\mathcal{E}_{\varepsilon}(m)<+\infty then supε>0ε(m)<+\sup_{\varepsilon>0}\hskip 1.69998pt\mathcal{F}_{\varepsilon}(m)<+\infty and hence, by [24, Theorem 5], that mH1(Ω;3)m\in H^{1}(\Omega;\mathbb{R}^{3}). ∎

Proof of Theorem 2.3.

We split the proof into two main steps. In Step 1, we prove (2.16), which, in particular, gives us (2.15) under the additional hypothesis that mC2(Ω¯;3)m\in C^{2}(\bar{\Omega};\mathbb{R}^{3}). Then, in Step 2, we use a density argument to show that if (2.15) holds for every mC2(Ω¯;3)m\in C^{2}(\bar{\Omega};\mathbb{R}^{3}) then it holds for every mH1(Ω;3)m\in H^{1}(\Omega;\mathbb{R}^{3}).

Step 1. Let yΩy\in\Omega. For R>0R>0 such that Rdist(y,Ω)R\leqslant\text{dist}(y,\partial\Omega), we split ε(mε)\mathcal{H}_{\varepsilon}(m_{\varepsilon}) as

ε(mε)\displaystyle\mathcal{H}_{\varepsilon}(m_{\varepsilon}) =Ω(BR(y)νε(xy)|xy|(mε(x)×mε(y))dx)dy\displaystyle=\int_{\Omega}\mathopen{}\mathclose{{\left(\int_{B_{R}(y)}\frac{\nu_{\varepsilon}(x-y)}{|x-y|}\cdot(m_{\varepsilon}(x)\times m_{\varepsilon}(y))\hskip 1.69998pt\mathrm{d}x}}\right)\mathrm{d}y
+Ω(ΩBR(y)νε(xy)|xy|(mε(x)×mε(y))dx)dy\displaystyle\qquad+\int_{\Omega}\mathopen{}\mathclose{{\left(\int_{\Omega\setminus B_{R}(y)}\frac{\nu_{\varepsilon}(x-y)}{|x-y|}\cdot(m_{\varepsilon}(x)\times m_{\varepsilon}(y))\hskip 1.69998pt\mathrm{d}x}}\right)\mathrm{d}y
=:ΩIε,R(mε,y)+Jε,R(mε,y)dy\displaystyle=:\int_{\Omega}I_{\varepsilon,R}(m_{\varepsilon},y)+J_{\varepsilon,R}(m_{\varepsilon},y)\mathrm{d}y (3.9)

Outside of the ball BR(y)B_{R}(y), we note that

|Jε,R(mε,y)|\displaystyle\mathopen{}\mathclose{{\left|J_{\varepsilon,R}(m_{\varepsilon},y)}}\right|\leqslant ΩBR(y)|νε(xy)||xy||mε(x)||mε(y)|dx\displaystyle\int_{\Omega\setminus B_{R}(y)}\frac{|\nu_{\varepsilon}(x-y)|}{|x-y|}|m_{\varepsilon}(x)||m_{\varepsilon}(y)|\hskip 1.69998pt\mathrm{d}x
1RmεL(Ω)23BR(0)|νε(x)|dx.\displaystyle\leqslant\frac{1}{R}\|m_{\varepsilon}\|^{2}_{L^{\infty}(\Omega)}\int_{\mathbb{R}^{3}\setminus B_{R}(0)}|\nu_{\varepsilon}(x)|\hskip 1.69998pt\mathrm{d}x.

Sending ε0\varepsilon\to 0 (for fixed R>0R>0), by assumption (H2) we find that limε0νεL1(3BR(0))=0\lim_{\varepsilon\to 0}\|\nu_{\varepsilon}\|_{L^{1}\mathopen{}\mathclose{{\left(\mathbb{R}^{3}\setminus B_{R}(0)}}\right)}=0 and, therefore, that

limε0|Jε,R(mε,y)|=0.\lim_{\varepsilon\to 0}\,\mathopen{}\mathclose{{\left|J_{\varepsilon,R}(m_{\varepsilon},y)}}\right|=0. (3.10)

We now take into account the contribution to ε(mε)\mathcal{H}_{\varepsilon}(m_{\varepsilon}) inside of the ball BR(y)B_{R}(y), i.e., the term Iε(mε,y)I_{\varepsilon}(m_{\varepsilon},y) in (3.9). Given that mεC2(Ω¯;3)m_{\varepsilon}\in C^{2}(\bar{\Omega};\mathbb{R}^{3}), we perform a first-order Taylor expansion inside BR(y)B_{R}(y) of the form mε(x)=mε(y)+Dmε(y)(xy)+Rε(x,y)m_{\varepsilon}(x)=m_{\varepsilon}(y)+\mathrm{D}m_{\varepsilon}(y)(x-y)+R_{\varepsilon}(x,y) with (up to a constant factor)

|Rε(x,y)|D(2)mεL|xy|2|R_{\varepsilon}(x,y)|\leqslant\|D^{(2)}m_{\varepsilon}\|_{L^{\infty}}\cdot|x-y|^{2}

to find

Iε,R(mε,y)\displaystyle I_{\varepsilon,R}(m_{\varepsilon},y) =BR(y)νε(xy)|xy|(Dmε(y)(xy)×mε(y))dx\displaystyle=\int_{B_{R}(y)}\frac{\nu_{\varepsilon}(x-y)}{|x-y|}\cdot(\mathrm{D}m_{\varepsilon}(y)(x-y)\times m_{\varepsilon}(y))\hskip 1.69998pt\mathrm{d}x (3.11)
+BR(y)νε(xy)|xy|(Rε(x,y)×mε(y))dx\displaystyle\qquad\qquad+\int_{B_{R}(y)}\frac{\nu_{\varepsilon}(x-y)}{|x-y|}\cdot(R_{\varepsilon}(x,y)\times m_{\varepsilon}(y))\hskip 1.69998pt\mathrm{d}x
=:Iε,R(1)(mε,y)+Iε,R(2)(mε,y).\displaystyle=:I_{\varepsilon,R}^{(1)}(m_{\varepsilon},y)+I_{\varepsilon,R}^{(2)}(m_{\varepsilon},y).

Let us focus on the first term on the right-hand side of (3.11). Simple algebra gives that

Iε,R(1)(mε,y)\displaystyle I_{\varepsilon,R}^{(1)}(m_{\varepsilon},y) =mε(y)BR(y)(νε(xy)|xy|×imε(y)(xiyi))dx\displaystyle=m_{\varepsilon}(y)\cdot\int_{B_{R}(y)}\mathopen{}\mathclose{{\left(\frac{\nu_{\varepsilon}(x-y)}{|x-y|}\times\partial_{i}m_{\varepsilon}(y)(x_{i}-y_{i})}}\right)\hskip 1.69998pt\mathrm{d}x
=i=13(BR(y)(xiyi)|xy|νε(xy)dx)(imε(y)×mε(y)).\displaystyle=\sum_{i=1}^{3}\mathopen{}\mathclose{{\left(\int_{B_{R}(y)}\frac{(x_{i}-y_{i})}{|x-y|}\nu_{\varepsilon}(x-y)\hskip 1.69998pt\mathrm{d}x}}\right)\cdot(\partial_{i}m_{\varepsilon}(y)\times m_{\varepsilon}(y)). (3.12)

Next, we observe that as a consequence of (H2) we have

limε0|3BR(y)νε(xy)|xy|(xiyi)dx|limε03BR(y)|νε(xy)|dx=0.\lim_{\varepsilon\to 0}\mathopen{}\mathclose{{\left|\int_{\mathbb{R}^{3}\setminus B_{R}(y)}\frac{\nu_{\varepsilon}(x-y)}{|x-y|}(x_{i}-y_{i})\hskip 1.69998pt\mathrm{d}x\hskip 1.69998pt}}\right|\leqslant\lim_{\varepsilon\to 0}\int_{\mathbb{R}^{3}\setminus B_{R}(y)}|\nu_{\varepsilon}(x-y)|\hskip 1.69998pt\mathrm{d}x=0. (3.13)

Therefore, combining (3.13) and (H3) we infer that for i=1,2,3i=1,2,3, there holds

limε0BR(y)(xiyi)|xy|νε(xy)dx\displaystyle\lim_{\varepsilon\to 0}\int_{B_{R}(y)}\frac{(x_{i}-y_{i})}{|x-y|}\nu_{\varepsilon}(x-y)\hskip 1.69998pt\mathrm{d}x =limε03(xiyi)|xy|νε(xy)dx\displaystyle=\lim_{\varepsilon\to 0}\int_{\mathbb{R}^{3}}\frac{(x_{i}-y_{i})}{|x-y|}\nu_{\varepsilon}(x-y)\hskip 1.69998pt\mathrm{d}x
=limε03xi|x|νε(x)dx\displaystyle=\lim_{\varepsilon\to 0}\int_{\mathbb{R}^{3}}\frac{x_{i}}{|x|}\nu_{\varepsilon}(x)\hskip 1.69998pt\mathrm{d}x
=di.\displaystyle=d_{i}. (3.14)

As a consequence, from (3.12) and (3.14) we conclude

limε0Iε,R(1)(mε,y)=i=13m(y)(di×im(y)).\lim_{\varepsilon\to 0}I_{\varepsilon,R}^{(1)}(m_{\varepsilon},y)=\sum_{i=1}^{3}m(y)\cdot(d_{i}\times\partial_{i}m(y)). (3.15)

Note that (3.15) does not depend on RR.

For the term Iε,R2(mε,y)I_{\varepsilon,R}^{2}(m_{\varepsilon},y), we observe that

|Iε,R(2)(mε,y)|\displaystyle\mathopen{}\mathclose{{\left|I_{\varepsilon,R}^{(2)}(m_{\varepsilon},y)}}\right| |mε(y)|BR(y)|νε(xy)||Rε(x,y)||xy|dx\displaystyle\leqslant|m_{\varepsilon}(y)|\int_{B_{R}(y)}|\nu_{\varepsilon}(x-y)|\frac{|R_{\varepsilon}(x,y)|}{|x-y|}\hskip 1.69998pt\mathrm{d}x
R|mε(y)|D(2)mεL3|νε(x)|dx\displaystyle\leqslant R|m_{\varepsilon}(y)|\|D^{(2)}m_{\varepsilon}\|_{L^{\infty}}\cdot\int_{\mathbb{R}^{3}}|\nu_{\varepsilon}(x)|\hskip 1.69998pt\mathrm{d}x
=R|mε(y)|D(2)mεL.\displaystyle=R|m_{\varepsilon}(y)|\|D^{(2)}m_{\varepsilon}\|_{L^{\infty}}.

For the latter equality we used assumption (H1). Thus, since mεmm_{\varepsilon}\to m strongly in C2(Ω¯;3)C^{2}(\overline{\Omega};\mathbb{R}^{3}),

limR0limε0|Iε,R(2)(mε,y)|=0.\lim_{R\rightarrow 0}\lim_{\varepsilon\to 0}\hskip 1.69998pt\mathopen{}\mathclose{{\left|I_{\varepsilon,R}^{(2)}(m_{\varepsilon},y)}}\right|=0. (3.16)

Overall, combining estimates (3.10), (3.12), (3.14), (3.15), and (3.16), we infer

limε0Ωνε(xy)|xy|(mε(x)×mε(y))dx\displaystyle\lim_{\varepsilon\rightarrow 0}\int_{\Omega}\frac{\nu_{\varepsilon}(x-y)}{|x-y|}\cdot(m_{\varepsilon}(x)\times m_{\varepsilon}(y))\hskip 1.69998pt\mathrm{d}x =limR0limε0(Iε,R(mε,y)+Jε,R(mε,y))\displaystyle=\lim_{R\rightarrow 0}\lim_{\varepsilon\rightarrow 0}\mathopen{}\mathclose{{\left(I_{\varepsilon,R}(m_{\varepsilon},y)+J_{\varepsilon,R}(m_{\varepsilon},y)}}\right) (3.17)
=limR0limε0Iε,R(mε,y)\displaystyle=\lim_{R\rightarrow 0}\lim_{\varepsilon\rightarrow 0}I_{\varepsilon,R}(m_{\varepsilon},y)
=limR0limε0(Iε,R(1)(mε,y)+Iε,R(2)(mε,y))\displaystyle=\lim_{R\rightarrow 0}\lim_{\varepsilon\rightarrow 0}\mathopen{}\mathclose{{\left(I_{\varepsilon,R}^{(1)}(m_{\varepsilon},y)+I_{\varepsilon,R}^{(2)}(m_{\varepsilon},y)}}\right)
=limε0Iε,R(1)(mε,y)\displaystyle=\lim_{\varepsilon\rightarrow 0}I_{\varepsilon,R}^{(1)}(m_{\varepsilon},y)
=i=13m(y)(di×im(y)).\displaystyle=\sum_{i=1}^{3}m(y)\cdot(d_{i}\times\partial_{i}m(y)). (3.18)

In view of (3.18), to prove

limε0ε(mε)=i=13Ωm(y)(di×im(y))dy,\lim_{\varepsilon\rightarrow 0}\mathcal{H}_{\varepsilon}(m_{\varepsilon})=\sum_{i=1}^{3}\int_{\Omega}m(y)\cdot(d_{i}\times\partial_{i}m(y))\mathrm{d}y, (3.19)

it is sufficient to show that the family of functions on the left-hand side of (3.17) falls under the hypotheses of the Dominated Convergence Theorem. But this is indeed the case because of the estimate

|Ωνε(xy)|xy|(mε(x)×mε(y))dx|\displaystyle\mathopen{}\mathclose{{\left|\int_{\Omega}\frac{\nu_{\varepsilon}(x-y)}{|x-y|}\cdot(m_{\varepsilon}(x)\times m_{\varepsilon}(y))\hskip 1.69998pt\mathrm{d}x}}\right| mεL(Ω)Ω|νε(xy)||mε(x)mε(y)||xy|dx\displaystyle\leqslant\|m_{\varepsilon}\|_{L^{\infty}(\Omega)}\int_{\Omega}|\nu_{\varepsilon}(x-y)|\frac{|m_{\varepsilon}(x)-m_{\varepsilon}(y)|}{|x-y|}\hskip 1.69998pt\mathrm{d}x
LmεmεL(Ω)3|νε(x)|dx\displaystyle\leqslant L_{m_{\varepsilon}}\hskip 1.69998pt\|m_{\varepsilon}\|_{L^{\infty}(\Omega)}\int_{\mathbb{R}^{3}}|\nu_{\varepsilon}(x)|\hskip 1.69998pt\mathrm{d}x
=LmεmεL(Ω),\displaystyle=L_{m_{\varepsilon}}\hskip 1.69998pt\|m_{\varepsilon}\|_{L^{\infty}(\Omega)}, (3.20)

where LmεL_{m_{\varepsilon}} is the Lipschitz constant of mεm_{\varepsilon}, which is uniformly bounded in ε\varepsilon.

Step 2. This second step concludes the proof by showing, through a density argument, that if (2.15) holds for every mC2(Ω¯;3)m\in C^{2}(\bar{\Omega};\mathbb{R}^{3}) then it holds for every mH1(Ω;3)m\in H^{1}(\Omega;\mathbb{R}^{3}).

We note that, for every m1,m2H1(Ω;3)m_{1},m_{2}\in H^{1}(\Omega;\mathbb{R}^{3}), there holds

ε(m1)ε(m2)\displaystyle\mathcal{H}_{\varepsilon}(m_{1})-\mathcal{H}_{\varepsilon}(m_{2}) =Ω×Ωνε(xy)|xy|((m1(x)m1(y))×m1(y))dxdy\displaystyle=\iint_{\Omega\times\Omega}\frac{\nu_{\varepsilon}(x-y)}{|x-y|}\cdot((m_{1}(x)-m_{1}(y))\times m_{1}(y))\mathrm{d}x\hskip 1.69998pt\mathrm{d}y
Ω×Ωνε(xy)|xy|((m2(x)m2(y))×m2(y))dxdy.\displaystyle\qquad-\iint_{\Omega\times\Omega}\frac{\nu_{\varepsilon}(x-y)}{|x-y|}\cdot((m_{2}(x)-m_{2}(y))\times m_{2}(y))\mathrm{d}x\hskip 1.69998pt\mathrm{d}y.

Thus, adding and subtracting the quantity (m1(x)m1(y))×m2(y)(m_{1}(x)-m_{1}(y))\times m_{2}(y), we infer

|ε(m1)ε(m2)|\displaystyle\mathopen{}\mathclose{{\left|\mathcal{H}_{\varepsilon}(m_{1})-\mathcal{H}_{\varepsilon}(m_{2})}}\right| Ω×Ω|νε(xy)||xy||m1(x)m1(y)||m1(y)m2(y)|dxdy\displaystyle\leqslant\iint_{\Omega\times\Omega}\frac{|\nu_{\varepsilon}(x-y)|}{|x-y|}|m_{1}(x)-m_{1}(y)||m_{1}(y)-m_{2}(y)|\mathrm{d}x\hskip 1.69998pt\mathrm{d}y
+Ω×Ω|νε(xy)||xy||(m1(x)m1(y))(m2(x)m2(y))||m2(y)|dxdy.\displaystyle+\iint_{\Omega\times\Omega}\frac{|\nu_{\varepsilon}(x-y)|}{|x-y|}|(m_{1}(x)-m_{1}(y))-(m_{2}(x)-m_{2}(y))||m_{2}(y)|\mathrm{d}x\hskip 1.69998pt\mathrm{d}y.
=:I1+I2.\displaystyle=:I_{1}+I_{2}. (3.21)

Let now m~1,m~2H1(3;3)\tilde{m}_{1},\tilde{m}_{2}\in H^{1}(\mathbb{R}^{3};\mathbb{R}^{3}) be extensions of m1,m2m_{1},m_{2} from Ω\Omega to 3\mathbb{R}^{3}. For the first integral I1I_{1} in (3.21) we have

I1\displaystyle I_{1} 3|νε(h)||h|(3|m~1(x+h)m~1(x)||m~1(x+h)m~2(x+h)|dx)dh\displaystyle\leqslant\int_{\mathbb{R}^{3}}\frac{|\nu_{\varepsilon}(h)|}{|h|}\mathopen{}\mathclose{{\left(\int_{\mathbb{R}^{3}}|\tilde{m}_{1}(x+h)-\tilde{m}_{1}(x)||\tilde{m}_{1}(x+h)-\tilde{m}_{2}(x+h)|\hskip 1.69998pt\mathrm{d}x}}\right)\mathrm{d}h
3|νε(h)|τhm~1m~1L2(3)|h|m~1m~2L2(3)dh\displaystyle\leqslant\int_{\mathbb{R}^{3}}|\nu_{\varepsilon}(h)|\hskip 1.69998pt\frac{\lVert\tau_{h}\tilde{m}_{1}-\tilde{m}_{1}\rVert_{L^{2}(\mathbb{R}^{3})}}{|h|}\hskip 1.69998pt\|\tilde{m}_{1}-\tilde{m}_{2}\|_{L^{2}(\mathbb{R}^{3})}\hskip 1.69998pt\mathrm{d}h
m~1L2(3)m~1m~2L2(3)\displaystyle\leqslant\|\nabla\tilde{m}_{1}\|_{L^{2}(\mathbb{R}^{3})}\lVert\tilde{m}_{1}-\tilde{m}_{2}\rVert_{L^{2}(\mathbb{R}^{3})}
Cm1H1(Ω)m1m2L2(Ω),\displaystyle\leqslant C\hskip 1.69998pt\|m_{1}\|_{H^{1}(\Omega)}\|m_{1}-m_{2}\|_{L^{2}(\Omega)}, (3.22)

for some constant C=C(Ω)>0C=C(\Omega)>0, where we have applied Hölder inequality and classical properties of Sobolev spaces, cf. [4, Prop. 9.3]. Analogously, for the second integral I2I_{2} in (3.21) we find

I2\displaystyle I_{2} 3|νε(h)|3|m~2(x+h)||(m~1m~2)(x)(m~1m~2)(x+h)||h|dxdh\displaystyle\leqslant\int_{\mathbb{R}^{3}}|\nu_{\varepsilon}(h)|\int_{\mathbb{R}^{3}}|\tilde{m}_{2}(x+h)|\frac{|(\tilde{m}_{1}-\tilde{m}_{2})(x)-(\tilde{m}_{1}-\tilde{m}_{2})(x+h)|}{|h|}\hskip 1.69998pt\mathrm{d}x\hskip 1.69998pt\mathrm{d}h
3|νε(h)|m~2L2(3)τh(m~1m~2)(m~1m~2)L2(3)|h|dh\displaystyle\leqslant\int_{\mathbb{R}^{3}}\mathopen{}\mathclose{{\left|\nu_{\varepsilon}(h)}}\right|\lVert\tilde{m}_{2}\rVert_{L^{2}(\mathbb{R}^{3})}\frac{\lVert\tau_{h}(\tilde{m}_{1}-\tilde{m}_{2})-(\tilde{m}_{1}-\tilde{m}_{2})\rVert_{L^{2}(\mathbb{R}^{3})}}{|h|}\hskip 1.69998pt\mathrm{d}h
(m~1m~2)L2(3)m~2L2(3)\displaystyle\leqslant\lVert\nabla(\tilde{m}_{1}-\tilde{m}_{2})\rVert_{L^{2}(\mathbb{R}^{3})}\lVert\tilde{m}_{2}\rVert_{L^{2}(\mathbb{R}^{3})}
Cm1m2H1(Ω)m2L2(Ω),\displaystyle\leqslant C\hskip 1.69998pt\lVert m_{1}-m_{2}\rVert_{H^{1}(\Omega)}\lVert m_{2}\rVert_{L^{2}(\Omega)}, (3.23)

for some constant C=C(Ω)>0C=C(\Omega)>0.

To sum up, estimating (3.21) through (3.22) and (3.23), we obtain that, for every m1,m2H1(Ω;3)m_{1},m_{2}\in H^{1}(\Omega;\mathbb{R}^{3}), there holds

|ε(m1)ε(m2)|C(m1H1(Ω)+m2H1(Ω))m1m2H1(Ω).\mathopen{}\mathclose{{\left|\mathcal{H}_{\varepsilon}(m_{1})-\mathcal{H}_{\varepsilon}(m_{2})}}\right|\leqslant C\hskip 1.69998pt(\|m_{1}\|_{H^{1}(\Omega)}+\|m_{2}\|_{H^{1}(\Omega)})\|m_{1}-m_{2}\|_{H^{1}(\Omega)}. (3.24)

Now, by density, for every mH1(Ω;3)m\in H^{1}(\Omega;\mathbb{R}^{3}) there exists a sequence (mk)k(m_{k})_{k\in\mathbb{N}} in Cc2(3;3)C^{2}_{c}(\mathbb{R}^{3};\mathbb{R}^{3}) such that

limkmmkH1(Ω)=0.\lim_{k\to\infty}\|m-m_{k}\|_{H^{1}(\Omega)}=0. (3.25)

Also, by Step 1, we know that for every mkCc2(3;3)m_{k}\in C^{2}_{c}(\mathbb{R}^{3};\mathbb{R}^{3}) one has

limε0ε(mk)=i=13Ωmk(x)(di×imk(x))dx=(mk)\lim_{\varepsilon\to 0}\mathcal{H}_{\varepsilon}(m_{k})=\sum_{i=1}^{3}\int_{\Omega}m_{k}(x)\cdot(d_{i}\times\partial_{i}m_{k}(x))\hskip 1.69998pt\mathrm{d}x=\mathcal{H}(m_{k}) (3.26)

Moreover, in view of (3.24), for every kk\in\mathbb{N}, there holds

lim supε0|ε(m)ε(mk)|C(mH1(Ω)+mkH1(Ω))mmkH1(Ω).\limsup_{\varepsilon\to 0}\hskip 1.69998pt\mathopen{}\mathclose{{\left|\mathcal{H}_{\varepsilon}(m)-\mathcal{H}_{\varepsilon}(m_{k})}}\right|\leqslant C(\|m\|_{H^{1}(\Omega)}+\|m_{k}\|_{H^{1}(\Omega)})\|m-m_{k}\|_{H^{1}(\Omega)}. (3.27)

Letting now k+k\to+\infty, by (3.25) and by (3.27),

limk+lim supε0|ε(m)ε(mk)|=0,\lim_{k\to+\infty}\limsup_{\varepsilon\to 0}\hskip 1.69998pt\mathopen{}\mathclose{{\left|\mathcal{H}_{\varepsilon}(m)-\mathcal{H}_{\varepsilon}(m_{k})}}\right|=0, (3.28)

thanks to the density (3.25) and the fact that mkH1(Ω)\|m_{k}\|_{H^{1}(\Omega)} is bounded in H1H^{1}.

In view of (3.25), we also have that di×imkdi×imd_{i}\times\partial_{i}m_{k}\to d_{i}\times\partial_{i}m in L2(Ω;3)L^{2}(\Omega;\mathbb{R}^{3}) for i=1,2,3i=1,2,3. As a consequence, we have that

limk(mk)=(m),\lim_{k\to\infty}\mathcal{H}(m_{k})=\mathcal{H}(m), (3.29)

that is, the limit functional \mathcal{H} is continuous with respect to the H1H^{1}-convergence.

To sum up, by (3.26), (3.28) and (3.29), we obtain that

lim supε0|ε(m)(m)|\displaystyle\limsup_{\varepsilon\to 0}\hskip 1.69998pt\mathopen{}\mathclose{{\left|\mathcal{H}_{\varepsilon}(m)-\mathcal{H}(m)}}\right| lim supε0(|ε(m)ε(mk)|+|ε(mk)(mk)|)+|(mk)(m)|\displaystyle\leqslant\limsup_{\varepsilon\to 0}\hskip 1.69998pt\mathopen{}\mathclose{{\left(\mathopen{}\mathclose{{\left|\mathcal{H}_{\varepsilon}(m)-\mathcal{H}_{\varepsilon}(m_{k})}}\right|+\hskip 1.69998pt\mathopen{}\mathclose{{\left|\mathcal{H}_{\varepsilon}(m_{k})-\mathcal{H}(m_{k})}}\right|}}\right)+\color[rgb]{0,0,0}|\mathcal{H}(m_{k})-\mathcal{H}(m)|\color[rgb]{0,0,0}
=(lim supε0|ε(m)ε(mk)|)+|(mk)(m)|.\displaystyle=\mathopen{}\mathclose{{\left(\limsup_{\varepsilon\to 0}\hskip 1.69998pt\mathopen{}\mathclose{{\left|\mathcal{H}_{\varepsilon}(m)-\mathcal{H}_{\varepsilon}(m_{k})}}\right|}}\right)+|\mathcal{H}(m_{k})-\mathcal{H}(m)|. (3.30)

Passing to the limit for kk\rightarrow\infty, we conclude that for every mH1(Ω;3)m\in H^{1}(\Omega;\mathbb{R}^{3}), there holds ε(m)(m)\mathcal{H}_{\varepsilon}(m)\rightarrow\mathcal{H}(m) as ε0\varepsilon\rightarrow 0, i.e.,

limε0Ω×Ωνε(xy)|xy|(m(x)×m(y))dxdy=i=13Ωm(x)(di×im(x))dx.\lim_{\varepsilon\to 0}\iint_{\Omega\times\Omega}\frac{\nu_{\varepsilon}(x-y)}{|x-y|}\cdot(m(x)\times m(y))\mathrm{d}x\hskip 1.69998pt\mathrm{d}y=\sum_{i=1}^{3}\int_{\Omega}m(x)\cdot(d_{i}\times\partial_{i}m(x))\mathrm{d}x.

This completes the proof. ∎

4. Compactness and Γ\Gamma-convergence (proof of Theorem 2.1)

Proof of Theorem 2.1.i, (Compactness).

By the proof of Theorem 2.2 (cf. (3.8)) we know that for every ε>0\varepsilon>0 there holds

12ε(mε)C(ε(mε)+mεL2(Ω)2),\frac{1}{2}\mathcal{F}_{\varepsilon}(m_{\varepsilon})\leqslant C\mathopen{}\mathclose{{\left(\mathcal{E}_{\varepsilon}(m_{\varepsilon})+\|m_{\varepsilon}\|_{L^{2}(\Omega)}^{2}}}\right), (4.1)

with C>0C>0 depending only on the homonymous constant in (A1). Since (mε)ε(m_{\varepsilon})_{\varepsilon} consists of 𝕊2\mathbb{S}^{2}-valued vector fields, i.e., |mε|=1|m_{\varepsilon}|=1 a.e. in Ω\Omega for every ε>0\varepsilon>0, from the assumption (2.9) and (4.1) we infer that

lim infε0ε(mε)C,\liminf_{\varepsilon\rightarrow 0}\mathcal{F}_{\varepsilon}(m_{\varepsilon})\leqslant C, (4.2)

for some C>0C>0 depending only on lim infε0ε(mε)\liminf_{\varepsilon\rightarrow 0}\hskip 1.69998pt\mathcal{E}_{\varepsilon}(m_{\varepsilon}), Ω\Omega, and on the homonymous constant in (A1). In particular, in view of assumption (G4), recalling the definition of the kernels (ρεrad)ε(\rho_{\varepsilon}^{\rm rad})_{\varepsilon} in (2.4) and setting

ρ~ε(x):=ρεrad(|x|)ρεrad(||)L1(3),\tilde{\rho}_{\varepsilon}(x):=\frac{\rho_{\varepsilon}^{\rm rad}(|x|)}{\|\rho_{\varepsilon}^{\rm rad}(|\cdot|)\|_{L^{1}(\mathbb{R}^{3})}},

from estimate (4.2) we infer

lim infε0Ω×Ωρ~ε(xy)|m(x)m(y)|2|xy|2dxdyCκ.\liminf_{\varepsilon\to 0}\iint_{\Omega\times\Omega}\tilde{\rho}_{\varepsilon}(x-y)\frac{|m(x)-m(y)|^{2}}{|x-y|^{2}}\hskip 1.69998pt\mathrm{d}x\hskip 1.69998pt\mathrm{d}y\leqslant\frac{C}{\kappa}. (4.3)

Estimate (4.3) allows us to invoke the compactness result for the symmetric exchange energy established in [23, Theorem 1.2] for the case of radial kernels, which implies that if (mε)ε(m_{\varepsilon})_{\varepsilon} is a bounded family in L2(Ω;3)L^{2}(\Omega;\mathbb{R}^{3}) such that ε(mε)C\mathcal{F}_{\varepsilon}(m_{\varepsilon})\leqslant C for some positive constant CC, then there exists mH1(Ω;3)m\in H^{1}(\Omega;\mathbb{R}^{3}) and a subsequence εj0\varepsilon_{j}\to 0 such that mεjmm_{\varepsilon_{j}}\to m strongly in L2(Ω;3)L^{2}(\Omega;\mathbb{R}^{3}). In particular, we obtain that (up to a further subsequence) mεjmm_{\varepsilon_{j}}\to m a.e. as jj\to\infty and, therefore, also the limiting function satisfies the constraint, that is, mL2(Ω;𝕊2)m\in L^{2}(\Omega;\mathbb{S}^{2}). ∎

Proof of Theorem 2.1.ii, (Γ\Gamma-convergence).

We split the proof into two classical steps. First, we derive the Γ\Gamma-limsup\mathrm{limsup} inequality, then we focus on the Γ\Gamma-liminf\mathrm{liminf} inequality.

Limsup inequality. It is sufficient to note that the constant family mε:=mm_{\varepsilon}:=m for every ε\varepsilon, is a recovery sequence. Indeed, on the one hand, if mL2(Ω;𝕊2)H1(Ω;𝕊2)m\in L^{2}(\Omega;\mathbb{S}^{2})\setminus H^{1}(\Omega;\mathbb{S}^{2}) then Theorem 2.2 gives that ε(m)+\mathcal{E}_{\varepsilon}(m)\rightarrow+\infty; on the other hand, if mH1(Ω;𝕊2)m\in H^{1}(\Omega;\mathbb{S}^{2}) then by Corollary 2.1, we conclude that

limε0ε(m)=(m).\lim_{\varepsilon\to 0}\mathcal{E}_{\varepsilon}(m)=\mathcal{E}(m).

Liminf inequality. Let (mε)εL2(Ω;𝕊2)(m_{\varepsilon})_{\varepsilon}\subset L^{2}(\Omega;\mathbb{S}^{2}) be such that mεmm_{\varepsilon}\to m strongly in L2(Ω;𝕊2)L^{2}(\Omega;\mathbb{S}^{2}) as ε0\varepsilon\to 0. If lim infε0ε(mε)=+\liminf_{\varepsilon\rightarrow 0}\mathcal{E}_{\varepsilon}(m_{\varepsilon})=+\infty, there is nothing to prove. If lim infε0ε(mε)<+\liminf_{\varepsilon\rightarrow 0}\mathcal{E}_{\varepsilon}(m_{\varepsilon})<+\infty then, by (4.1), we know that

lim infε0ε(mε)C(1+lim infε0ε(mε))<+,\liminf_{\varepsilon\to 0}\mathcal{F}_{\varepsilon}(m_{\varepsilon})\leqslant C\mathopen{}\mathclose{{\left(1+\liminf_{\varepsilon\to 0}\mathcal{E}_{\varepsilon}(m_{\varepsilon})}}\right)<+\infty, (4.4)

for some constant CC depending only on Ω\Omega and the homonymous constant in (A1). Also, by the compactness result (Theorem 2.1.i), mH1(Ω;𝕊2)m\in H^{1}\mathopen{}\mathclose{{\left(\Omega;\mathbb{S}^{2}}}\right), and from [24, Lemma 8] used in the case ω(t)=t2\omega(t)=t^{2}, we get the existence of a finite Radon measure μ(𝕊2)\mu\in\mathcal{M}(\mathbb{S}^{2}), with μ(𝕊2)=1\mu\mathopen{}\mathclose{{\left(\mathbb{S}^{2}}}\right)=1, such that, possibly up to the extraction of a non-relabelled subsequence, there holds

lim infε0ε(mε)Ω(𝕊2|σm(x)|2dμ(σ))dx.\liminf_{\varepsilon\to 0}\mathcal{F}_{\varepsilon}(m_{\varepsilon})\geqslant\int_{\Omega}\mathopen{}\mathclose{{\left(\int_{\mathbb{S}^{2}}|\partial_{\sigma}m(x)|^{2}\mathrm{d}\mu(\sigma)}}\right)\mathrm{d}x. (4.5)

Therefore, it only remains to show the analogous liminf\mathrm{liminf} inequality for the nonlocal antisymmetric exchange term, i.e., that

lim infε0ε(mε)(m).\liminf_{\varepsilon\to 0}\mathcal{H}_{\varepsilon}(m_{\varepsilon})\geqslant\mathcal{H}(m). (4.6)

We show something more, as stated in the next claim.

Claim:

limε0ε(mε)=(m).\lim_{\varepsilon\rightarrow 0}\mathcal{H}_{\varepsilon}(m_{\varepsilon})=\mathcal{H}(m). (4.7)

In order to prove our claim, we introduce a family of mollifiers (ηk)k(\eta_{k})_{k\in\mathbb{N}} in Cc(3)C^{\infty}_{c}(\mathbb{R}^{3}) build in the usual way: we consider an element ηCc(3)\eta\in C^{\infty}_{c}(\mathbb{R}^{3}), η0\eta\geqslant 0, ηL1(3)=1\|\eta\|_{L^{1}(\mathbb{R}^{3})}=1, suppηB1(0)\mathrm{supp}\eta\subset B_{1}(0), and we set ηk(x):=k3η(kx)\eta_{k}(x):=k^{3}\eta(kx). Also, for every sufficiently small δ>0\delta>0 we set Ωδ:={xΩ:dist(x,Ω)>δ}\Omega_{\delta}:=\{x\in\Omega:\mathrm{dist}(x,\partial\Omega)>\delta\}. For every k>1/δk>1/\delta the regularization of mεm_{\varepsilon} given by mε,k:=mεηkm_{\varepsilon,k}:=m_{\varepsilon}\ast\eta_{k} is well-defined on Ωδ\Omega_{\delta}. By classical properties of mollifiers, we know that for a given ε>0\varepsilon>0 there holds

mε,k:=mεηk\displaystyle m_{\varepsilon,k}:=m_{\varepsilon}\ast\eta_{k} k+mεstrongly in L2(Ωδ;3),\displaystyle\xrightarrow{k\rightarrow+\infty}m_{\varepsilon}\quad\text{strongly in }L^{2}(\Omega_{\delta};\mathbb{R}^{3}), (4.8)
mk:=mηk\displaystyle m_{k}:=m\ast\eta_{k} k+mstrongly in H1(Ωδ;3).\displaystyle\xrightarrow{k\rightarrow+\infty}m_{\phantom{\varepsilon}}\quad\text{strongly in }H^{1}(\Omega_{\delta};\mathbb{R}^{3}). (4.9)

In writing the previous relations, we agreed (with a safe abuse of notation) that the 𝕊2\mathbb{S}^{2}-valued vector fields mm and mεm_{\varepsilon} are extended by zero outside of Ω\Omega.

In what follows, it will be convenient to denote by ε,δ\mathcal{H}_{\varepsilon,\delta} and δ\mathcal{H}_{\delta} the analog of the functionals ε\mathcal{H}_{\varepsilon} and \mathcal{H} when defined on space of functions defined on the compact set Ω¯δΩ\bar{\Omega}_{\delta}\subset\Omega instead of Ω\Omega. Thus, for example,

ε,δ(mε,k)=Ω¯δ2νε(xy)|xy|mε,k(x)×mε,k(y)dxdy.\mathcal{H}_{\varepsilon,\delta}(m_{\varepsilon,k})=\iint_{\color[rgb]{0,0,0}\bar{\Omega}_{\delta}^{2}}\frac{\nu_{\varepsilon}(x-y)}{|x-y|}m_{\varepsilon,k}(x)\times m_{\varepsilon,k}(y)\mathrm{d}x\mathrm{d}y.

For the nonlocal antisymmetric energy ε,δ\mathcal{H}_{\varepsilon,\delta} we then observe that the uniform convergence result stated in Theorem 2.3 assures that

limε0ε,δ(mε,k)=δ(mk)=i=13Ωδmk(x)(di×imk(x))dx.\lim_{\varepsilon\to 0}\mathcal{H}_{\varepsilon,\delta}(m_{\varepsilon,k})=\mathcal{H}_{\delta}(m_{k})=\sum_{i=1}^{3}\int_{\Omega_{\delta}}m_{k}(x)\cdot(d_{i}\times\partial_{i}m_{k}(x))\mathrm{d}x. (4.10)

Indeed, since mεmm_{\varepsilon}\to m in L2(Ω;𝕊2)L^{2}(\Omega;\mathbb{S}^{2}) as ε0\varepsilon\to 0, we have that for every kk\in\mathbb{N} there holds mε,k=mεηkε0mk=mηkm_{\varepsilon,k}=m_{\varepsilon}\ast\eta_{k}\xrightarrow{\varepsilon\rightarrow 0}m_{k}=m\ast\eta_{k} in C2(Ω¯δ;3)C^{2}(\bar{\Omega}_{\delta};\mathbb{R}^{3}), and therefore, we are in the hypotheses of (2.16) in Theorem 2.3 applied to Ω¯δ\bar{\Omega}_{\delta}.

Note that, by (4.9) we have (m)=limδ0(limk+δ(mk))\mathcal{H}(m)=\lim_{\delta\rightarrow 0}(\lim_{k\rightarrow+\infty}\mathcal{H}_{\delta}(m_{k})) and, therefore, our goal (4.7) boils down to showing the equality

limε0ε(mε)=limδ0limk+limε0ε,δ(mε,k),\lim_{\varepsilon\rightarrow 0}\mathcal{H}_{\varepsilon}(m_{\varepsilon})=\lim_{\delta\rightarrow 0}\lim_{k\rightarrow+\infty}\lim_{\varepsilon\to 0}\mathcal{H}_{\varepsilon,\delta}(m_{\varepsilon,k}), (4.11)

because, by (4.10), the RHS of (4.11) is nothing but (m)\mathcal{H}(m). To show equality (4.11) we split ε(mε)\mathcal{H}_{\varepsilon}\mathopen{}\mathclose{{\left(m_{\varepsilon}}}\right) under the form

ε(mε)=ε,δ(mε,k)+(ε(mε)δ,ε(mε))+(δ,ε(mε)ε,δ(mε,k))\mathcal{H}_{\varepsilon}(m_{\varepsilon})=\mathcal{H}_{\varepsilon,\delta}\mathopen{}\mathclose{{\left(m_{\varepsilon,k}}}\right)+(\mathcal{H}_{\varepsilon}(m_{\varepsilon})-\mathcal{H}_{\delta,\varepsilon}(m_{\varepsilon}))+\mathopen{}\mathclose{{\left(\mathcal{H}_{\delta,\varepsilon}(m_{\varepsilon})-\mathcal{H}_{\varepsilon,\delta}\mathopen{}\mathclose{{\left(m_{\varepsilon,k}}}\right)}}\right) (4.12)

and then show that

limδ0limk+lim supε0|ε(mε)δ,ε(mε)|\displaystyle\lim_{\delta\rightarrow 0}\lim_{k\rightarrow+\infty}\limsup_{\varepsilon\rightarrow 0}|\mathcal{H}_{\varepsilon}(m_{\varepsilon})-\mathcal{H}_{\delta,\varepsilon}(m_{\varepsilon})| =0,\displaystyle=0, (4.13)
limδ0limk+lim supε0|δ,ε(mε)ε,δ(mε,k)|\displaystyle\lim_{\delta\rightarrow 0}\lim_{k\rightarrow+\infty}\limsup_{\varepsilon\rightarrow 0}\mathopen{}\mathclose{{\left|\mathcal{H}_{\delta,\varepsilon}(m_{\varepsilon})-\mathcal{H}_{\varepsilon,\delta}\mathopen{}\mathclose{{\left(m_{\varepsilon,k}}}\right)}}\right| =0.\displaystyle=0. (4.14)

Indeed, as soon as we achieve (4.13)–(4.14), passing to the limit limδ0limk+lim supε0\lim_{\delta\rightarrow 0}\lim_{k\rightarrow+\infty}\limsup_{\varepsilon\rightarrow 0} in (4.12) we conclude.

Step 1. Proof of (4.13).  Using the fact that the kernel νε\nu_{\varepsilon} is odd, we obtain that

ε(mε)δ,ε(mε)\displaystyle\mathcal{H}_{\varepsilon}(m_{\varepsilon})-\mathcal{H}_{\delta,\varepsilon}(m_{\varepsilon}) =Ω2Ωδ2νε(xy)|xy|(mε(x)×mε(y))dxdy\displaystyle=\iint_{\Omega^{2}\setminus\Omega_{\delta}^{2}}\frac{\nu_{\varepsilon}(x-y)}{|x-y|}\cdot\mathopen{}\mathclose{{\left(m_{\varepsilon}(x)\times m_{\varepsilon}(y)}}\right)\mathrm{d}x\mathrm{d}y

and, therefore,

|ε(mε)δ,ε(mε)|\displaystyle|\mathcal{H}_{\varepsilon}(m_{\varepsilon})-\mathcal{H}_{\delta,\varepsilon}(m_{\varepsilon})| 3(C×(Ω\Ωδ)|νε(xy)|dxdy)1/2(ε(mε))1/2\displaystyle\leqslant 3\mathopen{}\mathclose{{\left(C\iint_{\mathbb{R}\times(\Omega\backslash\Omega_{\delta})}\mathopen{}\mathclose{{\left|\nu_{\varepsilon}(x-y)}}\right|\mathrm{d}x\mathrm{d}y}}\right)^{1/2}\cdot\mathopen{}\mathclose{{\left(\mathcal{F}_{\varepsilon}(m_{\varepsilon})}}\right)^{1/2}
=3C1/2(ε(mε))1/2|Ω\Ωδ|1/2.\displaystyle=3C^{1/2}\mathopen{}\mathclose{{\left(\mathcal{F}_{\varepsilon}(m_{\varepsilon})}}\right)^{1/2}|\Omega\backslash\Omega_{\delta}|^{1/2}.

Passing to the limit in the previous inequality, we infer (4.13).

Step 2. Proof of (4.14). We observe that

δ,ε(mε)\displaystyle\mathcal{H}_{\delta,\varepsilon}(m_{\varepsilon}) ε,δ(mε,k)\displaystyle-\mathcal{H}_{\varepsilon,\delta}\mathopen{}\mathclose{{\left(m_{\varepsilon,k}}}\right)
=Ωδ2νε(xy)|xy|(mε(x)×mε(y)mε,k(x)×mε,k(y))dxdy\displaystyle=\iint_{\Omega_{\delta}^{2}}\frac{\nu_{\varepsilon}(x-y)}{|x-y|}\cdot\mathopen{}\mathclose{{\left(m_{\varepsilon}(x)\times m_{\varepsilon}(y)-m_{\varepsilon,k}(x)\times m_{\varepsilon,k}(y)}}\right)\mathrm{d}x\mathrm{d}y
=3×3(Ωδ2νε(xy)|xy|(mε(x)×mε(y))dxdy)ηk(z)ηk(w)dzdw\displaystyle=\iint_{\mathbb{R}^{3}\times\mathbb{R}^{3}}\mathopen{}\mathclose{{\left(\iint_{\Omega_{\delta}^{2}}\frac{\nu_{\varepsilon}(x-y)}{|x-y|}\cdot\mathopen{}\mathclose{{\left(m_{\varepsilon}(x)\times m_{\varepsilon}(y)}}\right)\mathrm{d}x\mathrm{d}y}}\right)\eta_{k}(z)\eta_{k}(w)\mathrm{d}z\hskip 1.69998pt\mathrm{d}w
3×3((z+Ωδ)×(w+Ωδ)νε(xy)|xy|(mε(x)×mε(y))dxdy)ηk(z)ηk(w)dzdw\displaystyle-\iint_{\mathbb{R}^{3}\times\mathbb{R}^{3}}\mathopen{}\mathclose{{\left(\iint_{(-z+\Omega_{\delta})\times(-w+\Omega_{\delta})}\frac{\nu_{\varepsilon}(x-y)}{|x-y|}\cdot\mathopen{}\mathclose{{\left(m_{\varepsilon}(x)\times m_{\varepsilon}(y)}}\right)\mathrm{d}x\mathrm{d}y}}\right)\eta_{k}(z)\eta_{k}(w)\mathrm{d}z\hskip 1.69998pt\mathrm{d}w
=3×3[(Ω2νε(xy)|xy|(mε(x)×mε(y))χ(z,w)(x,y)dxdy)]ηk(z)ηk(w)dzdw,\displaystyle=\iint_{\mathbb{R}^{3}\times\mathbb{R}^{3}}\mathopen{}\mathclose{{\left[\mathopen{}\mathclose{{\left(\iint_{\Omega^{2}}\frac{\nu_{\varepsilon}(x-y)}{|x-y|}\cdot\mathopen{}\mathclose{{\left(m_{\varepsilon}(x)\times m_{\varepsilon}(y)}}\right)\chi_{(z,w)}(x,y)\mathrm{d}x\mathrm{d}y}}\right)}}\right]\eta_{k}(z)\eta_{k}(w)\mathrm{d}z\hskip 1.69998pt\mathrm{d}w, (4.15)

where we set χ(z,w)(x,y):=1Ωδ(x)1Ωδ(y)1(z+Ωδ)(x)1(w+Ωδ)(y)\chi_{(z,w)}(x,y):=1_{\Omega_{\delta}}(x)1_{\Omega_{\delta}}(y)-1_{(-z+\Omega_{\delta})}(x)1_{(-w+\Omega_{\delta})}(y). Before continuing estimating (4.15), we need the following inequality, whose proof is postponed to Step 3 below:

3×3\displaystyle\iint_{\mathbb{R}^{3}\times\mathbb{R}^{3}} (Ω2|νε(xy)||χ(z,w)(x,y)|dxdy)ηk(z)ηk(w)dzdw\displaystyle\mathopen{}\mathclose{{\left(\iint_{\Omega^{2}}\mathopen{}\mathclose{{\left|\nu_{\varepsilon}(x-y)}}\right||\chi_{(z,w)}(x,y)|\mathrm{d}x\mathrm{d}y}}\right)\eta_{k}(z)\eta_{k}(w)\mathrm{d}z\mathrm{d}w
23Ω|1Ωδ(x)1(z+Ωδ)(x)|dxηk(z)dz.\displaystyle\qquad\qquad\qquad\qquad\leqslant 2\int_{\mathbb{R}^{3}}\int_{\Omega}|1_{\Omega_{\delta}}(x)-1_{(-z+\Omega_{\delta})}(x)|\mathrm{d}x\eta_{k}(z)\mathrm{d}z. (4.16)

Having (4.16) at our disposal, by combining Hölder inequality, (A1), as well as (4.15), we deduce

|δ,ε(mε)\displaystyle|\mathcal{H}_{\delta,\varepsilon}(m_{\varepsilon}) ε,δ(mε,k)|\displaystyle-\mathcal{H}_{\varepsilon,\delta}(m_{\varepsilon,k})|
\displaystyle\leqslant (2ε(mε))1/2(3)2(Ω2|νε(xy)|2ρε(xy)|χ(z,w)(x,y)|dxdy)1/2ηk(z)ηk(w)dzdw\displaystyle\mathopen{}\mathclose{{\left(2\mathcal{F}_{\varepsilon}(m_{\varepsilon})}}\right)^{1/2}\iint_{\mathopen{}\mathclose{{\left(\mathbb{R}^{3}}}\right)^{2}}\mathopen{}\mathclose{{\left(\iint_{\Omega^{2}}\frac{\mathopen{}\mathclose{{\left|\nu_{\varepsilon}(x-y)}}\right|^{2}}{\rho_{\varepsilon}(x-y)}|\chi_{(z,w)}(x,y)|\mathrm{d}x\mathrm{d}y}}\right)^{1/2}\eta_{k}(z)\eta_{k}(w)\mathrm{d}z\hskip 1.69998pt\mathrm{d}w
\displaystyle\leqslant C1/2(2ε(mε))1/2(3)2(Ω2|νε(xy)||χ(z,w)(x,y)|dxdy)1/2ηk(z)ηk(w)dzdw\displaystyle C^{1/2}\mathopen{}\mathclose{{\left(2\mathcal{F}_{\varepsilon}(m_{\varepsilon})}}\right)^{1/2}\iint_{\mathopen{}\mathclose{{\left(\mathbb{R}^{3}}}\right)^{2}}\mathopen{}\mathclose{{\left(\iint_{\Omega^{2}}\mathopen{}\mathclose{{\left|\nu_{\varepsilon}(x-y)}}\right||\chi_{(z,w)}(x,y)|\mathrm{d}x\mathrm{d}y}}\right)^{1/2}\eta_{k}(z)\eta_{k}(w)\mathrm{d}z\mathrm{d}w
\displaystyle\leqslant C1/2(2ε(mε))1/2((3)2(Ω2|νε(xy)||χ(z,w)(x,y)|dxdy)ηk(z)ηk(w)dzdw)1/2\displaystyle C^{1/2}\mathopen{}\mathclose{{\left(2\mathcal{F}_{\varepsilon}(m_{\varepsilon})}}\right)^{1/2}\mathopen{}\mathclose{{\left(\iint_{\mathopen{}\mathclose{{\left(\mathbb{R}^{3}}}\right)^{2}}\mathopen{}\mathclose{{\left(\iint_{\Omega^{2}}\mathopen{}\mathclose{{\left|\nu_{\varepsilon}(x-y)}}\right||\chi_{(z,w)}(x,y)|\mathrm{d}x\mathrm{d}y}}\right)\eta_{k}(z)\eta_{k}(w)\mathrm{d}z\hskip 1.69998pt\mathrm{d}w}}\right)^{1/2}
\displaystyle\leqslant 2C1/2(ε(mε))1/2(3Ω|1Ωδ(x)1(z+Ωδ)(x)|ηk(z)dxdz)1/2\displaystyle 2C^{1/2}\mathopen{}\mathclose{{\left(\mathcal{F}_{\varepsilon}(m_{\varepsilon})}}\right)^{1/2}\mathopen{}\mathclose{{\left(\int_{\mathbb{R}^{3}}\int_{\Omega}|1_{\Omega_{\delta}}(x)-1_{(-z+\Omega_{\delta})}(x)|\eta_{k}(z)\mathrm{d}x\mathrm{d}z}}\right)^{1/2}
=\displaystyle= 2C1/2(ε(mε))1/2(B11Ωδ()1Ωδ(z/k)L1(Ω)dz)1/2k0,\displaystyle 2C^{1/2}\mathopen{}\mathclose{{\left(\mathcal{F}_{\varepsilon}(m_{\varepsilon})}}\right)^{1/2}\mathopen{}\mathclose{{\left(\int_{B_{1}}\|1_{\Omega_{\delta}}(\cdot)-1_{\Omega_{\delta}}(\cdot-z/k)\|_{L^{1}(\Omega)}\mathrm{d}z}}\right)^{1/2}\xrightarrow{k\rightarrow\infty}0, (4.17)

the convergence of (4.17) to 0 being a consequence of Lebesgue’s Dominated Convergence Theorem, as well as of the continuity of the L2L^{2}-norm with respect to translations.

Step  3. Proof of (4.16). We first rewrite the function χ(z,w)(x,y)\chi_{(z,w)}(x,y) as

χ(z,w)(x,y)=[1Ωδ(x)1(z+Ωδ)(x)]1Ωδ(y)+1(z+Ωδ)(x)[1Ωδ(y)1(w+Ωδ)(y)],\chi_{(z,w)}(x,y)=[1_{\Omega_{\delta}}(x)-1_{(-z+\Omega_{\delta})}(x)]1_{\Omega_{\delta}}(y)+1_{(-z+\Omega_{\delta})}(x)[1_{\Omega_{\delta}}(y)-1_{(-w+\Omega_{\delta})}(y)],

from which we infer the following estimate:

|χ(z,w)(x,y)||1Ωδ(x)1(z+Ωδ)(x)|+|1Ωδ(y)1(w+Ωδ)(y)|.|\chi_{(z,w)}(x,y)|\leqslant|1_{\Omega_{\delta}}(x)-1_{(-z+\Omega_{\delta})}(x)|+|1_{\Omega_{\delta}}(y)-1_{(-w+\Omega_{\delta})}(y)|.

In particular,

Ω2|νε(xy)||χ(z,w)(x,y)|dxdy\displaystyle\iint_{\Omega^{2}}\mathopen{}\mathclose{{\left|\nu_{\varepsilon}(x-y)}}\right||\chi_{(z,w)}(x,y)|\mathrm{d}x\mathrm{d}y\leqslant Ω×3|νε(xy)||1Ωδ(x)1(z+Ωδ)(x)|dxdy\displaystyle\iint_{\Omega\times\mathbb{R}^{3}}\mathopen{}\mathclose{{\left|\nu_{\varepsilon}(x-y)}}\right||1_{\Omega_{\delta}}(x)-1_{(-z+\Omega_{\delta})}(x)|\mathrm{d}x\mathrm{d}y
+3×Ω|νε(xy)||1Ωδ(y)1(w+Ωδ)(y)|dxdy\displaystyle\qquad+\iint_{\mathbb{R}^{3}\times\Omega}\mathopen{}\mathclose{{\left|\nu_{\varepsilon}(x-y)}}\right||1_{\Omega_{\delta}}(y)-1_{(-w+\Omega_{\delta})}(y)|\mathrm{d}x\mathrm{d}y
=Ω|1Ωδ(x)1(z+Ωδ)(x)|+|1Ωδ(x)1(w+Ωδ)(x)|dx.\displaystyle=\int_{\Omega}|1_{\Omega_{\delta}}(x)-1_{(-z+\Omega_{\delta})}(x)|+|1_{\Omega_{\delta}}(x)-1_{(-w+\Omega_{\delta})}(x)|\mathrm{d}x.

The previous inequality, once integrated on 3×3\mathbb{R}^{3}\times\mathbb{R}^{3} with respect to the measure ηk(z)ηk(w)dzdw\eta_{k}(z)\eta_{k}(w)\mathrm{d}z\hskip 1.69998pt\mathrm{d}w yields (4.16). ∎

Acknowledgements

G.DiF. and R.G. acknowledge support from the Austrian Science Fund (FWF) through the project Analysis and Modeling of Magnetic Skyrmions (grant 10.55776/P34609). The research of E.D. has been supported by the Austrian Science Fund (FWF) through grants 10.55776/F65, 10.55776/V662, 10.55776/P35359 and 10.55776/Y1292, as well as from BMBWF through the OeAD/WTZ project CZ 09/2023. G.DiF. thanks TU Wien and MedUni Wien for their support and hospitality.

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E. Davoli, Institute of Analysis and Scientific Computing, TU Wien, Wiedner Hauptstraße 8-10, 1040 Vienna, Austria.

 

G. Di Fratta, Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università degli Studi di Napoli “Federico II”, Via Cintia, Complesso Monte S. Angelo, 80126 Naples, Italy.

 

R. Giorgio, Institute of Analysis and Scientific Computing, TU Wien, Wiedner Hauptstraße 8-10, 1040 Vienna, Austria and MedUni Wien, Währinger Gürtel 18-20, 1090 Vienna, Austria.