A Bourgain–Brezis–Mironescu formula accounting for nonlocal antisymmetric exchange interactions
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, -Convergence, antisymmetric exchange interactions, Micromagnetics, Dzyaloshinskii–Moriya interaction (DMI), Magnetic skyrmions.1991 Mathematics Subject Classification:
46E35; 49J45; 49S051. Introduction and motivation
The present paper investigates the short-range interaction limit of a family of nonlocal exchange energies of the form
(1.1) |
where the energy functionals and are given by
(1.2) | ||||
(1.3) |
and are both defined on a suitable (metric) subspace of . Here, denotes the space of vector-valued maps , where is a bounded Lipschitz domain of and the unit sphere of .
The energy functional is the result of two different types of interactions. The term in (1.2) accounts for the so-called symmetric exchange interactions, whereas the energy term in (1.3) accounts for antisymmetric exchange interactions. The scalar kernel and the vector-valued kernel model the strength and positional configuration of the exchange interactions at spatial scale . 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 and , the family converges, in the sense of -convergence in , and up to constant factors, to the local energy functional
(1.4) |
provided that . Here, the quantities denote constant vectors referred to as Dzyaloshinskii vectors in the Physics literature; their expression strongly depends on the limiting behavior of the family .
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 -convergence of the nonlocal family in (1.1) to the local energy functional 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 , the order parameter is the magnetization field . The modulus of , , is called spontaneous magnetization and is a function of the temperature that vanishes above the so-called Curie point : a critical value strongly depends on the specific crystal structure of the ferromagnet. When the specimen is at a fixed temperature well below , the function can be assumed constant in , and the magnetization can be conveniently written as , where is a vector field with values in the unit sphere of (cf. [7, 15]).
Despite being constant in , this is generally not the case for the direction of , 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.
(1.5) |
Here, , and is the extension by zero of to . The Dirichlet energy , i.e., the first term in (1.5), penalizes nonuniformities in the magnetization orientation, whereas the magnetostatic self-energy , i.e., the second term in (1.5), is the energy associated with the demagnetizing field generated by , which describes the long-range dipole interaction of the magnetic moments: for , the demagnetizing field can be characterized as the -projection of on the space of gradients , 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 , whose components are the Lifshitz invariants of (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
(1.6) |
The normalized constant 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 in (1.6) is what the energy term in (1.4) reduces to when the Dzyaloshinskii vectors have the form , with being the -th element of the standard basis of (observe that ).
1.3. The -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])
(1.7) |
Here, is the symmetric exchange constant between the spins , occupying the -th and -th site of the crystal lattice , and is the magnitude of the spin. The exchange constant 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])
(1.8) |
The Dzyaloshinskii vector is an axial vector that depends, other than from the relative distance between the spins and , 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 is symmetric, in the sense that magnetic moments with right-handed or left-handed alignment give the same contributions, the discrete antisymmetric exchange energy distinguishes between those two states via the local chirality imposed by the Dzyaloshinskii vectors .
In the limit of a continuous distribution of lattice sites occupying a region , the exchange energy (1.8), up to a constant term, can be expressed as
(1.9) |
with the normalized magnetization density, a symmetric exchange kernel with even symmetry, i.e., such that for every , and an antisymmetric exchange kernel with odd symmetry, i.e., such that for every .
The energy functional in (1.1) is a faithful analog of the energy in (1.9), and we found it more convenient to work with because of scaling reasons. Specifically, we set and . Overall, the exchange terms , , 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 , , in (1.2), (1.3), and the terms and in (1.4) is not evident, but it can be formally revealed through a first-order asymptotic expansion of the magnetization in a neighborhood of , i.e., by setting . The asymptotic analysis becomes more reliable the more one can neglect variations of around , i.e., the more the kernels and 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 in (1.1) is a short-range approximation of the Heisenberg -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 -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 to is established when is a family of radial mollifiers concentrating their mass at the origin as ; also, a technical lemma provides upper and lower bounds on the variational convergence of , but not sufficient to deduce -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 , 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 .
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 to reduce to , and we refer the reader to [8, 13] for some results about the class of admissible kernels.
In contrast to the symmetric energy functional , the variational convergence of 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 has been carried out thus far; this includes basic questions like the existence of -valued minimizers of 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 (). Balls with center and radius will be denoted by and characteristic functions of sets by , with the convention that if and otherwise. Whenever not explicitly mentioned otherwise, 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 , we denote by the subspace of where is finite:
(2.1) |
Also, since many of our results still hold when is not constrained to take values on , and even if is unbounded, it is convenient for us to denote by the analog unconstrained subspace of . Of course, the space on which the energy is finite depends on the particular choice of the kernels and . In what follows, we will make the following assumptions.
The symmetric exchange kernels
Driven by formal asymptotics, we consider a family of kernels 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).
-
(G1)
For every , in and with .
-
(G2)
For every
(2.2) -
(G3)
There exist linearly independent directions and such that for , and for any
(2.3) Here, for and , denotes the cone centered at the origin, whose projection on is given by .
-
(G4)
There exists a constant such that, denoting by the radial functions , defined for every as
(2.4) there holds
(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 (), the family has nontrivial support at least around three linearly independent directions. Condition (G3) has been introduced in [24] in order to prove a characterization of , . 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 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 is given by
for every and , where is such that in and , Hypothesis (G4) is directly satisfied as long as is strictly positive at the origin. In fact, assume that . By the regularity of , there exists a ball of radius such that for every . In particular, for every and , so that (2.5) holds with .
Finally, the regularity of the kernels in (G1) can be weakened to just in the case in which the kernels are radial. The further - regularity is only needed to provide a meaning in the sense of traces to the restriction of the kernels to spheres of radius , so that the definition in (2.4) is well-posed.
The antisymmetric exchange kernels
Motivated by formal asymptotics, we consider a family of vector-valued kernels satisfying the following assumptions.
-
(H1)
For every , is odd, i.e., for each , and with .
-
(H2)
For every there holds
(2.6) -
(H3)
For , the following limit exists
(2.7) Motivated by their physical significance, we will refer to the vectors as the Dzyaloshinskii vectors. Note that for every .
Later on, we will need the following condition (A1), which connects the integrability of the two families of kernels and .
-
(A1)
There exist a constant such that
(2.8)
Throughout the work, we specify in which cases assumption (A1) is needed.
Remark 2.2 (On the odd symmetry of ).
We emphasize that the requirement of 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 would give no contribution to due to the antisymmetric nature of the functional 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 -convergence analysis for our nonlocal exchange functionals in (1.1).
Theorem 2.1.
(Compactness and -convergence) Let be a bounded Lipschitz domain. Assume (G1)–(G4), (H1)–(H3), and (A1). The following assertions hold.
-
(i)
(Compactness) If is such that
(2.9) then, there exists such that, possibly up to a non-relabeled subsequence, strongly in .
-
(ii)
(-convergence) There exists a finite Radon measure , with , such that, possibly up to a non-relabeled subsequence,
(2.10) with if and
(2.11) if .
Remark 2.3 (The micromagnetic case).
We note that choosing kernels such that for in (H3), and kernels radial, from Theorem 2.1 we obtain that with respect to the strong topology of , there holds with
(2.12) |
if , and otherwise (if ). 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 and ).
The conditions imposed on the families of kernels and 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 and for every , where we have denoted by the characteristic function of the unit ball in . Note that is odd, , and is radial. Then, setting
for every and , Hypotheses (G1)–(G4), as well as (H1)–(H2) and (A1) are directly fulfilled. Additionally, (H3) holds with , .
Remark 2.5 (On the measure ).
Note that, on the one hand, as for the purely symmetric case, cf. [24, Lemma 8], the measure in Theorem 2.1 (ii) is, in principle, dependent on the choice of the extracted subsequence. For some specific choices of the kernels , 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 .
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 .
Theorem 2.2.
Remark 2.6 (The domain of the nonlocal functionals).
Remark 2.7 (On the assumptions of Theorem 2.2).
Estimate (2.13) guarantees that if (G1) and (H1) hold and , then . 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 , so that the partial derivatives of can be controlled over all directions. The further assumption (A1) expresses an integrability relation between the two families of kernels and 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 , contained in the following Theorem 2.3.
Theorem 2.3.
(Pointwise and uniform convergence of the antisymmetric exchange term) Assume (H1)–(H3) and let be bounded. For every there holds
(2.15) |
where, we recall (see (1.2) and (1.4))
and the are the Dzyaloshinskii vectors defined in (H3).
Moreover, if is a family in such that in , then
(2.16) |
Remark 2.8 (The case of unbounded ).
It will be evident from the proof that (2.15) holds even when is unbounded. In particular, when . 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 , with , such that, possibly up to a non-relabeled subsequence, for every , there holds
(2.17) |
with the being the Dzyaloshinskii vectors introduced in (H3).
Moreover, if the family consists of radial kernels, then for every we have
Remark 2.9 (Structure of the limiting symmetric term).
Since we are in the quadratic setting, one has and, therefore, one can rewrite the first term in (2.17) under the form
(2.18) |
where is the anisotropic matrix given by . If the kernels are radial, then is isotropic, in the sense that the resulting matrix is given by with being the 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 .
Step 1. Recall that, by definition, . For ease of computation, we treat the two energy terms separately. For , we denote by the extension of to the whole of given by the classical extension operator on .
For the symmetric exchange term we have
(3.1) | ||||
(3.2) |
for some constant depending only on through the linear extension operator. The equality (3.1) is the result of the change of variables for fixed , 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
(3.3) |
where in (3.3) we performed the change of variables for fixed and then used the odd symmetry of the kernel . 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
for some constant depending only on . This completes the proof of (2.13).
Step 2. We claim that it is sufficient to show that if , then . Indeed, as soon as we show that , we can then invoke [24, Theorem 5] which shows that if and the family satisfies (G1)–(G3), then and there exists (depending only on the family of kernels ) such that
(3.4) |
Since is a quantity without a definite sign, we use Young’s inequality for products to estimate the term from below. We get that for every there holds
(3.5) |
Adding to both sides of the previous estimate, we get that for every there holds
(3.6) |
Now, by (A1), and taking into account that , we infer
(3.7) |
with proportional to the constant from (A1). Hence, from (3.6) and (3.7) we deduce
(3.8) |
By taking we conclude that if then and hence, by [24, Theorem 5], that . ∎
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 . Then, in Step 2, we use a density argument to show that if (2.15) holds for every then it holds for every .
Step 1. Let . For such that , we split as
(3.9) |
Outside of the ball , we note that
Sending (for fixed ), by assumption (H2) we find that and, therefore, that
(3.10) |
We now take into account the contribution to inside of the ball , i.e., the term in (3.9). Given that , we perform a first-order Taylor expansion inside of the form with (up to a constant factor)
to find
(3.11) | ||||
Let us focus on the first term on the right-hand side of (3.11). Simple algebra gives that
(3.12) |
Next, we observe that as a consequence of (H2) we have
(3.13) |
Therefore, combining (3.13) and (H3) we infer that for , there holds
(3.14) |
As a consequence, from (3.12) and (3.14) we conclude
(3.15) |
Note that (3.15) does not depend on .
For the term , we observe that
For the latter equality we used assumption (H1). Thus, since strongly in ,
(3.16) |
Overall, combining estimates (3.10), (3.12), (3.14), (3.15), and (3.16), we infer
(3.17) | ||||
(3.18) |
In view of (3.18), to prove
(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
(3.20) |
where is the Lipschitz constant of , which is uniformly bounded in .
Step 2. This second step concludes the proof by showing, through a density argument, that if (2.15) holds for every then it holds for every .
We note that, for every , there holds
Thus, adding and subtracting the quantity , we infer
(3.21) |
Let now be extensions of from to . For the first integral in (3.21) we have
(3.22) |
for some constant , where we have applied Hölder inequality and classical properties of Sobolev spaces, cf. [4, Prop. 9.3]. Analogously, for the second integral in (3.21) we find
(3.23) |
for some constant .
To sum up, estimating (3.21) through (3.22) and (3.23), we obtain that, for every , there holds
(3.24) |
Now, by density, for every there exists a sequence in such that
(3.25) |
Also, by Step 1, we know that for every one has
(3.26) |
Moreover, in view of (3.24), for every , there holds
(3.27) |
Letting now , by (3.25) and by (3.27),
(3.28) |
thanks to the density (3.25) and the fact that is bounded in .
In view of (3.25), we also have that in for . As a consequence, we have that
(3.29) |
that is, the limit functional is continuous with respect to the -convergence.
4. Compactness and -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 there holds
(4.1) |
with depending only on the homonymous constant in (A1). Since consists of -valued vector fields, i.e., a.e. in for every , from the assumption (2.9) and (4.1) we infer that
(4.2) |
for some depending only on , , and on the homonymous constant in (A1). In particular, in view of assumption (G4), recalling the definition of the kernels in (2.4) and setting
from estimate (4.2) we infer
(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 is a bounded family in such that for some positive constant , then there exists and a subsequence such that strongly in . In particular, we obtain that (up to a further subsequence) a.e. as and, therefore, also the limiting function satisfies the constraint, that is, . ∎
Proof of Theorem 2.1.ii, (-convergence).
We split the proof into two classical steps. First, we derive the - inequality, then we focus on the - inequality.
Limsup inequality. It is sufficient to note that the constant family for every , is a recovery sequence. Indeed, on the one hand, if then Theorem 2.2 gives that ; on the other hand, if then by Corollary 2.1, we conclude that
Liminf inequality. Let be such that strongly in as . If , there is nothing to prove. If then, by (4.1), we know that
(4.4) |
for some constant depending only on and the homonymous constant in (A1). Also, by the compactness result (Theorem 2.1.i), , and from [24, Lemma 8] used in the case , we get the existence of a finite Radon measure , with , such that, possibly up to the extraction of a non-relabelled subsequence, there holds
(4.5) |
Therefore, it only remains to show the analogous inequality for the nonlocal antisymmetric exchange term, i.e., that
(4.6) |
We show something more, as stated in the next claim.
Claim:
(4.7) |
In order to prove our claim, we introduce a family of mollifiers in build in the usual way: we consider an element , , , , and we set . Also, for every sufficiently small we set . For every the regularization of given by is well-defined on . By classical properties of mollifiers, we know that for a given there holds
(4.8) | ||||
(4.9) |
In writing the previous relations, we agreed (with a safe abuse of notation) that the -valued vector fields and are extended by zero outside of .
In what follows, it will be convenient to denote by and the analog of the functionals and when defined on space of functions defined on the compact set instead of . Thus, for example,
For the nonlocal antisymmetric energy we then observe that the uniform convergence result stated in Theorem 2.3 assures that
(4.10) |
Indeed, since in as , we have that for every there holds in , and therefore, we are in the hypotheses of (2.16) in Theorem 2.3 applied to .
Note that, by (4.9) we have and, therefore, our goal (4.7) boils down to showing the equality
(4.11) |
because, by (4.10), the RHS of (4.11) is nothing but . To show equality (4.11) we split under the form
(4.12) |
and then show that
(4.13) | ||||
(4.14) |
Indeed, as soon as we achieve (4.13)–(4.14), passing to the limit in (4.12) we conclude.
Step 1. Proof of (4.13). Using the fact that the kernel is odd, we obtain that
and, therefore,
Passing to the limit in the previous inequality, we infer (4.13).
Step 2. Proof of (4.14). We observe that
(4.15) |
where we set . Before continuing estimating (4.15), we need the following inequality, whose proof is postponed to Step 3 below:
(4.16) |
Having (4.16) at our disposal, by combining Hölder inequality, (A1), as well as (4.15), we deduce
(4.17) |
the convergence of (4.17) to being a consequence of Lebesgue’s Dominated Convergence Theorem, as well as of the continuity of the -norm with respect to translations.
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.
E-mail: elisa.davoli@tuwien.ac.at.
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.
E-mail: giovanni.difratta@unina.it.
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.