We consider a vectorial energy functional proposed in the physical literature as a simplified model for thin films of ferronematics — composite materials formed by dispersing magnetic nanoparticles into a nematic liquid crystals host. The model features
two order parameters: a reduced \(\mathbf{Q}\)-tensor field \(\mathbf{Q}\), describing the nematic liquid crystal component, and a vector field \(\mathbf{M}\), accounting for the average polarisation vector field generated by the included particles. The energy contains a coupling term promoting alignment between \(\mathbf{Q}\) and \(\mathbf{M}\). It has been shown in [1], [2] that, as a small parameter \(\varepsilon\) tends to zero, the energy of critical pairs \((\mathbf{Q}_\varepsilon,\,\mathbf{M}_\varepsilon)\) concentrates on distinct singular sets: a finite set of points for the
\(\mathbf{Q}_\varepsilon\)-component and the support of a \(\mathscr{H}^1\)-rectifiable varifold for the \(\mathbf{M}_\varepsilon\)-component, with first
variation supported on the singular set for the \(\mathbf{Q}_\varepsilon\)-component. We show in this paper that, if the coupling constant is above a certain (explicit) threshold, then, up to rescaling its density by a
material constant, such a limiting varifold has integer multiplicity.
Over the past few decades, much effort has been devoted to the study of the Ginzburg-Landau functional of superconductivity and to the Allen-Cahn functional of phase separation, especially concerning the asymptotic behaviour, as a small parameter \(\varepsilon\) tends to zero, of their minimising and non-minimising critical points. Seminal works in these fields, such as, for instance, [3]–[12], revealed the emergence of
gradient-driven singular structures of codimension 2 (in the case of Ginzburg-Landau theory) and codimension 1 (in the Allen-Cahn case) as the sets where the energy of minimisers and critical points concentrates, at first order, as \(\varepsilon\) tends to zero. From the point of view of geometric measure theory, such energy-concentration phenomena are extremely interesting, as they provide a mechanism to construct non-trivial integer-multiplicity
rectifiable currents minimising the energy in an appropriate homology class and stationary varifolds of codimension 1 and 2, both in the Euclidean and in the Riemannian context (e.g., [13]–[22]).
Moreover, Pacard and Ritoré [23] and De Philippis and Pigati [20] have shown that any non-degenerate minimal submanifold of codimension 1 or 2 of a Riemannian manifold can be recovered as the energy-concentration set of critical points of the Allen-Cahn functional, if of codimension 1
(see [23]), or of the Ginzburg-Landau functional, if of codimension 2 (see [20]).
While the Ginzburg-Landau functional is inherently vectorial, the Allen-Cahn functional makes sense for both scalar and vectorial order parameters, depending on the application at hand (for instance, in the case of three or more phases, the vectorial
theory must be used). The asymptotic behaviour of minimisers has been studied within the framework of \(\Gamma\)-convergence, both in the scalar and the vectorial case [8], [9], [24], [25]. However, when coming to the analysis of the asymptotic behaviour of (non-minimising, in general) critical points, the scalar and the vectorial theories are profoundly different. The asymptotic behaviour of general
critical points in the scalar case was investigated by Hutchinson and the second author [11] through a careful PDE analysis in which a
key role is played by the so-called discrepancy function, defined as the difference between the potential energy density and the elastic energy density. By cleverly exploiting the maximum principle, Modica [8] showed that the discrepancy of entire scalar solutions is pointwise non-negative, already at the \(\varepsilon\)-level. In [11], it is proved that, in bounded domains, the discrepancy vanishes asymptotically in the \(L^1\)-sense for every sequence
of critical points with equibounded energy. In the vectorial case, however, there is no reason for these properties to hold (in fact, an example of entire solution whose discrepancy changes sign has been constructed in [26]) and the asymptotic behaviour of non-minimising critical points has eluded precise description up to the recent breakthrough paper [12] by Bethuel, in which new PDEs tools for the Allen-Cahn system on two-dimensional domains has been developed that avoid the discrepancy function and are instead ultimately based on refined energy
estimates.
In this paper we are concerned with a model for ferronematics which couples the Ginzburg-Landau theory with the vectorial Allen-Cahn theory. Ferronematics are composite materials obtained by dispersing magnetic nanoparticles into a nematic
liquid crystal hosts. The physics of such systems, firstly devised theoretically by Brochard and de Gennes in 1970 [27], and then realised
experimentally in various stages (see, e.g., [28] and references therein) is very rich, and the object of ongoing investigation.
A simplified model for thin films of ferronematics in the so-called superdilute regime (in which the density of magnetic particles is relatively low with respect to that of nematic molecules) has been proposed in Bisht2019?. The model features two order parameters: a reduced \(\mathbf{Q}\)-tensor field associated with the liquid crystal
host, which is a map \(\mathbf{Q}\) from the physical domain \(\Omega \subset \mathbb{R}^2\) to \(\mathscr{S}_0^{2\times 2}\), the space of \(2 \times 2\), symmetric, real matrices with trace equal to zero, and the average polarisation vector field \(\mathbf{M}: \Omega \to \mathbb{R}^2\) generated by the included particles. After
non-dimensionalisation, the model can be described as follows.
Let \(\Omega \subset \mathbb{R}^2\) be an open, bounded, and simply connected set with smooth boundary \(\partial \Omega\), let \(\varepsilon> 0\) be a
small parameter, and let \(G \subseteq \Omega\) be any measurable set. We define \[\label{eq:functional}
\mathscr{F}_\varepsilon(\mathbf{Q},\,\mathbf{M};\,G) := \int_G \left\{ \frac{1}{2} \left| \nabla \mathbf{Q} \right|^2 + \frac{\varepsilon}{2} \left| \nabla \mathbf{M} \right|^2 + \frac{1}{\varepsilon^2} f_\varepsilon(\mathbf{Q},\,\mathbf{M})\right\}
\,{\mathrm{d}}x,\tag{1}\] where the potential energy density \(f_\varepsilon(\mathbf{Q},\,\mathbf{M})\) is given by \[\label{eq:f}
f_\varepsilon(\mathbf{Q},\,\mathbf{M}) := \frac{1}{4}\left( \left| \mathbf{Q}_\varepsilon \right|^2 - 1 \right)^2 + \frac{\varepsilon}{4}\left( \left| \mathbf{M}_\varepsilon \right|^2 - 1 \right)^2 - \varepsilon\beta \mathbf{Q}\mathbf{M}\cdot \mathbf{M}+
\chi_\varepsilon.\tag{2}\] Here, \(\beta\) is a (strictly) positive parameter depending only on the material and the additive constant \(\chi_\varepsilon\) ensures that \(\inf f_\varepsilon= 0\) for each \(\varepsilon> 0\).
The mathematical study of the functional \(\mathscr{F}_\varepsilon\) started in [29] and then continued in [1], [2], [30], encompassing the asymptotic behaviour as \(\varepsilon\to 0\) of both minimising and non-minimising critical points under suitable boundary conditions. The limit \(\varepsilon\to 0\) should be understood as a large domain limit (the size
of the domain is much larger than the typical correlation length for the liquid crystal molecules, cf. [1], [29]).
In this paper, we refine some of the results obtained in [1], [2] for (non-minimising, in general) critical
points. To motivate the interest towards the study of non-minimising critical points, we recall that, while in variational theories for physical systems observable configurations are usually identified minimisers or local minimisers of the energy, it has
been argued on the basis of numerically and experimentally observed cases (see, e.g., [31] for a broader discussion) that non-minimising critical
points show up naturally, for instance, when the system decays from a higher to a lower energy state. Moreover, numerical simulations [29], Bisht2019? show an abundance of critical points of the functional 1 .
As in [1], [2], we assume to deal with sequences of (non-minimising, in general) critical pairs \((\mathbf{Q}_\varepsilon,\,\mathbf{M}_\varepsilon)\) of \(\mathscr{F}_\varepsilon\) in \(\Omega\) and that the boundary data are \(\varepsilon\)-independent maps \(\mathbf{Q}_{\mathrm{bd}}\in C^1(\partial \Omega,\,\mathscr{S}_0^{2\times 2})\), \(\mathbf{M}_{\mathrm{bd}}\in C^1(\partial
\Omega,\,\mathbb{R}^2)\) that satisfy either the ‘pure’ Dirichlet boundary conditions \[\label{hp:bc-dir} \left| \mathbf{M}_{\mathrm{bd}} \right| = \left(\sqrt{2}\beta + 1
\right)^{1/2}, \qquad \mathbf{Q}_{\mathrm{bd}}= \sqrt{2}\left(\frac{\mathbf{M}_{\mathrm{bd}}\otimes \mathbf{M}_{\mathrm{bd}}}{\sqrt{2}\beta + 1} - \frac{\mathbf{I}}{2}\right)\tag{3}\] or the ‘mixed’ boundary conditions
\[\label{hp:bc-mixed} \mathbf{Q}_{\mathrm{bd}}= \sqrt{2}\left( \mathbf{n}_{\mathrm{bd}}\otimes \mathbf{n}_{\mathrm{bd}}- \frac{\mathbf{I}}{2} \right), \qquad
\partial_{\boldsymbol{\nu}}\mathbf{M}_\varepsilon= 0,\tag{4}\] where \(\mathbf{n}_{\mathrm{bd}}\in C^1(\partial\Omega,\,\mathbb{S}^1)\) is an assigned, \(\varepsilon\)-independent vector field. Finally, we assume that there exists a positive constant \({\rm C}_{\rm pot}\) such that, for every \(\varepsilon>
0\), \[\label{hp:equibdd-f} \frac{1}{\varepsilon^2} \int_\Omega f(\mathbf{Q}_\varepsilon,\,\mathbf{M}_\varepsilon) \,{\mathrm{d}}x \leq {\rm C}_{\rm pot}.\tag{5}\]
Sufficient conditions for 5 are provided in [1].
The Euler-Lagrange equations of the functional \(\mathscr{F}_\varepsilon\) are \[\begin{align} &-\Delta \mathbf{Q}_\varepsilon+ \frac{1}{\varepsilon^2} \left( \left|
\mathbf{Q}_\varepsilon \right|^2 - 1 \right) \mathbf{Q}_\varepsilon- \frac{\beta}{\varepsilon}\left( \mathbf{M}_\varepsilon\otimes \mathbf{M}_\varepsilon- \frac{\left| \mathbf{M}_\varepsilon \right|^2}{2} \mathbf{I}\right) = 0, \tag{6} \\ &-\Delta
\mathbf{M}_\varepsilon+ \frac{1}{\varepsilon^2}\left(\left| \mathbf{M}_\varepsilon \right|^2 - 1 \right)\mathbf{M}_\varepsilon- \frac{2\beta}{\varepsilon^2}\mathbf{Q}_\varepsilon\mathbf{M}_\varepsilon= 0\tag{7}.
\end{align}\] Under the above assumptions, upon using 6 , 7 , it was shown in [1] that \[\label{eq:energy-bounds}
\mathscr{F}_\varepsilon(\mathbf{Q}_\varepsilon,\,\mathbf{M}_\varepsilon) \leq C \left| \log \varepsilon \right|, \qquad \int_\Omega \left| \nabla \mathbf{Q}_\varepsilon \right|^2\,{\mathrm{d}}x \leq C \left| \log \varepsilon \right|, \qquad
\varepsilon\int_\Omega \left| \nabla \mathbf{M}_\varepsilon \right|^2 \,{\mathrm{d}}x \leq C,\tag{8}\] where the constant \(C\) does not depend on \(\varepsilon\).
As in [1], [2], we define the energy densities \[\begin{align}
&\mu_\varepsilon:= \frac{1}{\left| \log\varepsilon \right|} \left( \frac{1}{2} \left| \nabla \mathbf{Q}_\varepsilon \right|^2 + \frac{\varepsilon}{2} \left| \nabla \mathbf{M}_\varepsilon \right|^2 + \frac{1}{\varepsilon^2}
f_\varepsilon(\mathbf{Q}_\varepsilon,\,\mathbf{M}_\varepsilon) \right), \\ &\nu_\varepsilon:= \frac{\varepsilon}{2} \left| \nabla \mathbf{M}_\varepsilon \right|^2 + \frac{1}{\varepsilon^2} f_\varepsilon(\mathbf{Q}_\varepsilon,\,\mathbf{M}_\varepsilon).
\end{align}\] From the assumption 5 and the energy bounds 8 , the sequences of positive functions \(\{\mu_\varepsilon\}_\varepsilon\), \(\{ \nu_\varepsilon\}_\varepsilon\) are bounded in \(L^1(\Omega)\), hence we can find limiting Radon measures \(\mu_\star\) and \(\nu_\star\) in \(\Omega\) such that, after possible extraction of a subsequence, \[\mu_\varepsilon\rightharpoonup^* \mu_\star, \qquad \nu_\varepsilon\rightharpoonup^*
\nu_\star \qquad as\varepsilon\to 0\] in the sense of Radon measures in \(\Omega\). In the limit, distinct (but related) gradient-driven singular structures show up. Indeed, it is shown in [1] that \(\mathop{\mathrm{spt}}\mu_\star\) is a finite set of points and in [2] that \(\mathop{\mathrm{spt}}\nu_\star\) is a union of segments with locally constant density. The relationship between \(\mathop{\mathrm{spt}}\mu_\star\) and \(\mathop{\mathrm{spt}}\nu_\star\) is expressed by item [item:balance-law] of Theorem 1 below and, in a more precise way, by [2].
These results are part of the outcome of a broader analysis on the systems 6 , 7 . At first sight, both 6 , 7 look like perturbed Ginzburg-Landau
systems, and this is true for 6 , although the relatively large perturbation term makes the analysis in [1] quite delicate. However, as
shown in [2], the structure of 7 is the one of a perturbed Allen-Cahn system. To see this, which is crucial to our purposes, we argue as
in [2] and, for any given pair \((\mathbf{Q},\,\mathbf{M}) \in \mathscr{S}_0^{2\times 2}\times \mathbb{R}^2\), we define the
functions \[\label{eq:ell} \ell(\mathbf{Q},\,\mathbf{M}) := \frac{1}{4}\left( \left| \mathbf{M} \right|^2 - 1 \right)^2 - \beta \mathbf{Q}\mathbf{M}\cdot\mathbf{M}+ \frac{1}{2}\left(\beta^2 +
\sqrt{2}\beta\right),\tag{9}\] and \[\label{eq:V} V(\mathbf{M}) := \ell(\mathbf{Q},\,\mathbf{M}) - \inf_{y \in \mathbb{R}^2} \ell(\mathbf{Q}, \, y).\tag{10}\] As shown
in [1], for any given \(\mathbf{Q}\neq 0\), the function \(\mathbf{M}\mapsto V(\mathbf{M})\) has
exactly two zeroes, at \(\mathbf{M}= \mathbf{M}_\pm\), where \[\mathbf{M}_\pm := \pm \left( 1 + \sqrt{2}\beta \left| \mathbf{Q} \right| \right)^{1/2} \mathbf{n},\] and where \(\mathbf{n}\) is a unit vector of \(\mathbf{Q}\) corresponding to its positive eigenvalue.
Denoting now \(\mathbf{Q}: \Omega \to \mathscr{S}_0^{2\times 2}\) a \(\mathbf{Q}\)-tensor field, \(\mathbf{M}: \Omega \to \mathbb{R}^2\) a vector field,
and \(G \subseteq \Omega\) a measurable set, we define the energy functional \[\label{eq:Eeps} E_\varepsilon(\mathbf{M};\,G) := \int_G\left\{
\frac{\varepsilon}{2} \left| \nabla \mathbf{M} \right|^2 + \frac{1}{\varepsilon} V(\mathbf{M}) \right\}\,{\mathrm{d}}x.\tag{11}\] By construction, if \((\mathbf{Q}_\varepsilon,\,\mathbf{M}_\varepsilon)\) is a
critical point of \(\mathscr{F}_\varepsilon\), then \(\mathbf{M}_\varepsilon\) satisfies the Euler-Lagrange equation of \(E_\varepsilon\), i.e.,
\[\label{eq:EL-Eeps} -\varepsilon\Delta \mathbf{M}_\varepsilon+ \frac{1}{\varepsilon}\nabla_\mathbf{M}V(\mathbf{M}_\varepsilon) = 0 \qquad in\Omega.\tag{12}\] It can be easily
checked that (cf. [2]) that \[\label{eq:Eeps-bound} \sup_{\varepsilon> 0}
E_\varepsilon(\mathbf{M}_\varepsilon) \leq \mathcal{E}_0 < +\infty,\tag{13}\] where the constant \(\mathcal{E}_0\) depends only on \(\Omega\), \(\beta\), and the \(L^1(\partial\Omega)\)- and \(L^2(\partial\Omega)\)-norm of \(\mathbf{Q}_{\mathrm{bd}}\times
\partial_{\boldsymbol{\tau}}\mathbf{Q}_{\mathrm{bd}}\).
The point of introducing the auxiliary functional \(E_\varepsilon\) is that the Equation 12 coincides exactly with 7 , so that both \(E_\varepsilon\) and 12 display an Allen-Cahn structure. However, the results of [12]cannot be applied
directly, because the wells of \(V\) are not fixed, but move with both \(x\) and \(\varepsilon\) due to their dependence on \(\mathbf{Q}_\varepsilon\). Nonetheless, one can still follow the strategy of [12], combining it with the estimates and convergence results
for the \(\mathbf{Q}\)-component from [1] to control the perturbation terms. Thus, upon defining the potential energy densities
\[\label{eq:zeta-eps} \zeta_\varepsilon:= \frac{1}{\varepsilon} V(\mathbf{M}_\varepsilon),\tag{14}\] which, due to 13 , are equibounded in
\(L^1(\Omega)\), one has \[\label{eq:zeta42} \zeta_\varepsilon\rightharpoonup^* \zeta_\star, \qquad as\varepsilon\to 0,\tag{15}\] and the
following theorem has been obtained in [2].
Theorem 1 ([2]). The set \(\mathop{\mathrm{spt}}\nu_\star\) is \(\mathscr{H}^1\)-rectifiable, with locally finite measure. Upon setting \[\label{eq:S42} \mathfrak{S}_\star := \mathop{\mathrm{spt}}\nu_\star \setminus
\mathop{\mathrm{spt}}\mu_\star,\tag{16}\] the following holds.
(i) The limiting potential energy measure \(\zeta_\star\) is absolutely continuous with respect to the measure \(\mathscr{H}^1
\mathbin{\vrule height 1.6ex depth 0pt width
0.13ex\vrule height 0.13ex depth 0pt width 1.3ex}\mathfrak{S}_\star\). In other words, there exists a function \(\mathfrak{v}_\star : \mathfrak{S}_\star \to \mathbb{R}^+\), locally integrable with respect to the
measure \(\mathscr{H}^1 \mathbin{\vrule height 1.6ex depth 0pt width
0.13ex\vrule height 0.13ex depth 0pt width 1.3ex}\mathfrak{S}_\star\), such that \[\zeta_\star = \mathfrak{v}_\star \,\mathscr{H}^1 \mathbin{\vrule height 1.6ex depth 0pt width
0.13ex\vrule height 0.13ex depth 0pt width 1.3ex}\mathfrak{S}_\star.\] The function \(\mathfrak{v}_\star\) is locally bounded both from above and from below in any compact set \(K
\subset \Omega\setminus\mathop{\mathrm{spt}}\mu_\star\).
(ii) The measure \(\zeta_\star\) is the weight measure of the \(\mathscr{H}^1\)-rectifiable varifold
\(\mathbb{V}_\star\) carried by \(\mathfrak{S}_\star\) with density function \(\mathfrak{v}_\star\).
(iii) The varifold \(\mathbb{V}_\star\) is stationary in \(\Omega \setminus \mathop{\mathrm{spt}}\mu_\star\) and its
first variation as a varifold in \(\Omega\) is concentrated on \(\mathop{\mathrm{spt}}\mu_\star\).
(iv) The set \(\mathop{\mathrm{spt}}\nu_\star\) is locally a union of segments, open relative to \(\Omega\), each of which
having constant density, given by the value of \(\mathfrak{v}_\star\) at any point of the segment. In addition, apart from an exceptional, \(\mathscr{H}^1\)-null set, around any point \(x_0\) of \(\mathfrak{S}_\star\), the singular set \(\mathop{\mathrm{spt}}\nu_\star\) consists of exactly one segment, with constant density \(\mathfrak{v}_\star(x_0)\).
(v) As \(\varepsilon\to 0\), \(\mathbf{M}_\varepsilon\to \mathbf{M}_\star\) in \(L^\infty_{\rm
loc}(\Omega \setminus (\mathop{\mathrm{spt}}\mu_\star \cup \mathop{\mathrm{spt}}\nu_\star))\).
Notice that Theorem 1 says nothing about the multiplicity of the limiting varifold \(\mathbb{V}_\star\). We recall that, in the scalar
Allen-Cahn theory [11], the integrality of the limiting interface where the energy critical points concentrates as \(\varepsilon\to 0\) is a characteristic feature, even for the perturbed equation, as first shown in [32], [33] for Sobolev type perturbations and then, in [34], [35] for \(L^2\) type perturbations. By way of contrast, integrality is generally false in the vectorial case [12], [36], and hence it requires specific analysis, depending on the particular form of the potential function at hand. Our main
result in this paper, Theorem 2 below, establishes that in the regime \[\sqrt{2}\beta > 1\] the varifold \(\mathbb{V}_\star\) has, after rescaling by the constant \(\sigma_\beta\) in 17 , integer multiplicity.
Theorem 2. Let \(\zeta_\star = \mathfrak{v}_\star\,\mathscr{H}^1 \mathbin{\vrule height 1.6ex depth 0pt width
0.13ex\vrule height 0.13ex depth 0pt width 1.3ex}\mathfrak{S}_\star\) be the limiting potential energy density and let \(\mathbb{V}_\star = \mathbf{v}(\mathfrak{S}_\star,\,\mathfrak{v}_\star)\) be the varifold
associated with \(\zeta_\star\) according to Theorem 1. Let \[\label{eq:sigma-beta} \sigma_\beta := \frac{\sqrt{2}}{3}\left( 1 + \sqrt{2}\beta \right)^{3/2}\tag{17}\] and assume that \(\sqrt{2}\beta > 1\). Then, the varifold \[\sigma_\beta^{-1} \mathbb{V}_* := \left( \mathfrak{S_*},\,\frac{\mathfrak{v}_\star}{ \sigma_\beta} \right)\] has integer multiplicity. In other words, the function \(\sigma_\beta^{-1}\mathfrak{v}_\star\) takes its values into \(\mathbb{N}\).
The condition \(\sqrt{2}\beta > 1\) ensures such a fast, quantitative decay with respect to \(\varepsilon\) of the component of \(\mathbf{M}_\varepsilon\) in the direction orthogonal to the eigenspace \(\mathbf{n}_\varepsilon\) of \(\mathbf{Q}_\varepsilon\) relative to its positive
eigenvalue, that this component cannot concentrate energy in the limit as \(\varepsilon\to 0\). This fast decay is obtained through the study of the equation satisfied by the orthogonal component \(u_{2,\varepsilon} := \mathbf{M}_\varepsilon\cdot \mathbf{n}^{\perp}_\varepsilon\) of \(\mathbf{M}_\varepsilon\), which undergoes a structural change exactly when \(\sqrt{2}\beta > 1\), see 63 below. As a result, although the problem remains fully vectorial at the \(\varepsilon\)-level, in the regime \(\sqrt{2}\beta > 1\) it is quantitatively close (in terms of \(\varepsilon\)) to the scalar problem obtained by formally neglecting the component \(u_{2,\varepsilon}\) of \(\mathbf{M}_\varepsilon\). More precisely, this quantitive decay allows us to look at the equation satisfied by \(u_{1,\varepsilon} :=
\mathbf{M}_\varepsilon\cdot \mathbf{n}_\varepsilon\) as a perturbed Allen-Cahn equation of the type considered in [34] and,
thus, to infer integrality from the results in [34].
Theorem 2 is, to the best of our knowledge, the first integrality result for the limiting varifold obtained in a vectorial Allen-Cahn problem, at least for
a non-trivial (i.e., not decoupled already at the \(\varepsilon\)-level), physically relevant potential energy, and for non-minimising solutions. In particular, it shows that integrality can occur if the potential energy
singles out a preferred direction, as it could be intuitively expected. On the other hand, even in this seemingly easy setting, proving that the components along the orthogonal directions decay sufficiently fast turns out to be a quite technical
matter.
It would be interesting to further study the behaviour of the multiplicity function \(\mathfrak{v}_\star\) in the opposite regime \(\sqrt{2}\beta \leq 1\). Particularly intriguing would
be a situation in which integrality does not hold in such a regime (or ceases to be true below a certain threshold). Indeed, besides the specific interest of such a deep structural change in this particular problem, it is conceivable that the phenomena
leading to the breakdown and the arguments involved in proving its occurrence could be helpful to the long-term goal of determining sufficient conditions for integrality in the vectorial Allen-Cahn problem.
In Section 1, we gather some results from [1], [2], as needed in this
paper. In Section 1.8 we obtain a quantitative rate with respect to \(\varepsilon\) for the convergence of \(\left| \mathbf{Q}_\varepsilon \right|\)
to zero in \(W^{1,2}_{\rm loc}(\Omega \setminus \mathop{\mathrm{spt}}\mu_\star)\) and of \(\frac{\left| \mathbf{Q}_\varepsilon \right| - 1}{\varepsilon}\) to the constant \(\kappa_\star\) defined in 18 in \(L^2_{\rm loc}(\Omega \setminus \mathop{\mathrm{spt}}\mu_\star)\) (see Lemma 1). This result is crucial to our purposes and improves on [1] (which, in turn, was a
crucial ingredient in [1], [2]). In Section 2, we show a quantitative decay
with respect to \(\varepsilon\) of the \(W^{1,2}\)-norm of \(u_{2,\varepsilon}\) away from \(\mathop{\mathrm{spt}}\mu_\star\). Finally, in Section 3, we employ the results of Section 2 and Lemma 1 to prove Theorem 2.
The main result of this section is Lemma 1, contained in Section 1.8. We will use it both in Section 2 and also directly in the proof of Theorem 2. Before arriving to Lemma 1, it will be necessary to establish some notation and gather some auxiliary results from [1], [2], which will be needed also in the rest of the paper. In particular, in Section 1.7 we define the auxiliary Allen-Cahn energy \(\mathcal{AC}_\varepsilon\). Although this functional was already considered in [1], [29], here we introduce the new decomposition 46 of its potential energy \(h\) in 41 , which will prove to be extremely useful to our
purposes.
We recall some basic terminology, as needed in this paper, concerning the theory of varifolds. For a detailed account of the theory, the reader is referred to [37].
A \(k\)-varifold \(\mathbb{V}\) on an open set \(U \subset \mathbb{R}^n\) is a Radon measure on \(G_k(U) := U \times
G(n,\,k)\), where \(G(n,\,k)\) denotes the Grassmanian manifold of \(k\)-dimensional planes in \(\mathbb{R}^n\). The weight measure\(\left\| \mathbb{V} \right\|\) of \(\mathbb{V}\) is the Radon measure defined by setting \[\int_U \phi(x) \,{\mathrm{d}} \left\| \mathbb{V} \right\|(x) := \int_{G_k(U)}
\phi(x)\,{\mathrm{d}}\mathbb{V}(x,\,S),\] for all \(\phi \in C_c(U)\). A \(k\)-varifold is \(\mathscr{H}^k\)-rectifiable if there exist a
countably \(\mathscr{H}^k\)-rectifiable set \(\mathfrak{S}\) and a locally \(\mathscr{H}^k\)-integrable function \(\theta\colon\mathfrak{S} \to \mathbb{R}^+\) (called density function or multiplicity) such that \[\int_{G_k(U)} \varphi(x,\,S) \,{\mathrm{d}} \mathbb{V}(x,\,S) = \int_{\mathfrak{S}}
\varphi(x,\,\mathrm{T}_x \mathfrak{S})\,\theta(x) \,{\mathrm{d}}\mathscr{H}^k(x)\] for all \(\varphi \in C_c(G_k(U))\), where \(\mathrm{T}_x \mathfrak{S}\) denotes the approximate
tangent space to \(\mathfrak{S}\) at \(x \in \mathfrak{S}\). Following the common convention, we extend \(\theta\) as \(0\)
in \(\mathbb{R}^n \setminus \mathfrak{S}\). Thus, if \(\mathbb{V}\) is \(\mathscr{H}^k\)-rectifiable, then \[\left\| \mathbb{V}
\right\| = \theta \mathscr{H}^k \mathbin{\vrule height 1.6ex depth 0pt width
0.13ex\vrule height 0.13ex depth 0pt width 1.3ex}\mathfrak{S}.\] If \(\theta\) takes only integer values, then \(\mathbb{V}\) is called integral.
Two pairs \((\mathfrak{S},\,\theta)\) and \((\mathfrak{S}',\,\theta')\) identify the same \(\mathscr{H}^k\)-rectifiable varifold if and only if
the symmetric difference \(\mathfrak{S} \Delta \mathfrak{S}'\) is a \(\mathscr{H}^k\)-null set and \(\theta = \theta'\) on \(\mathscr{H}^k\)-almost all of \(\mathfrak{S} \cap \mathfrak{S}'\). These conditions define an equivalence relation, so that rectifiable varifolds are actually equivalence classes of pairs
\((\mathfrak{S},\,\theta)\). We will write \(\mathbb{V} = \mathbf{v}(\mathfrak{S}, \, \theta)\) for the rectifiable varifold carried by \(\mathfrak{S}\) with
density function \(\theta\) (see e.g. [37]). The first variation of a varifold \(\mathbb{V}\) is the distribution \(\delta \mathbb{V}\) in \(U\) satisfying \[\delta\mathbb{V}(\mathbf{X}) := \int_{\mathfrak{S}}
\relax_{\mathrm{T}_x \mathfrak{S}} \mathbf{X}(x) \,{\mathrm{d}} \mathscr{H}^k(x)\] for all \(\mathbf{X}\in C^1_c(U,\,\mathbb{R}^n)\). If it happens that \(\delta{\mathbb{V}}(\mathbf{X}) =
0\) for any \(\mathbf{X}\in C^1_c(U,\,\mathbb{R}^n)\), then \(\mathbb{V}\) is said to be stationary in \(U\).
1.2 The zero set of the potential energy and the material constant \(\kappa_\star\)↩︎
It can be checked (see [29]) that, for any \(\varepsilon> 0\), we have \(f_\varepsilon(\mathbf{Q},\,\mathbf{M}) = 0\) if and only if \((\mathbf{Q},\,\mathbf{M}) = (\mathbf{Q}^{\rm pot}_\varepsilon,\,\mathbf{M}^{\rm pot}_\varepsilon)\), where \(\mathbf{Q}^{\rm pot}_\varepsilon,\,\mathbf{M}^{\rm pot}_\varepsilon\) satisfy \[\left| \mathbf{M}^{\rm pot}_\varepsilon \right| = \lambda_{\varepsilon,\beta}, \qquad \mathbf{Q}^{\rm pot}_\varepsilon=
\sqrt{2} s_{\varepsilon,\beta}\left( \frac{\mathbf{M}^{\rm pot}_\varepsilon\otimes \mathbf{M}^{\rm pot}_\varepsilon}{\lambda_{\varepsilon,\beta}^2} - \frac{\mathbf{I}}{2} \right),\] where \[s_{\varepsilon,\beta} = 1 +
\varepsilon\kappa_\star + {\rm O}_{\varepsilon\to 0}(\varepsilon^2), \qquad \lambda_{\varepsilon,\beta}^2 = 1 + \sqrt{2}\beta + \sqrt{2}\beta \varepsilon\kappa_\star + {\rm O}_{\varepsilon\to 0}(\varepsilon^2).\] Notice that \[s_{\varepsilon,\beta} \to 1, \qquad \lambda_{\varepsilon,\beta} \to \left( 1 + \sqrt{2}\beta \right)^{1/2} \qquad as\varepsilon\to 0.\] Here, \[\label{eq:kappa42} \kappa_\star := \frac{\beta}{2\sqrt{2}}\left( 1 + \sqrt{2}\beta \right)\tag{18}\] is a material-dependent, dimensionless constant, measuring the strength of the coupling between \(\mathbf{Q}\) and \(\mathbf{M}\) in terms of how much the interaction lifts the ground level of the potential at first order in \(\varepsilon\) from what it would
be with the interaction switched off.
Given boundary data as in 3 or as in 4 , one easily computes that \[\frac{1}{\varepsilon^2}f_\varepsilon(\mathbf{Q}_{\mathrm{bd}},\,\mathbf{M}_{\mathrm{bd}}) =
\kappa_\star^2 + \mathrm{o}_{\varepsilon\to 0}(\varepsilon^2).\]
By standard elliptic regularity, any critical pair \((\mathbf{Q}_\varepsilon,\,\mathbf{M}_\varepsilon)\) of \(\mathscr{F}_\varepsilon\) is real-analytic in \(\Omega\) and, under boundary conditions as in 3 or 4 , at least of class \(C^1\) in \(\overline{\Omega}\). Moreover, from [1] we have \[\begin{align} & \left\|
\mathbf{Q}_\varepsilon \right\|_{L^\infty(\Omega)} + \left\| \mathbf{M}_\varepsilon \right\|_{L^\infty(\Omega)} \leq C_\beta \tag{19} \\ & \left\| \nabla \mathbf{Q}_\varepsilon \right\|_{L^\infty(\Omega)} + \left\| \mathbf{M}_\varepsilon
\right\|_{L^\infty(\Omega)} \leq \frac{C_{\beta,\Omega}}{\varepsilon}\tag{20},
\end{align}\] where the constant \(C_\beta\) depends only on \(\beta\) and
the constant \(C_{\beta,\Omega}\) depends only on \(\beta\) and \(\Omega\). More precisely, the proof of [1] shows that, for any \(\varepsilon> 0\), the pointwise inequalities \[\begin{align} \left| \mathbf{Q}_\varepsilon \right|
&\leq 1 + \varepsilon\kappa_\star + c_\beta \varepsilon^2, \tag{21} \\ \left| \mathbf{M}_\varepsilon \right|^2 &\leq 1 + \sqrt{2}\beta \left| \mathbf{Q}_\varepsilon \right| \tag{22},
\end{align}\] where \(c_\beta\) depends only on \(\beta\), hold
globally in \(\Omega\), cf. [1].
Assume that \(G \subseteq \Omega\) is an open, bounded, and simply connected subset of \(\Omega\), with smooth boundary \(\partial G\). Assume that \(\mathbf{Q}: G \to \mathscr{S}_0^{2\times 2}\) is a smooth \(\mathbf{Q}\)-tensor field such that \(\mathbf{Q}(x) \neq 0\) for all \(x
\in G\). Then, by the spectral theorem and the simple connectedness of \(G\) and well-known lifting results (cf. [1] and
references therein), we can find a coherently defined and smoothly varying eigenframe of smooth unit eigenvectors for \(\mathbf{Q}\), denoted \((\mathbf{n},\,\mathbf{m})\), so that we may
write \[\label{eq:spectral-thm} \mathbf{Q}= \frac{\left| \mathbf{Q} \right|}{\sqrt{2}} \left( \mathbf{n}\otimes \mathbf{n}- \mathbf{m}\otimes \mathbf{m}\right) \qquad
inG.\tag{23}\] We will often use the following notation: \[\rho := \left| \mathbf{Q} \right|.\] We always assume that \(\mathbf{n}\) denotes a unit eigenvector relative to the
positive eigenvalue of \(\mathbf{Q}\).
Also, there exists a function \(\varphi : G \to \mathbb{R}\) such that \[\label{eq:neps-phieps-1} \mathbf{n}= \begin{pmatrix} \cos\varphi \\
\sin\varphi \end{pmatrix}.\tag{24}\] The following relations are immediately checked by straightforward computation: \[\label{eq:neps-phieps-2} \left| \nabla \mathbf{n}
\right| = \left| \mathbf{n}\times \nabla \mathbf{n} \right| = \left| \nabla \varphi \right|, \qquad \mathbf{n}\times \Delta \mathbf{n}= \Delta \varphi.\tag{25}\] Since \(\mathbf{m}\) is either \(\mathbf{n}^\perp = \begin{pmatrix} -n_2 \\ n_1 \end{pmatrix}\) or \(-\mathbf{n}^\perp\), the same relations as in 25 hold with \(\mathbf{n}\) replaced by \(\mathbf{m}\). Furthermore, by direct computation it is easily seen that \[\label{eq:dec-grad-Q-square} \left| \nabla \mathbf{Q}_\varepsilon \right|^2 = \left| \nabla \rho_\varepsilon \right|^2 + 4 \rho^2 \left| \nabla \varphi \right|^2.\tag{26}\] pointwise.
1.5 Convergence results for \(\mathbf{Q}_\varepsilon\)↩︎
Let \(\{(\mathbf{Q}_\varepsilon,\,\mathbf{M}_\varepsilon)\}_\varepsilon\) be a sequence of critical points of \(\mathscr{F}_\varepsilon\) subject to boundary conditions as in 3 or 4 and satisfying 5 . By [1], there exists a positive number \(\varepsilon_*\) such that, if \(B = B(x_0, \, R) \subset\!\subset\Omega \setminus \mathop{\mathrm{spt}}\mu_\star\), then \(\left| \mathbf{Q}_\varepsilon \right| \geq
1/2\) in \(B(x_0,\,3R/4)\). Consequently, in \(B\) we can decompose \(\mathbf{Q}_\varepsilon\) as in 23 , write
\(\mathbf{n}_\varepsilon\) as in 24 , and, for each \(\varepsilon\leq \varepsilon_* R\), we can find a function \(\varphi_\varepsilon: B(x_0,\,3R/4) \to \mathbb{R}\) satisfying 24 . By [1], we know that there
exists a constant \(C_p(x_0,\,R)\), depending only on \(\beta\), \(\Omega\), \(p\), \(x_0\), \(R\), \({\rm C}_{\rm pot}\), and the \(L_1(\partial\Omega)\)- and the \(L^2(\partial\Omega)\)-norm of \(\mathbf{Q}_{\mathrm{bd}}\times \partial_{{\boldsymbol{\tau}}}\mathbf{Q}_{\mathrm{bd}}\), such that \[\label{eq:varphi-Lp-bounds} \forall p \in [1,\,\infty), \qquad \left\| \nabla \varphi_\varepsilon \right\|_{L^p(B(x_0,\,R/2))} \leq C_{p}(x_0,\,R).\tag{27}\] By a standard covering argument, this implies that for every
compact set \(K \subset \Omega \setminus \mathop{\mathrm{spt}}\mu_\star\) there exists \(\varepsilon_K >0\) such that, for any \(\varepsilon\leq
\varepsilon_K\), there holds \(\left| \mathbf{Q}_\varepsilon \right| \geq 1/2\) in \(K\). In particular, both the considerations in Section 1.4 and
those above are valid in any simply connected, open subset \(G \subset\!\subset K\) with smooth boundary, for any \(\varepsilon\leq \varepsilon_K\).
Regarding \(\rho_\varepsilon\), we observe that it satisfies the equation \[\label{eq:rho-eps} -\frac{1}{2} \Delta \rho_\varepsilon^2 + \left| \nabla
\mathbf{Q}_\varepsilon \right|^2 + \frac{1}{\varepsilon^2}\left(\rho_\varepsilon^2 - 1 \right) \rho_\varepsilon^2 - \frac{\beta}{\varepsilon} \mathbf{Q}\mathbf{M}\cdot \mathbf{M}= 0,\tag{28}\] which is obtained by scalar multiplication of 6 with \(\mathbf{Q}_\varepsilon\). If \(G \subset \Omega\) is an open set such that \(\rho_\varepsilon> 0\) in \(G\), then, upon using 26 and dividing both sides by \(\rho_\varepsilon\), we can recast 28 as
\[\label{eq:rho-eps-bis} -\Delta \rho_\varepsilon+ 4 \rho^2 \left| \nabla \varphi \right|^2 + \frac{1}{\varepsilon^2}\left(\rho_\varepsilon^2 - 1 \right) \rho_\varepsilon^2 -
\frac{\beta}{\varepsilon} \mathbf{Q}\mathbf{M}\cdot \mathbf{M}= 0 \qquad inG.\tag{29}\] By using 27 and testing suitably 29 , it has been shown in [1] that \[\label{eq:GL-bounds-p} \left\| \nabla \mathbf{Q}_\varepsilon \right\|_{L^p(B(x_0,\,R/2))} + \left\|
\frac{\rho_\varepsilon- 1}{\varepsilon} \right\|_{L^p(B(x_0,\,R))} \leq C_{p}(x_0,\,R),\tag{30}\] for a constant \(C_p(x_0,\,R)\) depending only on \(\beta\), \(\Omega\), \(p\), \(x_0\), \(R\), \({\rm C}_{\rm pot}\), and the \(L_1(\partial\Omega)\)- and the \(L^2(\partial\Omega)\)-norm of \(\mathbf{Q}_{\mathrm{bd}}\times \partial_{{\boldsymbol{\tau}}}\mathbf{Q}_{\mathrm{bd}}\).
Furthermore, it has been shown in [1] that \[\label{eq:rho-unif-conv} \rho_\varepsilon\to 1
\qquad uniformly inB.\tag{31}\] for every ball \(B \subset\!\subset\Omega \setminus \mathop{\mathrm{spt}}\mu_\star\) and hence, by covering, in every compact set \(K \subset
\Omega\setminus \mathop{\mathrm{spt}}\mu_\star\). In addition, by [1], we have \[\label{eq:strong-conv-rho} \int_B \left\{\left| \nabla \rho_\varepsilon \right|^2 + \left(\frac{\rho_\varepsilon- 1}{\varepsilon} - \kappa_\star\right)^2\right\}\,{\mathrm{d}}x \to 0 \qquad as\varepsilon\to 0\tag{32}\]
for any ball \(B \subset\!\subset\Omega \setminus \mathop{\mathrm{spt}}\mu_\star\). Along with the uniform bounds in [1], 32 implies that for every \(p \in [1, +\infty)\) we have (cf. [1])
\[\label{eq:strong-conv-rho-p} \int_B \left\{\left| \nabla \rho_\varepsilon \right|^p + \left| \frac{\rho_\varepsilon- 1}{\varepsilon} - \kappa_\star
\right|^p\right\}\,{\mathrm{d}}x \to 0 \qquad as\varepsilon\to 0,\tag{33}\] for every \(p \in [1,\,+\infty)\) and every ball \(B \subset\!\subset\Omega \setminus
\mathop{\mathrm{spt}}\mu_\star\).
Among the most important consequences of 33 , there is the fact, crucially exploited in [2], that the limiting potential
energy density \(\zeta_\star\) defined in 15 contains, at first order, all the asymptotic information about the full potential energy density \(\frac{1}{\varepsilon^2}
f_\varepsilon(\mathbf{Q}_\varepsilon,\,\mathbf{M}_\varepsilon)\), in the sense that, for any \(p \in [1,\,+\infty)\), there holds \[\label{eq:f-V-p}
\int_K \left| \frac{1}{\varepsilon^2}f_\varepsilon(\mathbf{Q}_\varepsilon,\,\mathbf{M}_\varepsilon) - \frac{1}{\varepsilon}V(\mathbf{M}_\varepsilon) \right|^p \,{\mathrm{d}}x \to 0, \qquad as\varepsilon\to 0.\tag{34}\] Lemma 1 below improves on 32 , so as to promote the above convergence to a quantitative decay with respect to
\(\varepsilon\).
1.6 Decomposition of \(\mathbf{M}\) along an eigenframe of \(\mathbf{Q}\): the map \(\mathbf{u}\)↩︎
Let \(G \subseteq \Omega\) be a simply connected, open set with smooth boundary \(\partial G\). Assume that \(\mathbf{Q}: G \to \mathscr{S}_0^{2\times
2}\) is a \(\mathbf{Q}\)-tensor field such that \(\mathbf{Q}\neq 0\) in \(G\). Let \((\mathbf{n},\,\mathbf{m})\)
denote any eigenframe associated with \(\mathbf{Q}\) as in 23 . Then, we can project \(\mathbf{M}\) along the eigenframe \((\mathbf{n},\,\mathbf{m})\), obtaining the map \(\mathbf{u}: G \to \mathbb{R}^2\) defined by \[\label{eq:u} \mathbf{u}:=
\left(\mathbf{M}\cdot \mathbf{n},\,\mathbf{M}\cdot \mathbf{m}\right).\tag{35}\] We set \[\label{eq:u1u2} u_1 := \mathbf{M}\cdot \mathbf{n}, \quad u_2 := \mathbf{M}\cdot
\mathbf{m}\tag{36}\] and notice that \[\label{eq:mod-u} \left| \mathbf{u} \right| = \left| \mathbf{M} \right|\tag{37}\] pointwise in \(G\). In particular, by 37 and 19 , \[\label{eq:norm-u} \left\| \mathbf{u} \right\|_{L^\infty(G)} \leq
\left\| \mathbf{M} \right\|_{L^\infty(\Omega)} \leq C_\beta.\tag{38}\] We also observe that, if \(\mathbf{Q}\), \(\mathbf{M}\in C^\infty(G)\), then \(\mathbf{u}\in C^\infty(G)\). For later purposes, we explicitly observe that if \(\mathbf{Q}\in \mathscr{S}_0^{2\times 2}\setminus \{0\}\), then \[\mathbf{Q}\mathbf{M}\cdot \mathbf{M}= \frac{\rho}{\sqrt{2}}\left( (\mathbf{M}\cdot \mathbf{n})^2 - (\mathbf{M}\cdot \mathbf{m})^2 \right),\] i.e., \[\label{eq:QMM-u} \mathbf{Q}\mathbf{M}\cdot \mathbf{M}= \frac{\rho}{\sqrt{2}}\left( u_1^2 - u_2^2 \right)\tag{39}\]
1.7 The auxiliary energy functional \(\mathcal{AC}_\varepsilon\)↩︎
Let \(\{(\mathbf{Q}_\varepsilon,\,\mathbf{M}_\varepsilon)\}_\varepsilon\) be a sequence of critical points of \(\mathscr{F}_\varepsilon\) satisfying 5 . Let
\(K \subset \Omega \setminus \mathop{\mathrm{spt}}\mu_\star\) be any compact set, and let \(G \subset K\) be any simply connected, open set, with smooth boundary. As already seen above,
there exists \(\varepsilon_K\) so that, for any \(\varepsilon\leq \varepsilon_K\), there holds \(\left| \mathbf{Q}_\varepsilon \right| \geq 1/2\) in \(K\). Thus, for each \(\varepsilon\leq \varepsilon_K\), from \(\mathbf{M}_\varepsilon\) and \(\mathbf{n}_\varepsilon\), we can
construct a map \(\mathbf{u}_\varepsilon: G \to \mathbb{R}^2\) as in 35 . Upon defining \[\label{eq:ACeps}
\mathcal{AC}_\varepsilon(\mathbf{u}_\varepsilon;\,G) := \int_G \left\{ \frac{\varepsilon}{2}\left| \nabla \mathbf{u}_\varepsilon \right|^2 + \frac{1}{\varepsilon} h(\mathbf{u}_\varepsilon) \right\}\,{\mathrm{d}}x,\tag{40}\] where
\[\label{eq:h} h(\mathbf{u}) = h(u_1,\,u_2) := \frac{1}{4}\left( \left| \mathbf{u} \right|^2 - 1 \right)^2 - \frac{\beta}{\sqrt{2}}\left( u_1^2 - u_2^2 \right) + \frac{\beta^2 +
\sqrt{2}\beta}{2},\tag{41}\] we can rewrite (cf., e.g., [29]) \[\mathscr{F}_\varepsilon(\mathbf{Q}_\varepsilon,\,\mathbf{M}_\varepsilon;\,G) = \int_G \left\{ \frac{1}{2} \left| \nabla \mathbf{Q}_\varepsilon \right|^2 + \frac{1}{4\varepsilon^2}\left( \left| \mathbf{Q}_\varepsilon \right|^2 - 1
\right)^2\right\}\,{\mathrm{d}}x + \mathcal{AC_\varepsilon}(\mathbf{u}_\varepsilon;\,G) + R_\varepsilon,\] where \[R_\varepsilon\to 0 \qquad as\varepsilon\to 0.\] By [1], we know that \[\label{eq:ACeps-equibdd} \sup_{\varepsilon> 0} \mathcal{AC}_\varepsilon(\mathbf{u}_\varepsilon;\,G) \leq C <
+\infty,\tag{42}\] where the constant \(C\) does not depend on \(\varepsilon\) and on \(G\), but only on \(\beta\), \(\Omega\), \({\rm C}_{\rm pot}\), and the \(L^1(\partial\Omega)\)- and the \(L^2(\partial\Omega)\)-norm of \(\mathbf{Q}_{\mathrm{bd}}\times \partial_{\boldsymbol{\tau}}\mathbf{Q}_{\mathrm{bd}}\). Since, as it can be easily checked, \[h(\mathbf{u}) = h(\left| \mathbf{u} \right|,\,0) + \sqrt{2}\beta u_2^2, \qquad h(\left| \mathbf{u} \right|,\,0) \geq 0,\] for any vector \(\mathbf{u}= (u_1,\,u_2) \in \mathbb{R}^2\), an
immediate consequence of 42 is the bound \[\label{eq:decay-u2-basic} \frac{1}{\varepsilon}\int_G u_{2,\varepsilon}^2 \,{\mathrm{d}}x \leq
C_2,\tag{43}\] where the constant \(C_2\) does not depend on \(\varepsilon\) and on \(G\), but only on \(\beta\), \(\Omega\), \({\rm C}_{\rm pot}\), and the \(L^1(\partial\Omega)\)- and the \(L^2(\partial\Omega)\)-norm of \(\mathbf{Q}_{\mathrm{bd}}\times \partial_{\boldsymbol{\tau}}\mathbf{Q}_{\mathrm{bd}}\).
We point out some additional remarks on the function \(h\). By straightforward computations, we see that \(h(\mathbf{u})\) can be rewritten in terms of the components \(u_1\), \(u_2\) in the following ways (both of them will be used later on): \[\label{eq:h-ter} h(\mathbf{u}) = \frac{1}{4}\left(
\left| \mathbf{u} \right|^2 - 1 - \sqrt{2}\beta \right)^2 + \sqrt{2}\beta u_2^2\tag{44}\] and also \[\label{eq:h-bis} h(u_1,\,u_2) = \frac{1}{4}\left( u_1^2 - 1 - \sqrt{2} \beta
\right)^2 + u_2^2 \left( \frac{1}{2}\left( u_1^2 - 1 + \sqrt{2} \beta \;\right) + \frac{1}{4}u_2^2 \right).\tag{45}\] For \(u\), \(v \in \mathbb{R}\), we define
\[\label{eq:h1-h2} h_1(u) := \frac{1}{4}\left( u^2 - 1- \sqrt{2}\beta \right)^2, \qquad h_2(u,\,v) := v^2\left( \frac{1}{2}\left(u^2 - 1 + \sqrt{2}\beta \right) + \frac{1}{4}v^2
\right),\tag{46}\] so that for any \(\mathbf{u}= (u_1,\,u_2) \in \mathbb{R}^2\) we have \[h(\mathbf{u}) = h_1(u_1) + h_2(u_1,\,u_2).\] If \(\sqrt{2}\beta \geq 1\), then the function \(h_2\) is non-negative, for every \(u\), \(v \in \mathbb{R}\). Thus, if \(\sqrt{2}\beta \geq 1\), then \(h\) is the sum of two positive terms, one of which involving only \(u_1\). Moreover, besides 43 , we have other immediate consequences of 42 , that is, \[\begin{gather} \frac{1}{\varepsilon} \int_G \left( u_1^2 - 1 - \sqrt{2} \beta
\right)^2 \,{\mathrm{d}}x \leq C_1, \tag{47} \\ \frac{1}{\varepsilon} \int_G u_2^4 \,{\mathrm{d}}x \leq C_2. \tag{48}
\end{gather}\] For later use, we also observe that, by the definition 10 of \(V(\mathbf{M}_\varepsilon)\), the definition 41 of \(h(\mathbf{u})\), and elementary computations, it follows that (cf. [2]) \[\label{eq:V-h} V(\mathbf{M}) - h(\mathbf{u}) = \frac{\beta}{\sqrt{2}}\left(1 - \left| \mathbf{Q} \right| \right)\left(u_1^2 - u_2^2 -
\frac{\beta + \beta\left| \mathbf{Q} \right|}{\sqrt{2}} \right).\tag{49}\] If \((\mathbf{Q}_\varepsilon,\,\mathbf{M}_\varepsilon)\) is a critical pair for \(\mathscr{F}_\varepsilon\), then the equation satisfied by \(\mathbf{u}_\varepsilon\) in \(G\) reads as follows: \[\label{eq:EL-u} -\Delta \mathbf{u}_\varepsilon+ \frac{1}{\varepsilon^2}\left(\left| \mathbf{u}_\varepsilon \right|^2-1\right) \mathbf{u}_\varepsilon- \frac{\sqrt{2}\beta \left| \mathbf{Q}_\varepsilon \right|}{\varepsilon^2}
\bar{\mathbf{u}}_\varepsilon= \mathbf{u}_\varepsilon\left| \nabla \varphi_\varepsilon \right|^2 - \nabla \mathbf{u}_\varepsilon^\perp \cdot \nabla \varphi_\varepsilon- \mathbf{u}_\varepsilon^\perp \Delta \varphi_\varepsilon\qquad
inG,\tag{50}\] where we used the notation \[\mathbf{u}^\perp = \begin{pmatrix} -u_2 \\ u_1 \end{pmatrix}, \qquad \bar{\mathbf{u}} = \begin{pmatrix} u_1 \\ -u_2 \end{pmatrix}.\] Equation 50 is obtained by rewriting 7 in terms of \(\mathbf{u}\). Note, in particular, that 50 is not the Euler-Lagrange equation associated with
the functional \(\mathcal{AC}_\varepsilon\). In fact, it differs from it because the right-hand side of 50 is not zero (rather, it contains the interaction terms with the \(\mathbf{Q}_\varepsilon\)-components).
1.8 Quantitative convergence for the modulus of \({\mathbf{Q}_\varepsilon}\)↩︎
For our purposes in this paper, we need an improved version of 32 , which is contained in Lemma 1 below.
Lemma 1. For every compact set \(K \subset \Omega \setminus \mathop{\mathrm{spt}}\mu_\star\), there holds \[\label{eq:improved-decay-rho} \int_K \left\{ \left| \nabla \rho_\varepsilon \right|^2 + \left(\frac{\rho_\varepsilon-1}{\varepsilon} - \kappa_\star \right)^2 \right\}\,{\mathrm{d}}x \leq C_{\beta,K} \varepsilon\qquad as\varepsilon\to
0.\tag{51}\] The constant \(C_{\beta,K}\) depends only on \(\beta\), \(K\), \(\Omega\), \({\rm C}_{\rm pot}\), and the \(L^1(\partial\Omega)\)- and the \(L^2(\partial\Omega)\)-norm of \(\mathbf{Q}_{\mathrm{bd}}\times
\partial_{{\boldsymbol{\tau}}} \mathbf{Q}_{\mathrm{bd}}\).
Proof. Using a standard covering argument, it suffices to show that, for any ball \(B = B(x_0,\,R) \subset\!\subset\Omega \setminus \mathop{\mathrm{spt}}\mu_\star\), we have
\[\label{eq:improved-decay-rho-B} \int_{B'} \left\{ \left| \nabla \rho_\varepsilon \right|^2 + \left(\frac{\rho_\varepsilon-1}{\varepsilon} - \kappa_\star \right)^2
\right\}\,{\mathrm{d}}x \leq C_\beta(x_0,\,R) \varepsilon,\tag{52}\] where \(B' = B(x_0,\,R/2)\) and \(C_\beta(x_0,\,R)\) depends only on \(\beta\), \(x_0\), \(R\), \(\Omega\), \({\rm C}_{\rm pot}\), and the \(L^1(\partial\Omega)\)- and the \(L^2(\partial\Omega)\)-norm of \(\mathbf{Q}_{\mathrm{bd}}\times \partial_{{\boldsymbol{\tau}}} \mathbf{Q}_{\mathrm{bd}}\).
Step 1 (Using the clearing-out property). Since \(B\) stays at positive distance from \(\mathop{\mathrm{spt}}\mu_\star\), we know from [1] that there exists \(\varepsilon_*\) such that, for any \(\varepsilon\) with \(0 < \varepsilon\leq
\varepsilon_* R\), we have \(\rho_\varepsilon\geq 1/2\) in \(B\). From now on, to keep the notation as light as possible, we drop the subscript \(\varepsilon\), writing (for instance) \(\rho\) instead of \(\rho_\varepsilon\), and so on.
Since \(\rho \geq 1/2\) in \(B\), by 29 and 39 , the equation satisfied by \(\rho\) in \(B\) reads \[\label{eq:eq-rho} -\Delta \rho + 4\rho \left| \nabla \varphi \right|^2 + \frac{1}{\varepsilon^2}(\rho - 1) (\rho + 1)\rho =
\frac{\sigma}{\varepsilon},\tag{53}\] where we set \[\sigma := \beta \frac{\mathbf{Q}\mathbf{M}\cdot \mathbf{M}}{\rho} = \frac{\beta}{\sqrt{2}}\left(u_1^2 - u_2^2\right).\] As in [1], we subtract \(\frac{2}{\varepsilon} \kappa_\star\) from both sides of 53 , so as to obtain
\[\label{eq:eq-rho-bis} -\Delta \rho + 4 \rho \left| \nabla \varphi \right|^2 + \frac{1}{\varepsilon}\left(\frac{\rho - 1}{\varepsilon} (\rho + 1)\rho - 2 \kappa_\star \right) =
\frac{\sigma - 2 \kappa_\star}{\varepsilon}.\tag{54}\] We further observe that \[\label{eq:sigma-2k42} \begin{align} \sigma - 2 \kappa_\star &=
\frac{\beta}{\sqrt{2}}\left( u_1^2 - u_2^2 - 1 - \sqrt{2}\beta \right) \\ &=\frac{\beta}{\sqrt{2}}\left( \left| \mathbf{u} \right|^2 - 1 - \sqrt{2}\beta - 2 u_2^2 \right), \end{align}\tag{55}\] which will be used later on. We will also
use the trivial identity \[\label{eq:trivial-rho} (\rho - 1)(\rho + 1) \rho = 2(\rho - 1) + 3(\rho - 1)^2 + (\rho-1)^3.\tag{56}\]
Step 2. Let \(\zeta \in C^\infty_c(\mathbb{R}^2)\) be any cut-off function with support in \(B\) such that \[\mathop{\mathrm{spt}}\zeta \subset B,
\qquad 0 \leq \zeta \leq 1, \qquad \zeta \equiv 1 \quad inB', \qquad \left| \nabla \zeta \right| \leq \frac{4}{R} \quad in\mathbb{R}^2.\] We multiply both sides of 54 by \(\zeta^2 (\rho -
1 - \varepsilon\kappa_\star)\) and we integrate the result over \(B\). After rearrangement and upon using 56 , we obtain \[\label{eq:improved-decay-rho-compu1} \begin{align} \int_B \zeta^2 \left\{ \left| \nabla \rho \right|^2 + 2 \left(\frac{\rho-1}{\varepsilon} - \kappa_\star\right)^2 \right\} \,{\mathrm{d}}x &= \int_B \zeta^2 (\sigma - 2
\kappa_\star) \left(\frac{\rho-1}{\varepsilon} - \kappa_\star\right)\,{\mathrm{d}}x \\ &- \varepsilon\int_B 2 \zeta \, \nabla \rho \cdot \nabla \zeta \left(\frac{\rho-1}{\varepsilon} - \kappa_\star\right) \,{\mathrm{d}}x \\ &- \varepsilon\int_B 4
\zeta^2 \, \rho \left| \nabla \varphi \right|^2 \left(\frac{\rho-1}{\varepsilon} - \kappa_\star\right) \,{\mathrm{d}}x \\ &- \int_B \zeta^2\left( 2 (\rho-1)^2 + (\rho-1)^3 \right)\left(\frac{\rho-1}{\varepsilon} - \kappa_\star\right) \,{\mathrm{d}}x
\end{align}\tag{57}\] Upon using Young’s inequality, we can estimate \[\label{eq:improved-decay-rho-compu2} \varepsilon\int_B 2 \zeta \nabla \rho \cdot \nabla
\zeta \left(\frac{\rho-1}{\varepsilon} - \kappa_\star\right) \,{\mathrm{d}}x \leq \frac{1}{2} \int_B \zeta^2 \left| \nabla \rho \right|^2\,{\mathrm{d}}x + 8 \varepsilon^2 \int_B \left| \nabla \zeta \right|^2 \left(\frac{\rho-1}{\varepsilon} -
\kappa_\star\right)^2 \,{\mathrm{d}}x.\tag{58}\] Since \(\left| \nabla \zeta \right|^2 \leq 16/R^2\), we have \[\int_B \left| \nabla \zeta \right|^2 \left(\frac{\rho-1}{\varepsilon}
- \kappa_\star\right)^2 \,{\mathrm{d}}x \leq \frac{16}{R^2} \int_B \left(\frac{\rho-1}{\varepsilon} - \kappa_\star\right)^2 \,{\mathrm{d}}x.\] By 30 (with \(p = 2\)), the integral on
the right-hand side is bounded independently of \(\varepsilon\), in terms of a constant \(C_\beta(x_0,\,R)\) depending only on \(\beta\), \(\Omega\), \(x_0\), \(R\), \({\rm C}_{\rm pot}\), and the \(L_1(\partial\Omega)\)- and the \(L^2(\partial\Omega)\)-norm of \(\mathbf{Q}_{\mathrm{bd}}\times \partial_{{\boldsymbol{\tau}}}\mathbf{Q}_{\mathrm{bd}}\). Therefore, \[\varepsilon\int_B 2 \zeta \nabla
\rho \cdot \nabla \zeta \left(\frac{\rho-1}{\varepsilon} - \kappa_\star\right) \,{\mathrm{d}}x \leq \frac{1}{2} \int_B \zeta^2 \left| \nabla \rho \right|^2\,{\mathrm{d}}x + C_\beta(x_0,\,R) \varepsilon^2.\] Consequently, using again Young’s
inequality, from 57 we obtain , \[\label{eq:improved-decay-rho-compu3} \begin{align} \int_B \zeta^2 \left\{ \left| \nabla
\rho \right|^2 + \left(\frac{\rho-1}{\varepsilon} - \kappa_\star\right)^2 \right\} \,{\mathrm{d}}x \leq & 3 \int_B \zeta^2 (\sigma - 2 \kappa_\star)^2 \,{\mathrm{d}}x + 12 \varepsilon^2 \int_B \zeta^2 \rho^2 \left| \nabla \varphi \right|^4
\,{\mathrm{d}}x \\ &+ 3 \int_B \zeta^2 \left( 2 (\rho-1)^2 + (\rho-1)^3 \right)^2 \,{\mathrm{d}}x + C_\beta(x_0,\,R) \varepsilon^2 \end{align}\tag{59}\] From 30 (i.e., from [1]), we already know that the last term on the right-hand side in 57 tends to zero at least as fast as \(\varepsilon^4\), i.e., \[\label{eq:improved-decay-rho-compu4} \int_B \zeta^2 \left( 2 (\rho-1)^2 + (\rho-1)^3 \right)^2 \,{\mathrm{d}}x =
\mathrm{O}_{\varepsilon\to 0}(\varepsilon^4).\tag{60}\] Next, by 27 and 21 , we have \[\label{eq:improved-decay-rho-compu5} \varepsilon^2 \int_B 4 \zeta^2 \rho^2 \left| \nabla \varphi \right|^4 \,{\mathrm{d}}x = \mathrm{O}_{\varepsilon\to 0}(\varepsilon^2).\tag{61}\] Finally, by 55 and 44 , \[\frac{1}{4} (\sigma - 2 \kappa_\star)^2 \leq c_\beta\left[ \frac{1}{4}\left(\left| \mathbf{u} \right|^2 - 1 - \sqrt{2}\beta \right)^2 + \sqrt{2}\beta u_2^2
\right] = c_\beta h(\mathbf{u}),\] where \(c_\beta\) is a constant depending only on \(\beta\). Therefore, by 42 and since \(0 \leq \zeta \leq 1\), \[\label{eq:improved-decay-rho-compu6} \frac{1}{4} \int_B \zeta^2 (\sigma - 2 \kappa_\star)^2 \,{\mathrm{d}}x =
\mathrm{O}_{\varepsilon\to 0} (\varepsilon).\tag{62}\] Combining 59 , 60 , 61 , and 62 with 58 and recalling that \(\zeta^2 \equiv 1\) on \(B'\), we end up with 52 .
◻
2 Refined asymptotics in the regime \(\sqrt{2}\beta > 1\)↩︎
In this section, we show that, if \(\sqrt{2}\beta > 1\), then the structure of the equation satisfied by the component \(u_{2,\varepsilon}\) implies the decay of the \(W^{1,2}\)-norm of \(u_{2,\varepsilon}\) at a power-rate in \(\varepsilon\), far from \(\mathop{\mathrm{spt}}\mu_\star\). As a
consequence, in this regime the component \(u_{2,\varepsilon}\) does not concentrate energy on the limiting varifold. The main result of this Section is the following proposition.
Proposition 3. Let \(\{(\mathbf{Q}_\varepsilon,\,\mathbf{M}_\varepsilon)\}_\varepsilon\) be a sequence of critical points of \(\mathscr{F}_\varepsilon\) satisfying
boundary conditions either as in 3 or as in 4 , as well as assumption 5 . Assume, in addition, that \(\sqrt{2}\beta > 1\). Let
\(K \subset \Omega \setminus \mathop{\mathrm{spt}}\mu_\star\) be any compact set and let \(G \subset\!\subset K\) be any simply connected, open set, with smooth boundary \(\partial G\). Let \(\{\mathbf{u}_\varepsilon\}_\varepsilon\) be the sequence of maps defined in \(G\) as prescribed by 35 . Then, for any
\(\alpha \in (0,\,2)\), there hold \[\begin{gather} \int_G \left( \varepsilon\left| \nabla u_{2,\varepsilon} \right|^2 + \frac{1}{\varepsilon}\left(\left| \mathbf{u}_\varepsilon \right|^2 - 1 +
\sqrt{2}\beta \rho_\varepsilon\right) u_{2,\varepsilon}^2 \right)\,{\mathrm{d}}x \leq C_\alpha(\mathop{\mathrm{dist}}(G, \partial K)) \varepsilon^\alpha, \label{eq:decay-u2-goal1} \\ \int_G \left( \varepsilon\left| \nabla u_{2,\varepsilon} \right|^2 +
\frac{1}{\varepsilon} u_{2,\varepsilon}^2 \right)\,{\mathrm{d}}x \leq C_\alpha'(\mathop{\mathrm{dist}}(G, \partial K)) \varepsilon^\alpha \label{eq:decay-u2-goal2}.
\end{gather}\] {#eq: sublabel=eq:eq:decay-u2-goal1,eq:eq:decay-u2-goal2} The constants \(C_\alpha(\mathop{\mathrm{dist}}(G,\,\partial K))\), \(C'_\alpha(\mathop{\mathrm{dist}}(G,\,\partial K))\) depend only on \(\mathop{\mathrm{dist}}(G, \partial K)\), \(\alpha\), \(\beta\), \(\Omega\), \({\rm C}_{\rm pot}\), and the \(L^1(\partial\Omega)\)- and the \(L^2(\partial\Omega)\)-norms of \(\mathbf{Q}_{\mathrm{bd}}\times \partial_{\boldsymbol{\tau}}\mathbf{Q}_{\mathrm{bd}}\).
Proof. For notational convenience, we shall drop all subscripts \(\varepsilon\) within this proof.
We first argue locally, proving a version of ?? and ?? that hold in balls whose closure is well contained in \(\Omega \setminus \mathop{\mathrm{spt}}\mu_\star\). Then, we obtain the inequalities in the desired form by a
standard covering argument.
Let \(B_1 := B(x_0,\,R)\) be any ball in \(\Omega \setminus \mathop{\mathrm{spt}}\mu_\star\) such that \(B_2 := B(x_0,\,2R)\) is still contained in \(\Omega \setminus \mathop{\mathrm{spt}}\mu_\star\). For \(\ell > 0\), let us denote \(B_{\ell} := B(x_0,\,\ell R)\). By [1], we know that there exists \(\varepsilon_1 = \varepsilon_1(x_0,\,R)\) such that \(\left| \mathbf{Q}_\varepsilon \right|
\geq 1/2\) in \(B_1\), for every \(\varepsilon\leq \varepsilon_1\). Consequently, defining pointwise \(\mathbf{u}\) as in 35
gives rise to a well-defined map in \(B_1\) which satisfies 50 . The equation satisfied by the component \(u_2\) is obtained by projecting 50 along the direction \(\mathbf{e}_2 := (0,\,1)^{\rm t}\) and, after adding to both sides the term \(\frac{\sqrt{2}\beta(1+\varepsilon\kappa_\star)}{\varepsilon^2}
u_2\) and rearranging, it reads as follows: \[\label{eq:EL-u2} \begin{align} -\Delta u_2 &+ \frac{1}{\varepsilon^2} \left( \left| \mathbf{u} \right|^2 - 1 + \sqrt{2}\beta +
\sqrt{2}\beta \varepsilon\kappa_\star \right) u_2 \\ &= u_2 \left| \nabla \varphi \right|^2 + \nabla u_1 \cdot \nabla \varphi + u_1 \Delta \varphi + \sqrt{2}\beta \left(\kappa_\star - \frac{\rho_\varepsilon-
1}{\varepsilon}\right)\frac{u_2}{\varepsilon} \qquad inB_1. \end{align}\tag{63}\] By assumption, there exists some \(c > 0\) such that \(\sqrt{2}\beta = 1 + c\), hence \[\left( \left| \mathbf{u} \right|^2 - 1 + \sqrt{2}\beta + \sqrt{2}\beta \varepsilon\kappa_\star \right) \geq c > 0\] for every \(\varepsilon> 0\).
Step 3 (Basic bounds). Let us denote \[B = B_1, \qquad B' = B_{1/2}, \qquad B'' = B_{1/4}\] Let \(\zeta \in C^\infty_c(\mathbb{R}^2)\) be any cut-off function such
that \[\begin{gather} 0 \leq \zeta \leq 1, \qquad \zeta \equiv 1 \quad inB', \qquad \mathop{\mathrm{spt}}\zeta \subset B, \tag{64} \\ \left| \nabla \zeta \right| \leq 2\left(\frac{2}{R}\right), \qquad
\left| \nabla^2 \zeta \right| \leq 2\left(\frac{4}{R^2}\right) \qquad in\mathbb{R}^2. \tag{65}
\end{gather}\] After multiplying 63 by \(\varepsilon\zeta^2 u_2\) and
integrating the result over \(\Omega\) (or, which is the same, over \(B\)), we obtain the identity \[\label{eq:decay-u2-compu1} \begin{align} \int_\Omega & \zeta^2 \left( \varepsilon\left| \nabla u_2 \right|^2 + \frac{1}{\varepsilon} \left( \left| \mathbf{u} \right|^2 - 1 + \sqrt{2}\beta + \sqrt{2}\beta \varepsilon\kappa_\star
\right) u_2^2 \right)\,{\mathrm{d}}x \\ &= \varepsilon\int_\Omega \zeta^2 \left( u_2^2 \left| \nabla \varphi \right|^2 + u_2 \nabla u_1 \cdot \nabla \varphi + u_1 u_2 \Delta \varphi \right) \,{\mathrm{d}}x \\ &+ \varepsilon\int_\Omega 2 u_2 \zeta
\nabla u_2 \cdot \nabla \zeta \,{\mathrm{d}}x + \sqrt{2}\beta \int_\Omega \zeta^2 \left(\kappa_\star + \frac{1 -\rho_\varepsilon}{\varepsilon} \right) u_2^2\,{\mathrm{d}}x. \end{align}\tag{66}\] We now proceed to bound the terms on the
right-hand side of 66 .
Let \(\eta > 0\) be any number. By 43 , 38 , and 27 , and Hölder’s inequality, we have
\[\label{eq:decay-u2-compu2} \begin{align} \int_\Omega \varepsilon\zeta^2 u_2^2 \left| \nabla \varphi \right|^2\,{\mathrm{d}}x &\leq \varepsilon\left( \int_B
u_2^{2(1+\eta)}\,{\mathrm{d}}x \right)^{\frac{1}{1+\eta}} C_{\frac{2(\eta+1)}{\eta}}(x_0,\,R)^{\frac{\eta}{1+\eta}} \\ &\leq C_2^{\frac{2}{1+\eta}} C_{\frac{2(\eta+1)}{\eta}}(x_0,\,R)^{\frac{\eta}{1+\eta}} \varepsilon^{1+\frac{1}{1+\eta}},
\end{align}\tag{67}\] where \(C_2\) is the constant in 43 and \(C_{\frac{2(\eta+1)}{\eta}}(x_0,\,R)\) is the constant on the right-hand side
of 27 for \(p = \frac{2(\eta+1)}{\eta}\).
Next, integrating by parts, we rewrite \[\label{eq:decay-u2-compu3} \varepsilon\int_\Omega \zeta^2\left( u_2 \nabla u_1 \cdot \nabla \varphi + u_1 u_2 \Delta \varphi \right)
\,{\mathrm{d}}x = - \varepsilon\int_\Omega \zeta^2 u_1 \nabla u_2 \cdot \nabla \varphi \,{\mathrm{d}}x - \varepsilon\int_\Omega 2 \zeta u_1 u_2 \nabla \zeta \cdot \nabla \varphi \,{\mathrm{d}}x\tag{68}\] We bound the right-hand side of 68 as follows. First, we notice that \[\label{eq:decay-u2-compu4} \begin{align} \varepsilon\int_\Omega \zeta^2 \left( u_1 \nabla u_2 \cdot \nabla
\varphi \right) \,{\mathrm{d}}x &\leq \int_\Omega \frac{\varepsilon}{2} \zeta^2 \left| \nabla u_2 \right|^2 \,{\mathrm{d}}x + 2 \varepsilon\int_B \left| u_1 \right|^2 \left| \nabla \varphi \right|^2\,{\mathrm{d}}x \\ &\leq \int_\Omega
\frac{\varepsilon}{2} \zeta^2 \left| \nabla u_2 \right|^2 \,{\mathrm{d}}x + C_{\beta}(x_0,\,R) \varepsilon, \end{align}\tag{69}\] where we used Young’s inequality, 38 , and 27
. Next, by Hölder’s inequality, 38 , 43 , and 27 , \[\label{eq:decay-u2-compu5}
\begin{align} \varepsilon\int_\Omega 2 \zeta u_1 u_2 \nabla \zeta \cdot \nabla \varphi \,{\mathrm{d}}x &= 2 \varepsilon\int_B u_1 u_2 (\zeta \nabla \zeta \cdot \nabla \varphi)\,{\mathrm{d}}x \\ &\leq C_\beta \varepsilon\left( \int_B u_2^2
\,{\mathrm{d}}x \right)^{1/2} \left( \int_B \left| \nabla \zeta \right|^2 \left| \nabla \varphi \right|^2 \,{\mathrm{d}}x \right)^{1/2} \\ &\leq C_\beta(x_0,\,R) \frac{\varepsilon^\frac{3}{2}}{R}. \end{align}\tag{70}\] In both 69 and 70 , the constant \(C_\beta(x_0,\,R)\) depends only on \(x_0\), \(R\), \(\Omega\), \({\rm C}_{\rm pot}\), and the \(L^1(\partial\Omega)\)- and the \(L^2(\partial \Omega)\)-norm of \(\mathbf{Q}_{\mathrm{bd}}\times \partial_{\boldsymbol{\tau}}\mathbf{Q}_{\mathrm{bd}}\). For notational convenience, within this proof we avoid writing explicitly the dependence on \(\Omega\),
\({\rm C}_{\rm pot}\), and the \(L^1(\partial\Omega)\)- and the \(L^2(\partial \Omega)\)-norm of \(\mathbf{Q}_{\mathrm{bd}}\times
\partial_{\boldsymbol{\tau}}\mathbf{Q}_{\mathrm{bd}}\) of the constants.
Next, once again by Young’s inequality and 43 , we get \[\label{eq:decay-u2-compu6} \begin{align} 2 \varepsilon\int_\Omega\left( \zeta \nabla
u_2 \right) \cdot \left( u_2 \nabla \zeta \right) \,{\mathrm{d}}x &\leq \int_\Omega \zeta^2 \left( \frac{\varepsilon}{4} \left| \nabla u_2 \right|^2 \right) \,{\mathrm{d}}x + 16 \varepsilon\left\| \nabla \zeta \right\|_{L^\infty(K)}^2 \int_B
u_2^2\,{\mathrm{d}}x \\ &\leq \int_\Omega \zeta^2 \left( \frac{\varepsilon}{4} \left| \nabla u_2 \right|^2 \right) \,{\mathrm{d}}x + 256 C_2 \frac{\varepsilon^2}{R^2} \end{align}\tag{71}\] Finally, by Hölder’s inequality, Lemma 1, and 48 , \[\label{eq:decay-u2-compu6-bis} \begin{align} \int_B \zeta^2 \left(\kappa_\star + \frac{1 -\rho_\varepsilon}{\varepsilon} \right) u_2^2\,{\mathrm{d}}x &\leq \left( \int_B \left( \frac{\rho - 1}{\varepsilon} - \kappa_\star
\right)^2 \zeta^2 \,{\mathrm{d}}x \right)^\frac{1}{2} \left( \int_B \zeta^2 u_2^4 \,{\mathrm{d}}x \right)^\frac{1}{2} \\ &= \mathrm{O}_{\varepsilon\to 0}(\varepsilon) \end{align}\tag{72}\] Combining 67
for \(\eta=1\) with 68 , 69 , 70 , 71 , and 72 , we obtain \[\label{eq:decay-u2-step-1} \begin{align} \int_\Omega &\zeta^2 \left( \frac{\varepsilon}{4} \left| \nabla u_2 \right|^2 +
\frac{1}{\varepsilon}\left( \left| \mathbf{u} \right|^2 - 1 + \sqrt{2}\beta + \sqrt{2}\beta \varepsilon\kappa_\star \right) u_2^2 \right)\,{\mathrm{d}}x \\ &\leq C_\beta(x_0,\,R) \varepsilon+ C_{\beta}(x_0,\,R) \varepsilon^{3/2} + C_\beta(x_0,\,R)
\frac{\varepsilon^{3/2}}{R} + 256 C_2 \frac{\varepsilon^2}{R^2} \\ &\leq C_\beta(x_0,\,R) \varepsilon. \end{align}\tag{73}\] Notice that, up to this point, \(B = B(x_0,\,R)\) can be any ball such that
\(B(x_0,\,2R) \subset \Omega \setminus \mathop{\mathrm{spt}}\mu_\star\) and \(\beta\) can be any positive number (i.e., we have not used \(\sqrt{2}{\beta} >
1\) yet).
Step 4 (\(L^2\)-bounds on \(\Delta \varphi\)). Assume now, in addition to the previous assumptions, that \(\sqrt{2}\beta = 1 + c > 1\). We
have \[\label{eq:decay-u2-compu7} \left( \left| \mathbf{u} \right|^2 - 1 + \sqrt{2}\beta + \sqrt{2}\beta \varepsilon\kappa_\star \right) u_2^2 \geq c u_2^2 \qquad
inB,\tag{74}\] Combined with 73 , 74 yields \[\label{eq:decay-u2-intermediate} \int_B
u_2^2 \,{\mathrm{d}}x \leq C_\beta\left(x_0,\,R \right) \varepsilon^2,\tag{75}\] as well as \[\int_B \left( \varepsilon\left| \nabla u_2 \right|^2 + \frac{1}{\varepsilon} u_2^2 \right)\,{\mathrm{d}}x \leq
C_\beta\left(x_0,\,R \right) \varepsilon.\] Now, we take advantage of 75 to get \(L^2(B)\)-bounds on \(\zeta^2\Delta \varphi\). With such
bounds at hand, we can improve on the estimate of the right-hand side of 68 and, in turn, obtain a better decay for the left-hand side of 73 in the smaller ball \(B'\), see 81 . With 81 at hand, we obtain Lipschitz bounds on \(\varphi\) in \(B'\), and in turn this implies, as we shall in the next step, a still faster decay of \(u_2\) in the ball \(B''\).
To begin with, we recall that \(\varphi\) satisfies the equation \[\label{eq:varphi} -\relax(\rho^2 \nabla \varphi) = \frac{\beta \rho }{\sqrt{2}}\frac{u_1
u_2}{\varepsilon} \qquad inB,\tag{76}\] which is obtained by noticing that, by 6 and 7 , \[\mathbf{Q}\times \Delta \mathbf{Q}= \relax(\mathbf{Q}\times \nabla
\mathbf{Q}) = \frac{\varepsilon}{2}\relax(\mathbf{M}\times \nabla \mathbf{M}) = \frac{\varepsilon}{2} \mathbf{M}\times \Delta \mathbf{M} = - \frac{\beta}{\varepsilon} \mathbf{Q}\mathbf{M}\times \mathbf{M}\] and then that \[\relax\left(\frac{1}{2} \mathbf{Q}\times \nabla \mathbf{Q}\right) = \relax(\rho^2 \nabla \varphi) \qquad and \qquad \mathbf{Q}\mathbf{M}\times \mathbf{M} = \sqrt{2} \rho u_1 u_2 \qquad inB,\] where the first identity follows
from 23 and 24 while the second is an immediate consequence of the very definition 35 of \(\mathbf{u}\).
Let us set \[g_\varepsilon:= \frac{\beta \rho }{\sqrt{2}}\frac{u_1 u_2}{\varepsilon}.\] Thanks to 75 , we see that the sequence \(\{g_\varepsilon\}_\varepsilon\) is bounded in \(L^2(B)\). Now, since \(\varphi\) does not satisfy an homogeneous Dirichlet boundary condition, we cannot
conclude directly, by standard elliptic regularity and the \(W^{1,2}\)-bounds on \(\rho\) far from \(\mathop{\mathrm{spt}}\mu_\star\), that \(\Delta \varphi\) is bounded in \(L^2(B)\). However, looking at 68 , we see that we actually need to bound \(\zeta^2 \Delta
\varphi\), rather than \(\Delta \varphi\).
In order to bound \(\zeta^2 \Delta \varphi\), we observe that the function \(\zeta^2 \varphi\) vanishes on \(\partial B\) and satisfies
\[\label{eq:improved-bound-varphi-compu1} -\relax\left( \rho \nabla\left(\zeta^2 \varphi\right) \right) = g_\varepsilon\zeta^2 - \rho \nabla \varphi \cdot \nabla \zeta^2
- \relax\left(\rho \varphi \nabla \zeta^2\right) \qquad inB.\tag{77}\] Since \(\{g_\varepsilon\}_\varepsilon\) is bounded in \(L^2(B)\) and by 27 and 33 , we see that the right-hand side of 77 is bounded in \(L^2(B)\), uniformly with respect
to \(\varepsilon\) (and, indeed, by a constant depending only on \(x_0\), \(R\), \(\Omega\), \({\rm C}_{\rm pot}\), and the \(L^1(\partial\Omega)\)- and the \(L^2(\partial \Omega)\)-norm of \(\mathbf{Q}_{\mathrm{bd}}\times
\partial_{\boldsymbol{\tau}}\mathbf{Q}_{\mathrm{bd}}\)), which yields \[\left\| \zeta^2 \varphi \right\|_{W^{2,2}(B)} \leq C_\beta(x_0,\,R).\] In particular, \[\label{eq:Delta-varphi} \left\| \zeta^2 \Delta \varphi \right\|_{L^2(B)} \leq C_\beta(x_0,\,R).\tag{78}\]
Step 5 (Lipschitz bounds on \(\varphi\) in \(B'\)). Using 75 and 78 , we obtain
\[\label{eq:decay-u2-compu8} \varepsilon\int_B u_1 u_2 \zeta^2 \Delta \varphi \,{\mathrm{d}}x \leq C_\beta(x_0,\,R) \varepsilon^2.\tag{79}\] Let \(\eta > 0\) be any number. By Hölder’s inequality with \(p_1 = 1/2\), \(p_2 = 2 + \eta\), and \(p_3 = \frac{2(2+\eta)}{\eta}\)
(so that \(1/p_2 + 1/p_3 = 1/2\) and \(1/p_1 + 1/p_2 + 1/p_3 = 1\)), we get \[\label{eq:decay-u2-compu9}
\begin{align} \varepsilon\int_B \zeta^2 u_2 \nabla u_1 \cdot \nabla \varphi \,{\mathrm{d}}x &\leq \varepsilon\left( \int_B \left| u_2 \right|^2 \,{\mathrm{d}}x \right)^{\frac{1}{2}} \left( \int_B \left| \nabla u_1 \right|^{p_2} \,{\mathrm{d}}x
\right)^{\frac{1}{p_2}} \left( \int_B \left| \nabla \varphi \right|^{p_3} \,{\mathrm{d}}x \right)^{\frac{1}{p_3}} \\ &\leq C_\beta(x_0,\,R) C_{\beta,\eta}^{\frac{\eta}{2+\eta}} \,\varepsilon^{1 + \frac{1}{2 +\eta}} \\ &= C_{\beta,\eta}(x_0,\,R) \,
\varepsilon^{1+\frac{1}{2+\eta}}. \end{align}\tag{80}\] Thus, using 75 , 78 , and 79 , we may improve on 73 so as to obtain \[\label{eq:decay-u2-improved} \int_{B'}\left( \varepsilon\left| \nabla u_2 \right|^2 + \frac{1}{\varepsilon}\left( \left|
\mathbf{u} \right|^2 - 1 + \sqrt{2}\beta + \sqrt{2}\beta \varepsilon\kappa_\star \right)u_2^2\right)\,{\mathrm{d}}x \leq C_{\beta,\gamma}(x_0,\,R) \varepsilon^{1+\frac{\gamma}{2}},\tag{81}\] where \(\gamma =
\frac{1}{2+\eta}\). Since \(\eta > 0\) was arbitrary, this implies, in particular, \[\label{eq:decay-u2-improved-bis} \int_{B'}
\left( \frac{u_2}{\varepsilon} \right)^2 \,{\mathrm{d}}x \leq C_{\beta,\gamma}(x_0,\,R) \varepsilon^{\frac{\gamma}{2}}, \qquad \forall \gamma \in (0,\,1)\tag{82}\] In turn, 82 implies that
\[\label{eq:decay-L2-norm-geps} \left\| g_\varepsilon \right\|_{L^2(B')} \leq C_{\beta,\gamma}(x_0,\,R) \varepsilon^\frac{\gamma}{4}, \qquad \forall \gamma \in
(0,\,1)\tag{83}\] Plugged in 76 and along with 77 , 82 allows to conclude that
\[\label{eq:W2243est-varphi} \left\| \zeta^2 \varphi \right\|_{W^{2,2+\frac{\gamma}{2}}(B')} \leq C_{\beta,\gamma}(x_0,\,R), \qquad \forall \gamma \in
(0,\,1),\tag{84}\] so that, by Sobolev embedding, \[\label{eq:Lip-bound-varphi} \left\| \nabla \varphi \right\|_{L^\infty(B')} \leq
C_\beta(x_0,\,R).\tag{85}\] Notice that the constant on the right-hand side of 85 does not depend on \(\gamma\) because, since 84
holds for any \(\gamma \in (0,\,1)\), we can take the constant on the right-hand side of 85 to be the infimum over the constants \(C_{\beta,\gamma}(x_0,\,R)\) obtained by combining 84 and Sobolev embedding.
Step 6 (Improved decay of \(u_2\) in \(B''\)). In order to take advantage of 85 , we test 63 against
\(\varepsilon\psi^2 u_2\), where \(\psi \in C^\infty(\mathbb{R}^2)\) is any cut-off function such that \[\begin{gather} 0 \leq \psi \leq 1, \qquad \psi \equiv 1
\quad inB'', \qquad \mathop{\mathrm{spt}}\psi \subset B', \\ \left| \nabla \psi \right| \leq 4\left(\frac{2}{R}\right), \qquad \left| \nabla^2 \psi \right| \leq 4\left(\frac{4}{R^2}\right) \qquad in\mathbb{R}^2.
\end{gather}\] Then, upon using 85 and 75 , we obtain the improved estimate \[\label{eq:decay-u2-compu10} \begin{align} \varepsilon\int_{B'} \psi^2 u_2 \nabla u_1 \cdot \nabla \varphi \,{\mathrm{d}}x &\leq C_\beta(x_0,\,R)\,\sqrt{\varepsilon}\left( \int_B u_2^2 \,{\mathrm{d}}x \right)^{\frac{1}{2}}
\left( \varepsilon\int_B \left| \nabla u_1 \right|^2 \,{\mathrm{d}}x\right)^{\frac{1}{2}} \\ &\leq C_\beta(x_0,\,R) \varepsilon^{3/2}. \end{align}\tag{86}\] Notice that the integral on the left-hand side is over the smaller ball \(B'\) rather than over the original ball \(B\). In turn, since \(\psi \equiv 1\) in \(B''\), we obtain the improved
decay \[\label{eq:decay-u2-improved-ter} \int_{B''} \left( \varepsilon\left| \nabla u_2 \right|^2 + \frac{1}{\varepsilon}\left( \left| \mathbf{u} \right|^2 - 1 +
\sqrt{2}\beta \right) u_2^2 \right)\,{\mathrm{d}}x \leq C_\beta(x_0,\,R) \, \varepsilon^\frac{3}{2}.\tag{87}\]
Remark 1. Instead of 81 , we could employ 82 in the estimate 86 . This would give a factor \(\varepsilon^\frac{3+\gamma}{2}\) on the right-hand side of 87 , rather than \(\varepsilon^\frac{3}{2}\), making the decay of \(u_2\) slightly faster. However, as can be easily checked, since \(\gamma \in (0,\,1)\), this additional speed of convergence is not enough, according to the iteration argument presented in
Step 7 below, to yield, after iteration, a further improved decay of \(u_2\) with respect to the one achieved by
starting from 87 . For this reason, we preferred to keep things as simple as possible, avoiding the use of 82 and so the introduction of a further dependence on \(\gamma\) in 87 .
Step 7 (Iteration). Recall once again that we assume \(\sqrt{2}\beta = 1 + c > 1\). Under this assumption, 87 implies that
\[\label{eq:decay-u2-iteration-1} \int_{B_{1/4}} u_2^2\,{\mathrm{d}}x \leq C_\beta(x_0,\,R) \varepsilon^\frac{5}{2}.\tag{88}\] With 88 at hand, we repeat the argument from Step 3 to Step 6, but with \[B = B_{1/4}, \qquad B' = B_{1/8}, \qquad B'' = B_{1/16}.\] Correspondingly, we take a new cut-off function \(\zeta \in
C^\infty_c(\mathbb{R}^2)\) still as in 64 , but with 65 replaced by \[\left| \nabla \zeta \right| \leq \frac{16}{R}, \qquad \left| \nabla^2 \zeta \right| \leq
\frac{32}{R^2}, \qquad in\mathbb{R}^2.\] In general, at the \(n\)-th iteration step, we will take \[B = B_{2^{-2n}}, \qquad B' = B_{2^{-2n+1}}, \qquad B'' = B_{2^{-2n-2}}
.\] and \(\zeta = \zeta_n\) such that 64 holds, as well as \[\label{eq:cut-off-grad-n} \left| \nabla \zeta \right|
\leq 2^n\left(\frac{2}{R}\right), \qquad \left| \nabla^2 \zeta \right| \leq 2^n\left(\frac{4}{R^2}\right), \qquad in\mathbb{R}^2.\tag{89}\] At the first iteration, we obtain that 67 , 71 , 79 , and 80 can be replaced, respectively, as follows. First, by 87 and 85 , \[\label{eq:decay-u2-compu11} \varepsilon\int_B \zeta^2 u_2^2 \left| \nabla \varphi \right|^2 \,{\mathrm{d}}x \lesssim
\varepsilon^\frac{7}{2},\tag{90}\] By Hölder’s inequality, 78 , 83 , and 38 , we have \[\label{eq:decay-u2-compu12} \varepsilon\int_B u_1 u_2 \zeta^2 \Delta \varphi\,{\mathrm{d}}x \lesssim \varepsilon^2.\tag{91}\] By Hölder’s inequality, 88 , and 87 , we have \[\label{eq:decay-u2-compu13} \varepsilon\int_B (u_2 \nabla \zeta)\cdot (\zeta \nabla u_2) \,{\mathrm{d}}x \lesssim
\varepsilon^\frac{5}{2}.\tag{92}\] Finally, Hölder’s inequality, 85 , and 88 yield \[\label{eq:decay-u2-weakest-improv} \varepsilon\int_B \zeta^2 u_2 \nabla u_1\cdot \nabla\varphi\,{\mathrm{d}}x \lesssim \varepsilon^\frac{7}{4}.\tag{93}\] In the above inequalities, the implicit constant on the
right-hand side depends only on \(x_0\), \(R\), \(\beta\), \(\Omega\), \({\rm C}_{\rm
pot}\), and the \(L^1(\partial \Omega)\)- and the \(L^2(\partial \Omega)\)-norms of \(\mathbf{Q}_{\mathrm{bd}}\times
\partial_{\boldsymbol{\tau}}\mathbf{Q}_{\mathrm{bd}}\).
Since the decay on the right-hand side of 93 is the slowest (both at this step and at any other step), it provides the leading order in the improvement of the decay of the left-hand side of 87 . More precisely, combining 90 , 91 , 92 , and 93
with 66 , we see that, after the first iteration step, we obtain \[\label{eq:decay-u2-improved-4} \int_{B_{1/16}} \left( \varepsilon\left|
\nabla u_2 \right|^2 + \frac{1}{\varepsilon}\left( \left| \mathbf{u} \right|^2 - 1 + \sqrt{2}\beta \right) u_2^2 \right)\,{\mathrm{d}}x \leq C_\beta(x_0,\,R) \, \varepsilon^\frac{7}{4},\tag{94}\] as well as \[\int_{B_{1/16}} \left( \varepsilon\left| \nabla u_2 \right|^2 + \frac{1}{\varepsilon} u_2^2 \right)\,{\mathrm{d}}x \leq C'_\beta(x_0,\,R) \, \varepsilon^\frac{7}{4}.\] By a trivial induction argument, we see that at the
\(n\)-th step we have \[\label{eq:decay-u2-improved-n} \int_{B_{2^{-2n-2}}} \left( \varepsilon\left| \nabla u_2 \right|^2 +
\frac{1}{\varepsilon}\left( \left| \mathbf{u} \right|^2 - 1 + \sqrt{2}\beta \right) u_2^2 \right)\,{\mathrm{d}}x \leq C_{\beta,n}(x_0,\,R) \, \varepsilon^{\alpha_n},\tag{95}\] where the dependence on \(n\)
arises because of the dependence on \(n\) in 89 , and \[\label{eq:decay-u2-improved-n-bis}
\int_{B_{2^{-2n-2}}} \left( \varepsilon\left| \nabla u_2 \right|^2 + \frac{1}{\varepsilon} u_2^2 \right)\,{\mathrm{d}}x \leq C'_{\beta,n}(x_0,\,R) \, \varepsilon^{\alpha_n}.\tag{96}\] The exponent \(\alpha_n\) is given by the iterative formula \[\alpha_{n+1} = \frac{\alpha_n + 2}{2}, \qquad \alpha_0 = 1,\] i.e., \[\label{eq:alpha95n} \alpha_n := 2-2^{-(n+1)}.\tag{97}\]
Step 8 (Covering argument and conclusion). Let \(K \subset \Omega \setminus \mathop{\mathrm{spt}}\mu_\star\) be a compact set. Choose \(\alpha \in (0,\,3)\) arbitrarily and let
\(G \subset\!\subset K\) be any simply connected, open set with smooth boundary. Let \(n_\alpha \in \mathbb{N}\) be the smallest index such that \(\alpha \leq
\alpha_{n_\alpha}\), where \(\alpha_{n_\alpha}\) is given by 97 for \(n = n_\alpha\). Then, by 95 , we have
\[\int_{B_{2^{-2{n_\alpha}-2}}} \left( \varepsilon\left| \nabla u_2 \right|^2 + \frac{1}{\varepsilon} u_2^2 \right)\,{\mathrm{d}}x \leq C'_{\beta,n_\alpha}(x_0,\,R) \, \varepsilon^{\alpha}.\] Moreover, we can cover
\(G\) with finitely many balls of radius \(2^{{-2 n_\alpha} -2}R\), for \(R > 0\) sufficiently small (depending only on \(n_\alpha\) and on \(\mathop{\mathrm{dist}}(G,\,K)\)) that the corresponding concentric balls of radius \(2^{-2n_\alpha-1}R\) are still contained in \(K\). As a consequence, we obtain that ?? and ?? hold, for constants \(C_\alpha(\mathop{\mathrm{dist}}(G,\,K))\), \(C_\alpha'(\mathop{\mathrm{dist}}(G,\,K))\)
depending only on \(\mathop{\mathrm{dist}}(G, \partial K)\), \(\alpha\), \(\beta\), \(\Omega\), \({\rm C}_{\rm pot}\), and the \(L^1(\partial\Omega)\)- and the \(L^2(\partial \Omega)\)-norm of \(\mathbf{Q}_{\mathrm{bd}}\times
\partial_{\boldsymbol{\tau}}\mathbf{Q}_{\mathrm{bd}}\).
◻
Remark 2. As it can be checked immediately by inspection of the proof, due to the local uniform convergence \(\left| \mathbf{u} \right|^2 \to 1 + \sqrt{2}\beta\) in \(\Omega \setminus
(\mathop{\mathrm{spt}}\mu_\star \cup \mathop{\mathrm{spt}}\nu_\star)\) (which is an obvious consequence of item [item:unif-conv-M] of Theorem 1), Proposition 3 holds without restrictions on \(\beta\) if
we assume that \(K \subset \Omega \setminus (\mathop{\mathrm{spt}}\mu_\star \cup \mathop{\mathrm{spt}}\nu_\star)\).
Corollary 1. Let \(\{(\mathbf{Q}_\varepsilon,\,\mathbf{M}_\varepsilon)\}_\varepsilon\) be a sequence of critical points of \(\mathscr{F}_\varepsilon\) satisfying boundary
conditions either as in 3 or as in 4 , as well as assumption 5 . Assume, in addition, that \(\sqrt{2}\beta > 1\). Let \(K \subset \Omega \setminus \mathop{\mathrm{spt}}\mu_\star\) be a compact set. and let \(G \subset\!\subset K\) be any simply connected, open set with smooth boundary. Let \(\{\mathbf{u}_\varepsilon\}_\varepsilon\) be the sequence of maps defined in \(G\) as prescribed by 35 . Then, \[\label{eq:unif-decay-u2} \left\| u_{2,\varepsilon} \right\|_{L^\infty(G)} \lesssim \varepsilon^{\gamma/2}\tag{98}\] for every \(\gamma \in (0,\,1)\). In particular, as \(\varepsilon\to 0\), \[\label{eq:unif-conv-u2} u_{2,\varepsilon} \to 0 \quad uniformly inK.\tag{99}\]
Proof. Since \(G\) has smooth boundary, it follows from Morrey’s inequality that \[\left\| u_{2,\varepsilon} \right\|_{L^\infty(G)} \lesssim \left\| u_{2,\varepsilon}
\right\|_{W^{1,2}(G)}.\] Then, 98 follows immediately from ?? . Now, for any compact set \(K \subset \Omega \setminus \mathop{\mathrm{spt}}\mu_\star\), we can find another compact
\(K'\) and a simply connected, open set \(G\) with smooth boundary such that \(K \subset G \subset\!\subset K'\). Then, 98 holds in \(G\) and implies \(\left\| u_{2,\varepsilon} \right\|_{L^\infty(K)} \lesssim \varepsilon^{\gamma}\) for any \(\gamma
\in (0,\,1)\), so that 99 follows immediately by taking the limit as \(\varepsilon\to 0\). ◻
Remark 3. Besides specifying the rate of convergence, the point of Corollary 1 is that the component \(\{u_{2,\varepsilon}\}\) converges uniformly to zero locally everywhere away from \(\mathop{\mathrm{spt}}\mu_\star\), and so even across the energy concentration set for the \(\mathbf{M}\)-component, i.e., even on \(\mathop{\mathrm{spt}}\nu_\star\).
We now combine Proposition 3 and Corollary 1 so as to obtain an improved
decay for \(u_{2,\varepsilon}^4\), which will be useful in Section 3.
Corollary 2. Let \(\{(\mathbf{Q}_\varepsilon,\,\mathbf{M}_\varepsilon)\}_\varepsilon\) be a sequence of critical points of \(\mathscr{F}_\varepsilon\) satisfying boundary
conditions either as in 3 or as in 4 , as well as assumption 5 . Assume, in addition, that \(\sqrt{2}\beta > 1\). Let \(K \subset \Omega \setminus \mathop{\mathrm{spt}}\mu_\star\) be any compact set and let \(G \subset\!\subset K\) be any simply connected, open set, with smooth boundary \(\partial G\). Let \(\{\mathbf{u}_\varepsilon\}_\varepsilon\) be the sequence of maps defined in \(G\) as prescribed by 35 . Then, for any
\(\gamma \in (0,\,1)\) and any \(\varepsilon\) small enough, depending only on \(K\) and \(\beta\), we have
\[\label{eq:decay-u2-goal3} \int_G u_{2,\varepsilon}^4 \,{\mathrm{d}}x \leq C_\gamma(\mathop{\mathrm{dist}}(G,\,\partial K)) \varepsilon^{4\gamma}.\tag{100}\] where
\(C_\gamma(\mathop{\mathrm{dist}}(G,\,\partial K))\) depends only on \(\mathop{\mathrm{dist}}(G, \partial K)\), \(\gamma\), \(\beta\), \(\Omega\), \({\rm C}_{\rm pot}\), and the \(L^1(\partial\Omega)\)- and the \(L^2(\partial\Omega)\)-norms of \(\mathbf{Q}_{\mathrm{bd}}\).
Proof. Let \(K\) and \(G\) be as in the statement. Then, by Corollary 1, we have \[u_{2,\varepsilon}^2 \lesssim \varepsilon^\gamma, \qquad \forall \gamma \in (0,\,1).\] Thus, \[\int_G u_{2,\varepsilon}^4 \,{\mathrm{d}}x \lesssim \varepsilon^\gamma \int_G u_{2,\varepsilon}^2
\,{\mathrm{d}}x,\] and the conclusion follows from ?? . ◻
Before going to the details of the proof, we recall that, if \(\sqrt{2}\beta > 1\), then the function \(h_2\) in 46 is non-negative, for every value of \(u_1\), \(u_2\). Thus, if \(B = B(x_0,\,R) \subset\!\subset\Omega \setminus \mathop{\mathrm{spt}}\mu_\star\) is any ball and \(\{
(\mathbf{Q}_\varepsilon,\,\mathbf{M}_\varepsilon) \}_\varepsilon\) is any sequence of critical points of \(\mathscr{F}_\varepsilon\), then, by Proposition 3 and 38 , we have \[\label{eq:decay-h2} \frac{1}{\varepsilon}\int_B h_2(u_{1_\varepsilon}, u_{2,\varepsilon}) \,{\mathrm{d}}x
= {\rm O}_{\varepsilon\to 0}(\varepsilon^{\alpha}) \qquad for all\alpha \in (0,\,2).\tag{101}\] In analogy with the definition 14 , and taking 101 into account, we set \[\zeta_\varepsilon^{(h)} := \frac{1}{\varepsilon} h(u_{1,\varepsilon}), \qquad \zeta_\varepsilon^{(h_1)} := \frac{1}{\varepsilon} h_1(u_{1,\varepsilon}).\] By 42 , it then follows that \[\zeta_\varepsilon^{(h)} \rightharpoonup^* \zeta_\star^{(h)}, \qquad \zeta_\varepsilon^{(h_1)} \rightharpoonup^* \zeta_\star^{(h_1)} \qquad as\varepsilon\to 0.\] On the other hand, by 101 , we have
\[\int_K \left| \zeta_\varepsilon^{(h)} - \zeta_\varepsilon^{(h_1)} \right| \,{\mathrm{d}}x \to 0 \qquad as\varepsilon\to 0,\] for any compact set \(K \subset \Omega \setminus
\mathop{\mathrm{spt}}\mu_\star\), and thus the limiting varifold associated with the vectorial potential potential \(h\) is integral if and only if the one associated with the scalar potential \(h_1\) is. On the other hand, we already know that \(\zeta_\star^{(h)} = \zeta_\star\) in \(\Omega \setminus \mathop{\mathrm{spt}}\mu_\star\). In the proof of
Theorem 2 below, we prove that the density \(\mathfrak{h}_\star\) of \(\zeta_\star^{(h_1)}\)
exists at every point \(x_0\) in \(\Omega \setminus \mathop{\mathrm{spt}}\mu_\star\) and that \(\sigma_\beta^{-1} \mathfrak{h}_\star(x_0)\) is an integer.
Then, we conclude that \(\sigma_\beta^{-1} \mathfrak{v}_\star(x_0)\) takes only integer values by showing that the difference \(\zeta_\varepsilon- \zeta_\varepsilon^{(h_1)}\) vanishes in
\(L^1(K)\) as \(\varepsilon\to 0\), for every \(K \subset \Omega \setminus \mathop{\mathrm{spt}}\mu_\star\), and using that \(\zeta_\star(\mathop{\mathrm{spt}}\mu_\star) = 0\). (This last property follows from the monotonicity of \(\zeta_\star\) — see [2].)
Proof of Theorem 2. The equation satisfied by the component \(u_{1,\varepsilon}\) is obtained by projecting 50 along the direction \(\mathbf{e}_1 := (1,\,0)^{\rm t}\) and reads as follows: \[\label{eq:EL-u1} \begin{align} -\varepsilon\Delta
u_{1,\varepsilon} &+ \frac{1}{\varepsilon}\left(u_{1,\varepsilon}^2 - 1 - \sqrt{2}\beta \rho_\varepsilon\right)u_{1,\varepsilon} \\ &= \varepsilon\left( -u_{1,\varepsilon} \left| \nabla \varphi_\varepsilon \right|^2 - \nabla u_{2,\varepsilon} \cdot
\nabla \varphi_\varepsilon- u_{2,\varepsilon} \Delta \varphi_\varepsilon\right) - \frac{1}{\varepsilon}u_{2,\varepsilon}^2 u_{1,\varepsilon}. \end{align}\tag{102}\] This equation holds pointwise in any simply connected, open set \(G \subset\!\subset\Omega \setminus \mathop{\mathrm{spt}}\mu_\star\) with smooth boundary \(\partial G\). Since \(\sigma_\beta\) is a constant which depends only
on \(\beta\), the integrality claim in the statement is local in nature, and therefore we can localise the argument. More precisely, the strategy goes as follows. First, working in balls \(B =
B(x_0,\,R) \subset\!\subset\Omega \setminus \mathop{\mathrm{spt}}\mu_\star\) and relying on the results in Section 2, we recast 102 in the form considered [34], see 103 below, obtaining at the same time the required decay estimates for the perturbation term on the right-hand side. This
allows us to apply the machinery in [34] in any compact set \(K \subset \Omega \setminus
\mathop{\mathrm{spt}}\mu_\star\). Then, from [34] we infer that \(\mathfrak{v}_\star(x_0)\) is an
integer multiple of \(\sigma_\beta\) for any \(x_0 \in K\), so that the conclusion follows by letting \(K\) vary in \(\Omega
\setminus \mathop{\mathrm{spt}}\mu_\star\) and recalling that \(\mathfrak{v}_\star = 0\) on \(\mathop{\mathrm{spt}}\mu_\star\).
Step 9 (Rewriting 102 ). The aim of this step is to rewrite 102 in the form \[\label{eq:EL-u1-bis} -\varepsilon\Delta
\widetilde{u}_\varepsilon+ \frac{1}{\varepsilon}\left( \widetilde{u}_\varepsilon^2 - 1 - \sqrt{2}\beta \right)\widetilde{u}_\varepsilon = F_\varepsilon\left(\widetilde{u}_\varepsilon,\,u_{2,\varepsilon}\right),\tag{103}\] where
\[\label{eq:widetilde-u} \widetilde{u}_\varepsilon:= \lambda_\varepsilon u_{1,\varepsilon}, \qquad \lambda_\varepsilon:= \left(\frac{1 + \sqrt{2}\beta}{1+\sqrt{2\beta} +
\varepsilon\sqrt{2}\beta \kappa_\star}\right)^{1/2} ,\tag{104}\] and \(F_\varepsilon\left( \widetilde{u}_\varepsilon,\,u_{2,\varepsilon} \right)\), defined in 114 below, satisfies
the local estimate \[\label{eq:decay-capitolF} \frac{1}{\varepsilon} \int_K F_\varepsilon^2\,{\mathrm{d}}x \leq C_{\beta,K},\tag{105}\] in any compact set \(K \subset \Omega \setminus \mathop{\mathrm{spt}}\mu_\star\), where \(C_{\beta,K}\) is a constant depending only on \(\beta\), \(K\), \(\Omega\), \({\rm C}_{\rm pot}\), and the \(L^1(\partial \Omega)\)- and the \(L^2(\partial
\Omega)\)-norm of \(\mathbf{Q}_{\mathrm{bd}}\times \partial_{\boldsymbol{\tau}}\mathbf{Q}_{\mathrm{bd}}\).
Towards the announced purpose, we start by setting (for ease of notation, we drop the subscripts \(\varepsilon\) in the intermediate formulae below, writing \(u_1\), \(u_2\), \(\varphi\) in place of \(u_{1,\varepsilon}\), \(u_{2,\varepsilon}\), \(\varphi_\varepsilon\)) \[\begin{align} g_\varepsilon(u_1,\,u_2) &:= -\varepsilon\left(u_1 \left| \nabla \varphi \right|^2 + \nabla u_2 \cdot \nabla \varphi + u_2 \Delta \varphi\right), \\
f_\varepsilon&:= g_\varepsilon- \frac{u_2^2 u_1}{\varepsilon},
\end{align}\] which leads to \[\label{eq:u1-pert} -\varepsilon\Delta u_1 + \frac{1}{\varepsilon}\left(u_1^2 - 1 - \sqrt{2}\beta \rho \right)u_1 =
f_\varepsilon(u_1,\,u_2).\tag{106}\] We claim that, for any ball \(B = B(x_0,\,R) \subset\!\subset\Omega \setminus \mathop{\mathrm{spt}}\mu_\star\), \[\label{eq:decay-f} \frac{1}{\varepsilon}\int_B f_\varepsilon^2\,{\mathrm{d}}x \leq C_\beta(x_0,\,R) \varepsilon^\gamma, \qquad \forall \gamma \in (0,\,1),\tag{107}\] for a constant \(C_\beta(x_0,\,R)\) depending only \(\beta\), \(x_0\), \(R\), \(\Omega\), \({\rm C}_{\rm pot}\), and the \(L^1(\partial \Omega)\)- and the \(L^2(\partial \Omega)\)-norm of \(\mathbf{Q}_{\mathrm{bd}}\times
\partial_{\boldsymbol{\tau}}\mathbf{Q}_{\mathrm{bd}}\).
In order to prove the claim, we first observe that it follows from 100 and 38 that \[\label{eq:decay-u2944}
\frac{1}{\varepsilon}\int_B \frac{u_2^4 u_1^2}{\varepsilon^2}\,{\mathrm{d}}x \leq C_\beta(x_0,\,R) \varepsilon^\gamma, \qquad \forall \gamma \in (0,\,1),\tag{108}\] for every \(\varepsilon\) small enough,
depending on \(K\) and \(\beta\) only. In order to bound \(g_\varepsilon\), we observe that by an argument similar to the one in Step 4, Step 5, and Step 7 in the proof of Proposition 3, we may assume that \[\begin{gather}
\left\| \nabla \varphi \right\|_{L^\infty(B)} \leq C_\beta(x_0,\,R), \qquad \left\| \Delta \varphi \right\|_{L^2(B)} \leq C_\beta(x_0,\,R), \\ \left\| \nabla u_2 \right\|_{L^2(B)} \leq C_{\beta,\gamma}(x_0,\,R) \varepsilon^{\gamma}, \qquad \forall \gamma
\in (0,\,1),
\end{gather}\] for constants \(C_\beta(x_0,\,R)\), \(C_{\beta,\gamma}(x_0,\,R)\) depending only \(\beta\), \(\gamma\), \(x_0\), \(R\), \(\Omega\), \({\rm C}_{\rm pot}\), and the \(L^1(\partial \Omega)\)- and the \(L^2(\partial \Omega)\)-norm of \(\mathbf{Q}_{\mathrm{bd}}\times \partial_{\boldsymbol{\tau}}\mathbf{Q}_{\mathrm{bd}}\). From
these bounds, it follows that \[\label{eq:decay-geps} \left\| g_\varepsilon \right\|_{L^2(B)} \leq C_\beta(x_0,\,R)\,\varepsilon.\tag{109}\] From 108 and 109 , 107 follows.
Now, we take care of the fact that, as written, the potential term on the right-hand side of 106 depends on \(\rho\) (and thus implicitly on both \(\varepsilon\) and \(x\)). To this purpose, we further rewrite 106 as follows: \[\label{eq:u1-pert-bis} -\varepsilon\Delta u_1 + \frac{1}{\varepsilon}\left(u_1^2 - 1 - \sqrt{2}\beta - \varepsilon\sqrt{2}\beta \kappa_\star \right)u_1 = \sqrt{2}\beta \left( \frac{\rho-1}{\varepsilon} - \kappa_\star \right) u_1 +
f_\varepsilon(u_1,\,u_2).\tag{110}\] Notice that, by Lemma 1, \[\label{eq:decay-rho-pot} \frac{1}{\varepsilon} \int_B \left( \frac{\rho-1}{\varepsilon} - \kappa_\star \right)^2 u_1^2 \,{\mathrm{d}}x \lesssim \frac{1}{\varepsilon} \int_B \left( \frac{\rho-1}{\varepsilon} - \kappa_\star \right)^2
\,{\mathrm{d}}x \leq C_\beta(x_0,\,R).\tag{111}\] Next, as anticipated in 104 , we set \[\widetilde{u}_\varepsilon:= \lambda_\varepsilon u_{1,\varepsilon}, \qquad
\lambda_\varepsilon:= \left(\frac{1 + \sqrt{2}\beta}{1+\sqrt{2\beta} + \varepsilon\sqrt{2}\beta \kappa_\star}\right)^{1/2}.\] Rewriting 110 in terms of \(\widetilde{u} =
\widetilde{u}_\varepsilon\), we obtain \[\label{eq:u1-pert-ter} \begin{align} -\varepsilon\Delta \widetilde{u} + \frac{1}{\varepsilon}\left(\widetilde{u}^2 - 1 - \sqrt{2}\beta
\right)\widetilde{u} = &\left( 1- \frac{1}{\lambda_\varepsilon^2} \right)\frac{1}{\varepsilon}\left(\widetilde{u}^2 - 1 - \sqrt{2}\beta \right)\widetilde{u} \\ &+ \sqrt{2}\beta \lambda_\varepsilon\left( \frac{\rho-1}{\varepsilon} - \kappa_\star
\right) \widetilde{u} \\ &+ \lambda_\varepsilon^2 f_\varepsilon(u_1,\,u_2). \end{align}\tag{112}\] By a straightforward computation, we have \[\frac{1}{\varepsilon} \left( 1- \frac{1}{\lambda_\varepsilon^2}
\right)^2 \int_B \frac{1}{\varepsilon^2}\left(\widetilde{u}^2 - 1 - \sqrt{2}\beta \right)^2\widetilde{u}^2 \,{\mathrm{d}}x = \frac{\beta^2}{8} \int_B \frac{1}{\varepsilon}\left(\widetilde{u}^2 - 1 - \sqrt{2}\beta \right)^2\widetilde{u}^2 \,{\mathrm{d}}x
.\] Recalling that \(\left| \mathbf{u} \right|^2\) is uniformly bounded depending only on \(\beta\), that \(\lambda_\varepsilon\to 1\) as \(\varepsilon\to 0\), and that \(\kappa_\star = \frac{\beta}{2\sqrt{2}}\left( 1 + \sqrt{2}\beta \right)\), we see that \[\label{eq:decay-widetilde-u} \frac{1}{\varepsilon} \left( 1- \frac{1}{\lambda_\varepsilon^2} \right)^2 \int_B \frac{1}{\varepsilon^2}\left(\widetilde{u}^2 - 1 - \sqrt{2}\beta \right)^2\widetilde{u}^2 \,{\mathrm{d}}x \leq C(\beta)
\int_B \frac{1}{\varepsilon}h(\mathbf{u})\,{\mathrm{d}}x \stackrel{\eqref{eq:h-bis}}{\leq} C_\beta(x_0,\,R),\tag{113}\] where the constant \(C(\beta)\) depends only on \(\beta\) and \(C_\beta(x_0,\,R)\) depends only on \(\beta\), \(\gamma\), \(x_0\), \(R\), \(\Omega\), \({\rm C}_{\rm pot}\), and the \(L^1(\partial \Omega)\)- and the \(L^2(\partial
\Omega)\)-norm of \(\mathbf{Q}_{\mathrm{bd}}\times \partial_{\boldsymbol{\tau}}\mathbf{Q}_{\mathrm{bd}}\). Upon setting \[\label{eq:capitolF}
F_\varepsilon\left( \widetilde{u},\,u_2\right) := \frac{\beta}{2\sqrt{2}}\left(\widetilde{u}^2 - 1 - \sqrt{2}\beta \right)\widetilde{u} + \sqrt{2}\beta\lambda_\varepsilon\left( \frac{\rho-1}{\varepsilon} - \kappa_\star \right) \widetilde{u} +
f_\varepsilon(u_1,\,u_2).\tag{114}\] from 107 , 111 , and 113 we see that \[\label{eq:decay-capitolF-local} \frac{1}{\varepsilon} \int_B F_\varepsilon^2\,{\mathrm{d}}x \leq C_\beta(x_0,\,R),\tag{115}\] where the constant \(C_\beta(x_0,\,R)\) depends only on
\(\beta\), \(\gamma\), \(x_0\), \(R\), \(\Omega\), \({\rm C}_{\rm
pot}\), and the \(L^1(\partial \Omega)\)- and the \(L^2(\partial \Omega)\)-norm of \(\mathbf{Q}_{\mathrm{bd}}\times
\partial_{\boldsymbol{\tau}}\mathbf{Q}_{\mathrm{bd}}\). Finally, let \(K \subset \Omega \setminus \mathop{\mathrm{spt}}\mu_\star\) be any compact set. Then, by 115 and a
standard covering argument, we have \[\frac{1}{\varepsilon}\int_K F_\varepsilon^2\,{\mathrm{d}}x \leq C_{\beta,K}\] where the constant \(C_{\beta,K}\), depends only on \(\beta\), \(K\), \(\Omega\), \({\rm C}_{\rm pot}\), and the \(L^1(\partial \Omega)\)- and the
\(L^2(\partial \Omega)\)-norm of \(\mathbf{Q}_{\mathrm{bd}}\times \partial_{\boldsymbol{\tau}}\mathbf{Q}_{\mathrm{bd}}\). This proves 105 . Combining 112 and 114 , we obtain 103 . This concludes the proof of the claim.
Step 10 (Inferring integrality from [34], locally in \(\Omega \setminus
\mathop{\mathrm{spt}}\mu_\star\)). Let \(K \subset \Omega \setminus \mathop{\mathrm{spt}}\mu_\star\) be any compact set. We observe that Equation 103 has the same form as [34] with \(F_\varepsilon\) that, as a function of \(x \in K\), satisfies [34]. By the definition of \(\widetilde{u}_{1,\varepsilon}\), the energy bound [34] is satisfied because of 42 and 120 below. We consider the energy densities
(since at this point the component \(u_2\) has been ruled out, we further simplify the notation writing \(u\) in place of \(u_1\)) \[\widetilde{\eta}_\varepsilon:= \frac{\varepsilon\left| \nabla \widetilde{u}_{\varepsilon} \right|^2}{2} + \frac{\left(\widetilde{u}_\varepsilon^2-1-\sqrt{2}\beta\right)^2}{4 \varepsilon},\] as well as \[\eta_\varepsilon:= \frac{\varepsilon\left| \nabla u_{\varepsilon} \right|^2}{2} + \frac{\left(u_\varepsilon^2-1-\sqrt{2}\beta\right)^2}{4 \varepsilon}.\] In view of the definition 104 of \(\widetilde{u}_\varepsilon\), for every \(\varepsilon> 0\), we have \[\label{eq:tilde-zeta-h1-eps-zeta-h1-eps} \begin{align} \widetilde{\zeta}^{(h_1)}_\varepsilon- \zeta^{h_1}_\varepsilon&= \frac{1}{\varepsilon}( h_1(\widetilde{u}_\varepsilon) - h_1(u_\varepsilon) ) \\ &= \frac{1}{4\varepsilon}
\left\{ \left(\lambda_\varepsilon^2 u^2_\varepsilon- 1 - \sqrt{2}\beta \right)^2 - \left(u^2_\varepsilon- 1 - \sqrt{2}\beta \right)^2 \right\} \\ &= \frac{u_\varepsilon^2}{4\varepsilon}\left\{ \left(\lambda_\varepsilon^2 - 1\right)^2 u_\varepsilon^2 +
2\left(1-\lambda_\varepsilon^2\right)^2 \left(u_\varepsilon^2-1-\sqrt{2}\beta\right) \right\}, \end{align}\tag{116}\] so that, by the definition of \(\lambda_\varepsilon\) in 104 and 42 , \[\label{eq:tilde-zeta-h1-eps-zeta-h1-eps-limit} \int_K \left|
\widetilde{\zeta}^{(h_1)}_\varepsilon- \zeta^{(h_1)}_\varepsilon \right|\,{\mathrm{d}}x \to 0, \qquad as\varepsilon\to 0.\tag{117}\] On the other hand, \[\label{eq:tilde-nabla-u-nabla-u} \left| \nabla \widetilde{u}_\varepsilon \right|^2 - \left| \nabla u_\varepsilon \right|^2 = \left( \lambda_\varepsilon^2 - 1 \right) \left| \nabla u_\varepsilon \right|^2,\tag{118}\]
whence \[\label{eq:tilde-nabla-u-nabla-u-limit} \int_K \left( \left| \nabla \widetilde{u}_\varepsilon \right|^2 - \left| \nabla u_\varepsilon
\right|^2\right)\,{\mathrm{d}}x \to 0, \qquad as\varepsilon\to 0.\tag{119}\] From 117 and 119 it follows that
\[\label{eq:tildeetaeps-etaeps-pointwise} \widetilde{\eta}_\varepsilon- \eta_\varepsilon = \varepsilon\lambda\left(\lambda_\varepsilon^2-1\right)\left| \nabla
u_\varepsilon \right|^2 - \frac{u^2_\varepsilon}{4\varepsilon}\left\{ \left(\lambda_\varepsilon^2 - 1\right)^2 u_\varepsilon^2 + 2\left(1-\lambda_\varepsilon^2\right)^2 \left(u_\varepsilon^2-1-\sqrt{2}\beta\right) \right\},\tag{120}\] and
thus \[\label{eq:tildeetaeps-etaeps} \int_K \left| \widetilde{\eta}_\varepsilon- \eta_\varepsilon \right| \,{\mathrm{d}}x \to 0, \qquad as\varepsilon\to
0.\tag{121}\] Since the sequences of positive functions \(\left( \widetilde{\eta}_\varepsilon\right)_{\varepsilon> 0}\), \(\left( \eta_\varepsilon\right)_{\varepsilon>
0}\) are bounded in \(L^1(K)\), for every \(K \subset \Omega \setminus \mathop{\mathrm{spt}}\mu_\star\), by standard compactness properties of Radon measures, they converge in the
sense of Radon measures along a subsequence to limiting measures \(\widetilde{\eta}_\star\), \(\eta_\star\) defined on the whole \(\Omega \setminus
\mathop{\mathrm{spt}}\mu_\star\). In view of 121 , we have \[\label{eq:tildeeta42-eta42} \eta_\star \mathbin{\vrule height 1.6ex depth
0pt width
0.13ex\vrule height 0.13ex depth 0pt width 1.3ex}K = \widetilde{\eta}_\star \mathbin{\vrule height 1.6ex depth 0pt width
0.13ex\vrule height 0.13ex depth 0pt width 1.3ex}K\tag{122}\] independently of the chosen subsequence.
Theorem 4.1 in [34] gives that \(\widetilde{\eta}_\star\) is \(\mathscr{H}^1\)-rectifiable, i.e., \(\widetilde{\eta}_\star = \widetilde{\theta}_\star \mathscr{H}^1 \mathbin{\vrule height 1.6ex depth 0pt width
0.13ex\vrule height 0.13ex depth 0pt width 1.3ex}\Sigma\), where \(\Sigma\) is \(\mathscr{H}^1\)-rectifiable and \(\widetilde{\theta}_\star\) denotes
the density of \(\widetilde{\eta}_\star\). In particular, the limit \[\widetilde{\theta}_\star(x_0) := \lim_{r \to 0} \frac{\widetilde{\eta}_\star(B(x_0,\,r))}{2r}\] exists at any point
\(x_0 \in \Sigma\). Since rectifiability is a local property, the rectifiability of \(\widetilde{\eta}_\star\) and 122 imply that \(\eta_\star\) is rectifiable as well and, moreover, that \[\theta_\star(x_0) = \widetilde{\theta}_\star(x_0), \qquad \forall x_0 \in K.\] Furthermore, if we set \[\sigma_\beta := \int_{-\sqrt{1+\sqrt{2}\beta}}^{\sqrt{1+\sqrt{2}\beta}} \sqrt{\frac{h_1(s)}{2}}\,{\mathrm{d}}s = \frac{\sqrt{2}}{3}\left(1 + \sqrt{2}\beta\right)^{3/2},\] then applying [34] and using 116 and 118 leads to the fact that \((2 \sigma_\beta)^{-1}{\theta_\star(x_0)}\) is an integer for all \(x_0 \in K\). In addition, thanks to 117 and 118 , [34] tells us that the discrepancy functions \[\xi_\varepsilon^{(h_1)}
:= \frac{1}{\varepsilon} h_1\left(u_\varepsilon\right) - \frac{\varepsilon}{2}\left| \nabla u_\varepsilon \right|^2\] satisfy \[\lim_{\varepsilon\to 0} \int_K \left| \xi^{(h_1)}_\varepsilon \right| \,{\mathrm{d}}x =
0,\] i.e., equipartition holds in the limit. More precisely, if we define the density of the potential energy functions (dually seen as measures) \[\label{eq:def-tilde-zeta-eps-h1} {\zeta}_{\varepsilon}^{(h_1)} := \frac{1}{\varepsilon} h_1\left({u}_\varepsilon\right),\tag{123}\] we see that there exists a limiting potential energy measure \({\zeta}^{(h_1)}_\star\), which satisfies \[{\eta}_\star \mathbin{\vrule height 1.6ex depth 0pt width
0.13ex\vrule height 0.13ex depth 0pt width 1.3ex}K = 2 {\zeta}_\star^{(h_1)} \mathbin{\vrule height 1.6ex depth 0pt width
0.13ex\vrule height 0.13ex depth 0pt width 1.3ex}K,\] so that, in particular, \[\mathop{\mathrm{spt}}\left( {\eta}_\star \mathbin{\vrule height 1.6ex depth 0pt width
0.13ex\vrule height 0.13ex depth 0pt width 1.3ex}K \right) = \mathop{\mathrm{spt}}\left( {\zeta}^{(h_1)}_\star \mathbin{\vrule height 1.6ex depth 0pt width
0.13ex\vrule height 0.13ex depth 0pt width 1.3ex}K \right).\] Moreover, the density \({\mathfrak{h}}_\star(x_0)\) of \({\zeta}_\star^{(h_1)}\) exists at every point \(x_0 \in K\) and satisfies \[\lim_{r \to 0} \frac{{\zeta}^{(h_1)}_\star(B(x_0,\,r))}{2r} = {\mathfrak{h}}_\star(x_0) = \frac{{\theta}_\star(x_0)}{2},\] so that \(\sigma_\beta^{-1} {\mathfrak{h}}_\star(x_0)\) is an integer, for any \(x_0 \in K\). Finally, by letting \(K\) vary in \(\Omega
\setminus \mathop{\mathrm{spt}}\mu_\star\), we obtain \[\label{eq:h42-integer} \sigma_\beta^{-1} \mathfrak{h}_\star(x_0) \in \mathbb{N}, \qquad \forall x_0 \in \Omega \setminus
\mathop{\mathrm{spt}}\mu_\star.\tag{124}\]
Step 11 (Conclusion). In order to conclude the proof, we have to show that the integrality of the rescaled density \(\sigma_\beta^{-1} \mathfrak{h}_\star\) implies that the density \(\mathfrak{v}_\star\) of \(\zeta_\star\) is integer-valued as well after rescaling, i.e, that \(\sigma_\beta^{-1} \mathfrak{v}_\star\) takes only integer values.
In fact, we are going to prove that \[\label{eq:v4261h42} \mathfrak{v}_\star(x_0) = \mathfrak{h}_\star(x_0), \qquad \forall x_0 \in \Omega \setminus
\mathop{\mathrm{spt}}\mu_\star.\tag{125}\] To this purpose, it suffices to show that the difference \(\zeta_\varepsilon- \widetilde{\zeta}^{(h_1)}_\varepsilon\) tends to zero in \(L^1\left(K\right)\) as \(\varepsilon\to 0\). By the definition 14 of \(\zeta_\varepsilon\), the definition 123 of \({\zeta}^{(h_1)}_\varepsilon\), 49 , and 45 , we have \[\label{eq:diff-potentials} \begin{align} \zeta_\varepsilon- \zeta^{(h_1)}_\varepsilon &= \frac{1}{\varepsilon}\left( h(\mathbf{u}) + \frac{\beta}{\sqrt{2}} (1-\rho)\left(u_1^2 - u_2^2 - 1 - \frac{\beta + \beta \rho}{\sqrt{2}}
\right) \right) - \frac{1}{\varepsilon}h_1(u_{1}) \\ &= \frac{1}{\varepsilon} h_2(u_1,\,u_2) + \frac{\beta}{\sqrt{2}} \left(\frac{1-\rho}{\varepsilon}\right)\left(u_1^2 - 1 - \sqrt{2}\beta\right) + \frac{\beta}{\sqrt{2}}\left(
\frac{1-\rho}{\varepsilon} \right) \left( \frac{\beta}{\sqrt{2}}(\rho - 1) - u_2^2 \right) \end{align}\tag{126}\] All terms on the right-hand side of 126 vanish in \(L^1\left( K
\right)\) as \(\varepsilon\to 0\). Indeed, by 101 (recall that \(h_2(u_1,\,u_2) \geq 0\) pointwise, if \(\sqrt{2}\beta \geq
1\)) and a covering argument, \[\frac{1}{\varepsilon} \int_K h_2(u_1,\,u_2) \,{\mathrm{d}}x \lesssim \varepsilon^\alpha, \qquad as\varepsilon\to 0,\] for every \(\alpha \in
(0,\,2)\). Next, by Hölder’s inequality and Lemma 1, \[\begin{align} \int_K \left|
\left(\frac{1-\rho}{\varepsilon}\right)\left(u_1^2 - 1 - \sqrt{2}\beta\right) \right| &\leq \left(\int_K \left(\frac{1-\rho}{\varepsilon}\right)^2 \,{\mathrm{d}}x \right)^{1/2} \left( \int_K \left(u_1^2 - 1 - \sqrt{2}\beta\right)^2 \,{\mathrm{d}}x
\right)^{1/2} \\ &\lesssim \sqrt{\varepsilon}, \qquad as\varepsilon\to 0. \end{align}\] Finally, by [1] and Proposition 3, \[\frac{1}{\varepsilon} \int_K \left\{ (\rho - 1)^2 + u_2^2 \right\}\,{\mathrm{d}}x \lesssim \varepsilon, \qquad as\varepsilon\to 0.\] Combining the last
three inequalities with 126 , it follows that \[\int_K \left| \zeta_\varepsilon- \zeta_\varepsilon^{(h_1)} \right|\,{\mathrm{d}}x \lesssim \sqrt{\varepsilon}, \qquad as\varepsilon\to
0.\] As a consequence, \[\zeta_\star \mathbin{\vrule height 1.6ex depth 0pt width
0.13ex\vrule height 0.13ex depth 0pt width 1.3ex}K = \zeta_\star^{(h_1)} \mathbin{\vrule height 1.6ex depth 0pt width
0.13ex\vrule height 0.13ex depth 0pt width 1.3ex}K.\] In particular, \[\mathfrak{v}_\star = \mathfrak{h}_\star \qquad onK.\] Since \(K\) was arbitrary in \(\Omega \setminus \mathop{\mathrm{spt}}\mu_\star\), by letting \(K\) vary in \(\Omega \setminus \mathop{\mathrm{spt}}\mu_\star\), we have \[\mathfrak{v}_\star(x_0) = \mathfrak{h}_\star(x_0), \qquad \forall x_0 \in \Omega \setminus \mathop{\mathrm{spt}}\mu_\star,\] i.e., we have proved 125 . In particular, by 125 and 124 , it follows that \(\sigma_\beta^{-1} \mathfrak{v}_\star\) is an integer-valued function on \(\Omega \setminus
\mathop{\mathrm{spt}}\mu_\star\). Finally, the conclusion follows because, by [2], \(\mathfrak{v}_\star(a) = 0\) for every
\(a \in \mathop{\mathrm{spt}}\mu_\star\).
◻
3.0.0.1 Acknowledgments
F.L.D. would like to thank Institute of Science Tokyo for warm hospitality. F.L.D. is a member of GNAMPA-INdAM and he is partially supported by INdAM-GNAMPA Project CUP E53C25002010001. Y.T. is partially supported by JSPS Grant-in-aid for scientific
research (A) #23H00085.
Federico Luigi Dipasquale Scuola Superiore Meridionale
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Yoshihiro Tonegawa Department of Mathematics, Institute of Science Tokyo
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