The Yang–Mills–Higgs Functional on Complex Line Bundles: Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma $$\end{document}-Convergence and the London Equation

We consider the Abelian Yang–Mills–Higgs functional, in the non-self dual scaling, on a complex line bundle over a closed Riemannian manifold of dimension n≥3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\ge 3$$\end{document}. This functional is the natural generalisation of the Ginzburg–Landau model for superconductivity to the non-Euclidean setting. We prove a Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma $$\end{document}-convergence result, in the strongly repulsive limit, on the functional rescaled by the logarithm of the coupling parameter. As a corollary, we prove that the energy of minimisers concentrates on an area-minimising surface of dimension n-2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n-2$$\end{document}, while the curvature of minimisers converges to a solution of the London equation.


Introduction
Let (M, g) be a smooth, compact, connected, oriented, Riemannian manifold without boundary, of dimension n ≥ 3. Let E → M be a Hermitian complex line bundle on M , equipped with a (smooth) reference connection D 0 . For any 1-form A ∈ W 1,2 (M, T * M ), we denote by D A := D 0 −iA the associated connection and by F A the curvature 2-form of D A . Let ε > 0 be a small parameter. For any section u ∈ W 1,2 (M, E) of E and any 1-form A ∈ W 1,2 (M, T * M ), we consider the Ginzburg-Landau or Abelian Yang-Mills-Higgs functional In this paper, we prove a convergence result for minimisers in the limit as ε → 0: given a sequence of minimisers {(u min ε , A min ε )}, the energy density of {(u min ε , A min ε )} concentrates on an (n − 2)-dimensional surface S * , which is area-minimising in a distinguished homology class, determined by the topology of the bundle E → M (i.e., the Poincaré-dual to the first Chern class c 1 (E) ∈ H 2 (M ; Z)). On the other hand, the curvature of A min ε converges to a solution of the London equation, with a singular source term carried by S * . This convergence result for minimisers is stated in Corollary B and it is deduced from our main result, Theorem A below, which provides a full Γ-convergence theorem for the functionals G ε . Moreover, and in agreement with known results in the Euclidean setting (c.f., e.g., [41,2]), Theorem A shows that energy concentration on topological singular sets is not unique to minimisers but is a general feature for sequences {(u ε , A ε )} satisfying a (natural) logarithmic asymptotic bound on the energy. Functionals of the form (1) were originally proposed by V. Ginzburg and L. Landau in 1950 [33] as a model of superconductors subject to a magnetic field (where u is an order parameter such that |u| 2 is proportional to the density of electronic Cooper pairs, while A is the vector potential for the magnetic field). The theory accounts for most commonly observed effects (such as the quantisation of magnetic flux, the Meissner effect, and the emergence of Abrikosov vortex lattices, see e.g. [62]); moreover, it can be justified as a suitable limit of Bardeen, Cooper and Schrieffer's microscopic theory [8]. Ginzburg-Landau functionals, or variants thereof, arise in other areas of physics -for instance, superfluidity (see e.g. [29]) and particle physics, for (1) is the Abelian version of Yang-Mills-Higgs action functional in gauge theory (see e.g. [39]).
Being invariant under gauge transformations is, indeed, one of the most prominent features of the functional (1). In a Euclidean domain Ω ⊆ R n , the functional (1) reduces to where u ∈ W 1,2 (Ω, C) is a complex-valued map and A ∈ W 1,2 (Ω, T * Ω) is a real-valued one-form on Ω. Assuming for simplicity that Ω is simply connected, the invariance of (2) under gaugetransformations can be expressed as follows: for any (u, A) ∈ W 1,2 (Ω, C) × W 1,2 (Ω, T * Ω) and any θ ∈ W 2,2 (Ω, R), there holds The property (3) suggests that Ginzburg-Landau functionals are naturally set in the context of complex line bundles over a manifold. Indeed, the gauge group S 1 ≃ U(1) acts on a pair (u, A) precisely in the same way as transition functions of a bundle act (locally) on sections and connection forms. In this general setting, gauge-invariance takes the following general form: for any (u, A) ∈ W 1,2 (M, E) × W 1,2 (M, T * M ) and Φ ∈ W 2,2 (M, S 1 ), consider the transformation where Φu is defined by the fibre-wise action of the structure group U(1) ≃ S 1 on E. Then there holds Gauge-invariance will play a crucial rôle in this paper. Physically, each observable quantity must be gauge-invariant and the energy is only one of them. For instance, each term in the energy density of G ε is gauge-invariant, as well as F A (whose physical counterpart in superconductivity is, in fact, the magnetic field). Minimisers of gauge-invariant functionals on manifolds, such as (1), include several examples of physically relevant objects (such as Bogomolnyi monopoles, vortices, instantons, Hermite-Einstein connections on Kälher manifolds). Moreover, such objects play an important rôle in topology and complex geometry. For example, Yang-Mills minimisers are used in the classification of stable holomorphic vector bundles on complex manifolds (this is the content of Donaldson, Uhlenbeck and Yau's theorem [28,63]; for generalisations to Yang-Mills-Higgs minimisers, see e.g. [18,19,31]). Functionals such as (1) may also be coupled to Einstein equations; this approach is relevant to cosmology, as it provides a model for gravity-driven spontaneous symmetry breaking [60].
We are interested in the asymptotic regime as ε → 0, which is known as the London limit within the context of superconductivity, or the strongly repulsive limit, within the context of particle physics. This limit is characterised by the emergence of topological singularitiesthe energy of minimisers concentrates, to leading order, on (n − 2)-dimensional sets, whose global structure depends on the topology of the bundle E → M . We are interested in providing a variational characterisation, in the sense of Γ-convergence, of those singularities as being area-minimising in a homology class. We expect that this variational characterisation will provide indications on the dynamics of singularities arising in the limit of the corresponding timedependent models. For instance, for a generalisation of (1) on a Lorentzian manifold, we expect that the energy concentrate on time-like relativistic strings or M -branes (see e.g. [40,9,10] for the analysis of related problems in Minkowski space-time). Moreover, we expect the heat flow of (1) to be related to motion by mean curvature (see [14] for the asymptotics of a non-gaugeinvariant problem, with no dependence on A, in the Euclidean setting).
The asymptotic analysis of (1) in the limit as ε → 0 heavily relies on analogous results for the simplified Ginzburg-Landau functional, i.e.
In case E is the tangent bundle on a closed Riemann surface and the connection D A is fixed, asymptotics for minimisers and gradient-flow solutions are available in [37,23]. Minimisers of (1) in the limit as ε → 0 were studied in case the base manifold M is two-dimensional [49,54]. Other results address a self-dual variant of the functional (1), in which the curvature term |F A | 2 is replaced by ε 2 |F A | 2 -see e.g. [36] for the two-dimensional case and [53,52] for the higherdimensional case. In a way, self-duality is a sort of additional symmetry in the Yang-Mills-Higgs functionals, with applications in the theory of minimal surfaces, as explored e.g. in the recent papers [53,25,52]. However, the non self-dual scaling (1) appears to be more closely related to the original physical motivation of the Ginzburg-Landau functional. To better explain this point, let us introduce the gauge-invariant Jacobian J(u, A) of a pair (u, A), defined pointwise (when the right-hand-side of (7) below makes sense) as the 2-form for any smooth test field X, Y on M . Here ·, · is the real-valued scalar product induced by the Hermitian form on E. Plainly, J(u, A) is well-defined for any (u, A) ∈ W 1,2 (M, E) × W 1,2 (M, T * M ) such that G ε (u, A) < +∞. In superconductivity, there is a physical observable associated with gauge-invariant Jacobians: the supercurrent vorticity (cf., e.g., [62,Chapter 5] and [39,Chapter 1]).
As it is easy to check, for minimisers (u min ε , A min ε ) of the non-self dual functional (1), the curvatures F ε := F A min ε satisfy the (gauge-invariant) London equation a distinctive feature of superconductivity (see, e.g., [62,Chapters 1 and 5]). In Corollary B below, we show that the curvatures F ε = F A min ε converge, up to extraction of a subsequence, to a solution F * of the (limiting) London equation The right-hand side J * of (9) is the limit of the Jacobians, i.e. J ε → πJ * (cf. Theorem A). J * is a singular measure, with values in 2-forms, carried by an area-minimising set of dimension n − 2.
By contrast, minimisers of the self-dual functionals do not converge to solutions of (9); in the limit as ε → 0, Moreover, while minimisers of the self-dual functionals have uniformly bounded energy as ε → 0, minimisers (u min ε , A min ε ) of (1) on a non-trivial bundle must satisfy G ε (u min ε , A min ε ) → +∞ as ε → 0 -in fact, G ε (u min ε , A min ε ) is of the order of |log ε|, as we will see below (cf. Remark 2). Such a large amount of energy allows for wild oscillations in the phases of the maps u ε , preventing compactness even in weak topologies in Sobolev spaces and making the proof of Theorem A quite challenging, as in fact is for its Euclidean counterpart in [41,2].
One might expect that critical points of (1) converge, in a suitable sense, to minimal surfaces, as is the case for (14) (see [12]) and for the self-dual Yang-Mills-Higgs energies (see [53]). Here, we prove this result for sequences of minimisers of (1) (see Corollary B below); we plan to investigate non-minimising critical points in a forthcoming work [22].
We denote by ⋆ the Hodge dual operator, regarded as a map from k-forms to (n − k)-currents and from k-currents to (n − k)-forms (upon composition with the natural isomorphism between vectors and covectors, induced by the metric -see (1.1) below and Appendix B for more details). We consider a distinguished, non-empty class C of integer-multiplicity, rectifiable (n−2)-currents with no boundary. The class C is uniquely defined in terms of the topology of the bundle E → M ; more precisely, C ∈ H n−2 (M ; Z) is Poincaré-dual to the first Chern class c 1 (E) ∈ H 2 (M ; Z) of the bundle (see (1.6) below for more details).
Theorem A. The following statements hold.
(ii) Let S * be an integer-multiplicity, rectifiable (n − 2)-cycle in the class C and let J * := ⋆S * be the dual 2-form. Then, there exists a sequence {(u ε , A ε )} ε such that J(u ε , A ε ) → πJ * in W −1,p (M ) for any p with 1 ≤ p < n/(n − 1) and it may or may not be true that |D 0 u| = |D A u + iAu| ∈ L 2 (M )). For simplicity, we have chosen to state our results in terms of W 1,2 -pairs, but our results extend to all Sobolev pairs of finite energy, with no significant change in the proofs.

Remark 2.
Since the class C is always not empty, a straightforward consequence of Theorem A is that the energy of sequences of minimisers of G ε is automatically of order |log ε| as ε → 0.
As a corollary, we obtain a variational characterisation for the limit of a sequence of minimisers. Given (u, A) ∈ W 1,2 (M, E) × W 1,2 (M, T * M ), we denote the (rescaled) energy density of (u, A) as Corollary B. Let (u min ε , A min ε ) be a minimiser of G ε in W 1,2 (M, E) × W 1,2 (M, T * M ). Then, there exist bounded measures J * , F * , with values in 2-forms, and a (non-relabelled) subsequence such that for any p < n/(n − 1), and µ ε (u min ε , A min ε ) ⇀ π |J * | weakly * as measures. Moreover, the current ⋆J * belongs to C and has minimal mass among all the currents in C, while F * satisfies the London equation In other words, the energy of the minimisers (u min ε , A min ε ) concentrate, to leading order, on the support of a (n − 2)-dimensional current, which is dual to the limit Jacobian J * and minimises of the area in its homology class. Moreover, the limit curvature F * is uniquely determined from J * , via the London equation. Due to gauge-invariance (5), we cannot expect compactness for the minimisers themselves, u min ε , A min ε . However, it seems plausible that compactness (in suitable norms) should be restored if we make a suitable choice of the gauge. We plan to address this point in a forthcoming paper [22].
As an intermediate step towards the proof of Theorem A, we prove a Γ-convergence result for a simpler functional that only depends on the variable u. For any u ∈ W 1,2 (M, E), we consider The functional E ε is analogous to the non-gauge-invariant version of the Ginzburg-Landau functional, (6). We define J(u) := J(u, 0), i.e. J(u) is the Jacobian of u with respect to the reference connection D 0 .
Theorem C. The following statements hold.
Then, there exists a (non-relabelled) subsequence and a bounded measure J * , with values in 2-forms, such that J(u ε ) → πJ * in W −1,p (M ) for any p with 1 ≤ p < n/(n − 1) and Moreover, ⋆J * is an integer-multiplicity, rectifiable (n − 2)-cycle in the class C.
(ii) Let S * be an integer-multiplicity, rectifiable (n − 2)-cycle in the class C and let J * := ⋆S * be the dual 2-form. Then, there exists a sequence The proof of Theorem C depends heavily on analogous Γ-convergence results obtained in the Euclidean setting [41,2]. The most delicate point is, probably, showing that the limit Jacobian J * satisfies ⋆J * ∈ C, for this is a global topological property that cannot be deduced from localisation arguments. The proof of this fact is contained in Section 2.1.3.
Once Theorem C is proved, the upper bound (ii) in the statement of Theorem A follows immediately. On the other hand, the proof of the lower bound in Theorem A relies on the London equation, as well as Theorem C. More precisely, given a sequence {(u ε , A ε )} that satisfies the logarithmic energy bound (10), we construct a sequence of bounded sections {v ε } and a sequence of 1-forms {B ε } in such a way that is small in W −1,p (M ) and, most importantly, the curvatures F Bε satisfy the London equation (8). Specifically, the sections v ε are obtained from the corresponding u ε by a truncation argument while the 1-forms B ε are obtained by minimising the auxiliary energy functionals in a suitable class, while keeping v ε fixed (see Section 3 for full details). By exploiting the ellipticity and the gauge-invariance of the London equation, and up to a suitable choice of gauge, we can then show that the difference G ε (v ε , B ε ) − E ε (v ε ) is of order smaller than |log ε|. Once this is done, the conclusion follows by Theorem C.
The paper is organised as follows. In Section 1, we set some notation and recall a few useful properties of the Jacobian, J(u, A). Section 2 is devoted to the proof of Theorem C, while Section 3 contains the proof of Theorem A and Corollary B. The paper is completed by a series of appendices, which contain a review of some background material on Sobolev spaces for sections of a vector bundle and currents, as well as the proof of a few technical results.

Preliminaries
We will frequently encounter Sobolev spaces of sections of a bundle. For the convenience of the reader, we provide the definitions and state some basic properties in Appendix A. The notation we use is fairly self-explanatory. However, we stress that the symbols such as W 1,p (M, E) denote Sobolev spaces of W 1,p -sections of the bundle E → M . In case E is a trivial bundle, The notation we use for differential forms is rather standard, too. For instance, we denote as # : Λ k T * M → Λ k TM , ♭ : Λ k TM → Λ k T * M the isometric isomorphisms between vectors and forms induced by the metric on M , and by * : Λ k T * M → Λ n−k T * M the Hodge dual operator induced by the metric and the orientation of M . We define an operator as follows: for any k-form ω and any k-vector v, The operator ⋆ can be extended to an operator between currents and form-valued distributions; see Appendix B for details. We denote by d, d * the exterior differential and codifferential of forms, respectively. The codifferential of a k-form ω is defined by d * ω : We recall a few basic notions about currents in Appendix B. Given a current S, we will write ∂S for its boundary (defined as in (B.1) below), M(S) for its mass (see (B.4) below) and F(S) for its (integer-multiplicity) flat norm (see (B.7) below).
It is convenient to revisit the definition of Jacobian, (7). Given a pair (u, A) ∈ W 1,2 (M, E)× W 1,2 (M, T * M ) such that D A u ∈ L 2 (M, T * M ⊗ E), we define the gauge-invariant pre-Jacobian as the 1-form Under the assumptions above, j(u, A) is integrable, so it makes sense to consider its differential in the sense of distributions. We define the (distributional) gauge-invariant Jacobian as In case the pair (u, A) is smooth, an explicit computation shows that the Jacobian as defined by (1.3) agrees with (7). The same remains true if (u, A) satisfies G ε (u, A) < +∞, by a truncation and density argument. However, (1.3) allows us to define the Jacobian for a broader class of pairs (u, A) -for instance, when u ∈ W 1,1 (M, E) ∩ L ∞ (M, E) and A ∈ L 1 (M, T * M ). , we see that also J(u, A) is a local operator and commutes with restrictions.
The distributional Jacobian is continuous with respect to suitable notions of weak convergence. For the convenience of the reader, we recall a result in this direction. We denote by j(u), J(u) the pre-Jacobian and Jacobian with respect to the reference connection D 0 , i.e.
Proof. Thanks to (1.5), it suffices to show that j(u ε ) ⇀ * j(u) weakly * in the sense of distributions. This follows from the Hölder inequality.
In fact, the Jacobian J(u, A) satisfies suitable continuity properties as a function of both u and A (see e.g. [52,Proposition 3.1]).
Finally, we introduce a distinguished (homology) class of (n − 2) currents, C ∈ H n−2 (M ; Z), as follows. Let w : M → E be a smooth section of E. We assume that w is transverse to the zero-section of E; that means, for any point p ∈ M such that w(p) = 0, the differential d p w induces a surjective linear map d p w : where E p is the fibre of E at p. Such a section w exists; in fact, by Thom's transversality theorem (see e.g. [20,Theorem 14.6]), any smooth section can be approximated (e.g., uniformly) by transverse sections. Transversality implies that the inverse image Z := w −1 (0) is a smooth manifold without boundary, of dimension n − 2. As both M and E are oriented manifolds (the orientation on E is the one induced by the complex structure on each fibre), the manifold Z can be given an orientation, in a natural way [17,Proposition 12.7]. As a consequence, there is a well-defined (n − 2)-current Z , carried by Z, with unit multiplicity. We have ∂ Z = 0, because Z is manifold without boundary. We define The class C does not depend on the choice of w (see [17,Proposition 12.8]). Moreover, by the boundary rectifiability theorem (see e.g. [58,Theorem 6.3]), all the elements of C are integer-multiplicity, rectifiable (n − 2)-currents with no boundary. In the topological jargon, C ∈ H n−2 (M ; Z) is Poincaré-dual to the first Chern class c 1 (E) ∈ H 2 (M ; Z). Equivalently, C is Poincaré-dual to the Euler class of E, regarded as a real bundle over M with the orientation induced by the complex structure. The following statement motivates our interest in the class C. Sketch of the proof of Proposition 1.4. Let w be a smooth section of E → M that is transverse to the zero section of E, as above. Let u 0 := w/ |w|. The section u 0 is well-defined and smooth away from Z := w −1 (0). Moreover, u 0 ∈ W 1,p (M, E) for any p ∈ [1, 2). Indeed, since w is transverse to the zero section of E, the differential d w restricted to the normal bundle NZ of Z is a fibre-wise isomorphism NZ → E. As M is compact, it follows that there exists a constant C such that, for any x ∈ M , dist(x, Z) ≥ C |w(x)| (1.7) Therefore, The integral at the right-hand side is finite for any p ∈ [1, 2), because Z has codimension 2 (see e.g. [2,Lemma 8.3]). Moreover, for a suitable orientation of Z (as in [17,Proposition 12.7]), there holds ⋆ J(u 0 ) = π Z (1.9) As the Jacobian is a local operator (by Remark 1.1), it suffices to check that (1.9) is satisfied in an arbitrary coordinate neighbourhood U ⊆ M . Due to Remark 1.2, J(u 0|U ) is equal to the Jacobian of u 0|U with respect to the flat connection,J := 1 2 d d u 0 , iu 0 . For the latter, we have ⋆J = π Z in U (this computation is similar to, e.g., [42,Example 3.4]) and hence (1.9) follows.
2 Γ-convergence for the functional E ε The aim of this section is to prove Theorem C.

A truncation argument
In some of the arguments below, it will be useful to assume that sequence u ε is uniformly bounded in L ∞ (M, E). Fortunately, it is possible to reduce to this case by means of a classical truncation argument. We formulate this argument in terms of pairs (u ε , A ε ), for later use.
In the proof of Lemma 2.1 and in the sequel, we will make repeated use of the following observation.
Proof. In {u = 0}, the pair (u, iu) is an orthogonal frame for (the real bundle associated with) E. Therefore, keeping in mind that the connection D A is compatible with the metric (i.e., (A.21)), we obtain Proof of Lemma 2.1. Towards the proof of (2.3), we observe that Lemma 2.2 implies in the set {|u ε | > 1}. As a consequence, By the energy estimate (2.1), we deduce that j(u ε , Let us consider the 2-form σ(u ε , A ε ) defined by σ(u ε , A ε )[X, Y ] := i D Aε,X u ε , D Aε,Y u ε for any smooth vector fields X, Y . By a direct computation, we have From (2.9) and the energy estimate (2.1), we deduce We work with the reference connection D 0 on E and denote by j(u ε ) := j(u ε , 0), J(u ε ) := J(u ε , 0) the pre-Jacobian and Jacobian of u ε with respect to D 0 (see (1.5)). Our first goal is to prove that {J(u ε )} is compact in W −1,p (M ), for any p < n/(n − 1). Lemma 2.3. Let u ε ∈ W 1,2 (M, E) be a sequence that satisfies (2.11). Then, there exists a (nonrelabelled) subsequence and a bounded measure J * , with values in 2-forms, such that J(u ε ) → πJ * in W −1,p (M ) for any p such that 1 ≤ p < n/(n − 1).
In the Euclidean setting, compactness results analogue to Lemma 2.3 are well-known [41,2]. We will deduce Lemma 2.3 from its Euclidean counterparts by a localisation argument. Let U ⊂ M be a smooth, contractible domain in M , all contained in a coordinate chart of M . By working in local coordinates, we identify U with a subset of R n , equipped with a smooth Riemannian metric g. Moreover, as U is contractible, the bundle E → M trivialises over U . Therefore, we can (and do) identify sections of E with maps U → C. The reference connection D 0 , restricted to U , may be written as where d is the Euclidean connection on R n (that is, d u = d ℜ(u) + i d ℑ(u) for any u : U → C) and γ 0 : U → (R n ) * is a smooth real-valued 1-form. It will be useful to compare the restriction of E ε to U , that is with its Euclidean counterpart, (2.14) The integral in (2.14) is taken with respect to the Lebesgue measure d x, not the volume form vol g induced by the metric g. The functional (2.14) is precisely the Ginzburg-Landau functional, in the simplified form that was introduced by Bethuel, Brezis and Hélein [11]. Given u ∈ W 1,2 (U, C), we denote the pre-Jacobian and the Jacobian of u with respect to the flat connection d as The quantities j(u),(u) and J(u),J(u), respectively, are related to each other by where γ 0 is given by (2.12). We recall a well-known compactness result for the Euclidean Ginzburg-Landau functional (2.14). For any α ∈ (0, 1), we let C 0,α 0 (U ) be the space of α-Hölder continuous functions ϕ : U → R such that ϕ = 0 on ∂U . We let (C 0,α 0 (U )) ′ denote the topological dual of C 0,α 0 (U ).
We deduce Lemma 2.3 from Theorem 2.5.
Step 1 (Local convergence). Let U ⊂ M be a contractible, smooth, open subset of M , which we identify with a subset of R n . Since the manifold M is compact and smooth, there exists a constant C (depending on M only) such that d By Theorem 2.5, we may extract a subsequence and find a bounded measure J U on U , with values in 2-forms, such that Step 2 (Covering argument and, on the other hand, ρ k J(u ε ) = ρ k J(u ε ) |U k for each k = 1, . . . , N . According to Remark 2.4, J(u ε ) is local and commutes with restrictions, i.e., J(u ε ) |U k = J(u ε|U k ) where J * is a bounded measure on M with values into 2-forms which is well-defined because, as it can be easily checked, it is independent of the chosen partition of unity and of the chosen trivialization.

Identifying the homology class of ⋆J *
Our next goal is to show that the limit of the Jacobians, ⋆J * , belongs to the homology class C (the Poincaré dual of the first Chern class of E; see (1.6)).
The proof of Proposition 2.7 relies on the following result.
Proposition 2.7 follows immediately from Proposition 1.4 and Lemma 2.8. It only remains to prove Lemma 2.8. We will need an auxiliary result, again borrowed from [52]. We denote by Harm 1 (M ) the space of harmonic 1-forms on M and by vol S 1 the volume form of S 1 . Remark 2.9. If Φ ∈ W 1,1 (M, S 1 ), then Φ * (vol S 1 ) has a pointwise a.e. meaning, and we have the pointwise a.e. equality Lemma 2.10. For any ϕ ∈ W 1,2 (M, R) and ξ ∈ Harm 1 (M ), there exist a map Φ : M → S 1 , as regular as ϕ, and a formξ ∈ Harm 1 (M ) such that where C M > 0 is a constant that depends only on M .
Remark 2.11. By Hodge theory, the space Harm 1 (M ) has finite dimension. Therefore, the difference ξ −ξ is bounded not only in L 2 (M ), but also in any other norm.
Proof. The proof follows very closely the argument of [52,Lemma 3.4]. Since, however, that lemma is designed to deal with a slightly different situation and cannot be used directly in our case, we provide full details for reader's convenience.
We notice that, for all smooth maps f, g : M → S 1 , the map φ := f g : M → S 1 is still smooth, and we can pull back vol S 1 by φ. Next, since S 1 is an Abelian Lie group with an invariant volume form, it holds φ * (vol S 1 ) = f * (vol S 1 ) + g * (vol S 1 ). For any smooth function ψ : M → R and f : M → S 1 harmonic, if we set g := e iψ , we have φ = f e iψ and, by the previous formula and the translational invariance of vol S 1 , we end up with φ * (vol . Following [52] and [3, Example 3.3.8], we observe that f can be chosen so that , the conclusion is however still true thanks to a standard density argument. Finally, by definition, it easily seen that Φ is as regular as ϕ. Proof of Lemma 2.8. We can assume without loss of generality that {u ε } is bounded in L ∞ (M ), independently of ε -for otherwise, we replace each map u ε by the truncated map v ε defined in (2.2) and apply Lemma 2.1 (with A ε = 0). Then, it follows that j(u ε ) ∈ L 2 (M, T * M ) and that due to the energy estimate (2.11). We are going to construct a sequence of maps of the form Then, we will obtain a map w * with the desired properties by passing to the (weak) limit in the w ε 's. We split the proof into several steps.
Step 1. We consider the Hodge decomposition of j(u ε ) -that is, we write (in a unique way) for some (co-exact) form ϕ ε ∈ W 1,2 (M, R), some (exact) form ψ ε ∈ W 1,2 (M, Λ 2 T * M ), and some ξ ε ∈ Harm 1 (M ). This decomposition (recalled in Proposition A.13) is orthogonal in L 2 (M ). Then, ψ ε is closed and L 2 -orthogonal to all harmonic 2-forms. By taking the differential in (2.23), we obtain where F 0 is the curvature of the reference connection D 0 . By Lemma 2.3, we know that J(u ε ) in bounded in W −1,p (M ), independently of ε. By applying Lemma C.4, we deduce for some constant C p that depends on p (and on F 0 ), but not on ε.
Step 2. In this step, we construct suitable maps Φ ε : M → S 1 , in such a way that j(Φ ε u ε ) is bounded in L p (M ). We do so by applying Lemma 2.10. By Lemma 2.10, for each ε > 0 there exist a map Φ ε : M → S 1 and a formξ ε ∈ Harm 1 (M ) such that where C M is a constant depending only on M . We consider the section Φ ε u ε . We have and hence, recalling the energy estimate (2.11), for some constant C p depending on p, M and D 0 , but not on ε. Indeed, due to (2.23) and (2.26), we have Let q > 2 be such that 1/p = 1/q + 1/2. As u ε L ∞ (M ) ≤ C, by generalised Hölder's inequality and interpolation we obtain up to a multiplicative constant which depends only on M and p. Hence, recalling (2.11) and (2.28), (2.33) Step 3. The estimate (2.30) is not enough to guarantee that Φ ε u ε is bounded in W 1,p (M ): we also need to control the differential of |u ε |. Although we cannot make sure that d |u ε | L p (M ) is bounded, in general (see Remark 2.13 below), we can construct suitable functions ρ ε : M → R so that w ε := ρ ε Φ ε u ε satisfies the desired estimates. For any ε > 0, we take a smooth nonnegative function f ε : as ε → 0. In particular, (2.33), (2.38) and the fact that The second term at the right-hand side is bounded by (2.29). To estimate the other term, we recall (2.35) and observe that d(|u ε |) L 2 (M ) ≤ D 0 u ε L 2 (M ) , by Lemma 2.2. Then, Finally, using (2.35) and the fact that |u ε | → 1 in L 2 (M ) (due to our assumption (2.11)), we deduce as ε → 0.
As a byproduct of the arguments above, we have proved the following statement.
Thus, any sequence {u ε } bounded in L ∞ (M ) and with E ε -energy of order |log ε| can be split into a compact and a non-compact part (with respect to the weak W 1,p -topology). Moreover, Corollary 2.12 asserts that the compact part stores the information necessary to determine the topological energy-concentration set, while the non-compact part of the sequence is, in this sense, "topologically irrelevant".
On a qualitative level, the lack of compactness is due to wild oscillations in the phases made possible by the large amount of energy at disposal. At first glance, this inconvenient seems to be "cured" by the gauge transformations Φ ε . However, we must emphasise that the "penalisations" ρ ε , even if small in uniform norm as ε → 0, play a subtle rôle. Indeed, Remark 2.13 below points out that gauge transformations alone are in general not sufficient to perform the splitting and obtain compactness.
Remark 2.13. Let u ε ∈ W 1,2 (M, E) be a sequence that satisfies (2.48). In general, it may not be possible to find maps Φ ε : . However, if u ε are solutions of the Ginzburg-Landau equations (i.e., critical points of E ε ) in Euclidean domains, then it is possible to obtain compactness for a sequence of the form Φ ε u ε -see [

Lower bounds
Again, we consider a sequence u ε ∈ W 1,2 (M, E) that satisfies the energy bound (2.11). By Lemma 2.3, we know that J(u ε ) → πJ * in W −1,p (M ) for any p < n/(n − 1). The aim of this section is to prove the following We stress that, in the statement of Proposition 2.14, the open set V may be chosen arbitrarily; it does not need to be contained in a coordinate chart. For instance, we may take V = M . Therefore, once Proposition 2.14 is proved, Statement (i) of Theorem C follows at once.
Proof of Theorem C, Statement (i). The statement follows from Lemma 2.3, Proposition 2.7 and Proposition 2.14.
In the rest of this section, we deduce Proposition 2.14 from its Euclidean counterpart [41,2], by means of a localisation argument.
Lemma 2.15. Let δ > 0 be smaller than the injectivity radius of M . Let U ⊂ M be a smooth, contractible domain, entirely contained in a geodesic ball of radius δ. Then, Proof. We identify U with a subset of R n , by local coordinates, and write D 0 = d −iγ 0 , where d is the flat connection on U and γ 0 is a smooth 1-form that depends on D 0 only. We consider again the functionalĒ ε given by (2.14). Thanks to (A.1) (which we can apply as U is supposed to be contained in a geodesic ball), we can writē The last line follows by Young's inequality (which gives, for each choice of σ > 0, the point- and then the previous inequality follows by choosing σ = δ 2 ). Consequently, we obtain Since this holds for every ε > 0, we can pass both sides to the lim inf as ε → 0 keeping the inequality. On doing so, the last term on the right above vanishes, because the energy estimate (2.11) implies that u ε is bounded in L 2 (M ). Therefore, we get the desired estimate recalling (iii) of Theorem 2.5.
We deduce Proposition 2.14 from Lemma 2.15 by applying a Vitali-Besicovitch-type covering theorem, which we recall here.
The proof of Theorem 2.16 can be found in [30, Theorem 2.8.14, Corollary 2.8.15]. The statements in [30] apply not only to compact Riemannian manifolds, but also to a more general class of metric spaces, i.e. 'directionally limited' metric spaces. (Moreover, they apply to outer measures as well as measures.) However, the statement given here is sufficient for our purposes.
The next lemma is an immediate consequence of Theorem 2.16.
Proof. We apply Theorem 2.16 to the bounded measure µ := |J * | and the collection of (closed) balls Since |J * | is a finite measure, we have |J * | (∂B r (x 0 )) = 0 for a.e. r ∈ (0, δ). Then, the assumptions of Theorem 2.16 are satisfied, and the lemma follows.

Upper bounds
The goal of this section is to prove Statement (ii) of Theorem C. First, we introduce some notation. Let u : M → E be a section of the bundle E → M , and let D 0 be our reference (smooth) connection on E. Let X ⊂ M be a closed set. Following [1,2], we say that u has a nice singularity at X (with respect to D 0 ) if |u| = 1 in M \ X, u is locally Lipschitz on M \ X and there exists a constant C > 0 such that If u has a nice singularity at X with respect to D 0 , then u has a nice singularity at X with respect to any smooth connection D, because D can be written as D = D 0 −iA for some smooth 1-form A. Therefore, there is no ambiguity in saying that u has a nice singularity at X without specifying the reference connection D 0 . If X is a finite union of submanifolds of dimension q or less and u has a nice singularity at X, then u ∈ (L ∞ ∩ W 1,p )(M, E) for any p < n − q. This is a consequence of the following lemma: Proof. When M = R n , the proof may be found e.g. in [2,Lemma 8.3]. When M is a compact, smooth manifold, we reduce to the Euclidean case by working in coordinate charts.
Remark 2.19. If u has a nice singularity on a closed Lipschitz set X of dimension n−2 at most, for any smooth vector fields v, w.
We consider smooth triangulations on M , as defined e.g. in [47,Definition 8.3]. Given a triangulation T of M and an integer q ∈ {0, 1, . . . , n}, we call T q the set of all q-dimensional simplices of T. We denote by Sk q T the q-dimensional skeleton of T, that is, When (2.51) holds, we say that the current S is carried by the triangulation T. To construct a recovery sequence for E ε , we borrow a result by Parise, Pigati and Stern [52,Proposition 4.2]. For the convenience of the reader, we reproduce the statement here, using the notation we introduced above. Moreover, if S is a polyhedral (n − 2)-cycle in the class C and T is a triangulation that carries S, then there exists a section u : M → E that has a nice singularity at Sk n−2 T and satisfies ⋆J(u) = πS.
Remark 2.21. In the statement of [52,Proposition 4.2], the authors do not say explicitly that the approximating chains S j and S are homologous, but this fact is contained in the proof.
Let T be a triangulation of M and let γ > 0 be a small parameter. For any simplex K ∈ T n−2 , we define The set V K is the closure of a Lipschitz domain in M . For γ small enough, V K is contained in a tubular neighbourhood of K and it retracts by deformation onto K. In particular, V K is contractible for γ small enough.
Lemma 2.22. If γ > 0 is small enough, then, for any K ∈ T n−2 and K ′ ∈ T n−2 , the intersec- Proof. We claim that, for any K ∈ T n−2 and K ′ ∈ T n−2 , there exists a constant C > 0 such that for any x ∈ K. Indeed, let x 0 ∈ K be given. It suffices to find a constant C that satisfies (2.53) for any x close enough to x 0 ; we will then able to find C that satisfies (2.53) for any x ∈ K, as claimed, because K is compact. If x 0 ∈ K \ K ′ , then the quotient dist(·, ∂K)/ dist(·, K ′ ) is bounded in a neighbourhood of x 0 , and (2.53) follows. Suppose now that x 0 ∈ K ∩ K ′ . By definition (see [47,Definition 8.3], T is the homeomorphic image of a simplicial complex in R ℓ (for some ℓ ≥ n), via a piecewise smooth map with injective differential. (As the differential of the parametrisation is injective, it follows that K, K ′ meet at a non-zero angle at x 0 .) Upon composition with the parametrisation, we may assume without loss of generality that K, K ′ are affine simplices in an affine plane of dimension n − 1. Then, a routine computation shows that (2.53) is satisfied in a neighbourhood of x 0 , for some constant C that depends on the angle between K and K ′ at x 0 . This completes the proof of (2.53). As there are only finitely many simplices in T, the constant C in (2.53) may be chosen uniformly with respect to K, K ′ .
This proves the lemma.
We can now prove Statement (ii) of Theorem C.
Thanks to Proposition 2.20, and up to a diagonal argument, we may assume without loss of generality that S * is a polyhedral (n − 2)-cycle in the homology class C. Let T be a triangulation that carries S * . Up to a subdivision, we may assume without loss of generality that each simplex K ∈ T has diameter diam K ≤ δ. For each (n − 2)-simplex K ∈ T n−2 , we define V K as in (2.52). Construction of the recovery sequence out of V . We claim that u is of class for any K ∈ T n−2 and hence, dist(x, Sk n−2 T) ≥ γ dist(x, Sk n−3 T). As u has a nice singularity at Sk n−2 T, we obtain Construction of the recovery sequence on each V K . Let K ∈ T n−2 . As diam T ≤ δ, we have diam V K ≤ Cδ, for some constant C that depends only on γ. When δ small enough, the diameter of V K is smaller than the injectivity radius of V K . By working in geodesic normal coordinates, we may identify V K with a subset of R n . We also identify sections V K → E with complex-valued maps defined on a subset of R n . The restriction u |∂V K has a nice singularity at ∂K and hence, u ∈ W 1,p (∂V K , E) for any p < 2, because of Lemma 2.18. By Sobolev embeddings, it follows that u ∈ W 1/2,2 (∂V K , E). Therefore, we are in position to apply Γ-convergence results for the Ginzburg-Landau functional in Euclidean settings. Thanks to [2, Theorem 5.5 p. 32 and Remark i p. 33], there exists a sequence of complex-valued maps u K ε , of class W 1,2 in the interior of V K , such that u K ε = u on ∂V K (in the sense of traces), Here, as in (2.15),J(u K ε ) := 1 2 d d u K ε , iu K ε and d x denotes the Lebesgue measure on V K . The reference connection D 0 can be written as D 0 = d −iγ 0 on V K , for some smooth 1-form γ 0 . Therefore, J(u K ε )−J(u K ε ) = 1 2 d(γ 0 (1−|u K ε | 2 )). The energy estimate (2.58) implies that J(u K ε )− J(u K ε ) → 0 in W −1,2 (M ) and hence, We recall that the volume form vol g , in geodesic coordinates on a ball of radius r, satisfies vol g ≤ (1 + Cr 2 ) d x, where C is a constant that depends only on the curvature of M , not on r.
Then, keeping in mind that the difference D 0 − d is bounded independently of ε, from (2.58) we deduce lim sup for δ small enough.
Conclusion. Recall that we have defined u ε := u in M \ K. Now, we take u ε := u K ε on each V K . Since the V K 's have pairwise disjoint interiors, the section u ε is well-defined and is of class W 1,2 . As the Jacobian is a local operator, from (2.56) and (2.59) we deduce that J(u ε ) → πJ * in W −1,1 (M ). The inequality (2.54) follows from (2.55) and (2.60). This completes the proof.
3 Γ-convergence for the functional G ε In this section, we prove Theorem A relying on Theorem C and Corollary B relying, in turn, on Theorem A. Since the recovery sequence in the proof of (ii) of Theorem C essentially works as well for the functional G ε , almost all the work in this section is addressed to the proof of Statement (i) of Theorem A. The key idea is that the given sequence {(u ε , A ε )} in Statement (i) of Theorem A can always be replaced, in the proof, with a better sequence to the purpose, which we call an optimised sequence (Definition 3.1). The relevant properties of optimised sequences are listed in Lemma 3.5. They allow, in the end, both to find a lower bound for the lim inf of the rescaled energies (see Corollary 3.9 and Remark 3.10) and to show the flat convergence of gauge-invariant Jacobians (see Lemma 3.5, Remark 3.6, and Corollary 3.9) with much less effort than needed dealing directly with the "original" sequence {(u ε , A ε )} in Statement (i) of Theorem A.
We begin this section dealing with an auxiliary problem, involved in the construction of optimised sequences. As a preliminary step, recall from Remark A.10 that an exact 2-form for every ε > 0.
Proof. To shorten notations, let us omit the dependence on u, A, and D 0 and simply write F in place of F( · ; u, A, D 0 ). From Lemma 3.1, [A] is not empty and sequentially weakly closed. Obviously, F is bounded below, and thus, to prove existence of minimisers of F over [A], we only have to prove that F is sequentially coercive and sequentially lower semicontinuous over [A]. Once this is done, the conclusion follows immediately by the direct method in the calculus of variations.
Step 1: F is sequentially coercive. As [A] is sequentially weakly closed in the reflexive space . Then, B = A + d * ψ for some exact 2-form ψ ∈ W 2,2 (M, Λ 2 T * M ). Notice that F B = F A + d d * ψ and that d d * ψ = −∆ψ because ψ is an exact form belonging to W 2,2 (M, Λ 2 T * M ). Also, recall that ∆ψ L 2 (M ) is a norm equivalent to ψ W 2,2 (M ) on exact forms of class W 2,2 . Thus, using Young's inequality, where C 1 , C 2 > 0 are constants independent of B (that is, independent of ψ). Thus, F is sequentially coercive over [A], for every choice of D 0 , A, and u.
Step 2: F is sequentially weakly lower semicontinous. Let {B j } be a sequence in [A]. Then, for each j ∈ N there is an exact 2-form ψ j ∈ W 2,2 (M, is compact, and therefore d * ψ j → d * ψ strongly in L 2 (M, T * M ) as j → ∞ and almost everywhere on a (not relabelled) subsequence. From Fatou's lemma it then follows that On the other hand, ∆ψ j → ∆ψ weakly in L 2 (M, Λ 2 T * M ), and hence we have (recall that . To see that F B satisfies the London equation, take any η ∈ C ∞ (M, Λ 2 T * M ) and set B t := B + t d * η for t ∈ R. By Hodge decomposition, only the exact part of η contributes to B t , whence F(B) ≤ F(B t ) for every t ∈ R and every choice of η ∈ C ∞ (M, Λ 2 T * M ). Thus, we must have d d t t=0 F(B t ) = 0, and so  Since we want to prove a Γ-convergence result, and since |log ε| is the energy scaling of minimisers of G ε according to Remark 2, it is natural to be concerned with sequences We are now going to explore the consequences of this additional assumption.
By (3.7) and the minimality of B ε , we have up to a constant depending only on M , and therefore −∆ψ ε L 2 (M ) |log ε| 1/2 .
for all ε > 0, and the right hand side tends to zero sending ε → 0. Consequently, recalling (1.4), it follows which is the conclusion.
as follows. First, we replace each u ε with v ε , the essentially bounded section associated with u ε by (2.2). Secondly, we replace each A ε with a corresponding minimiser
Remark 3.6. In view of (3.11) and (3.12), it is always possible to pass to an optimised sequence in the proof of Statement (i) of Theorem A (i.e., to replace, in the proof, the given sequence {(u ε , A ε )} by any associated optimised sequence). As we will see later, a decisive advantage of doing so is that the curvature of the connections of an optimised sequence satisfy the London equation (8).
It is convenient to introduce the following notation: if Φ ∈ W 2,2 (M, S 1 ) is a gauge transformation and A ∈ W 1,2 (M, T * M ), we set (cf. Remark 2.9) (3.15) The next technical lemma produces a family of gauge transformations playing a crucial rôle in the whole rest of this section.
Remark 3.8. In general, we cannot assert that the original sequence {A ε } satisfies (3.16), as we have no control on the co-exact part of the Hodge decomposition of the maps A ε (i.e., on ϕ ε ).
We immediately make use of Lemma 3.7 to prove the following corollary. . Then, up to extraction of a (not relabelled) subsequence, J(u ε , A ε ) → πJ * in W −1,p (M ) for any 1 ≤ p < n n−1 as ε → 0, where J * is a bounded measure with values in 2-forms. In addition, ⋆J * is an integer-multiplicity rectifiable (n − 2)-cycle belonging to C.
Proof. Let {Φ ε } be the family of gauge transformations in Lemma 3.7. In view of (3.17), we can apply Theorem C to the sequence {Φ ε u ε }. By Statement (i) of Theorem C, we obtain a (not relabelled) subsequence {Φ ε u ε } and a bounded measure J * with values in 2-forms such that J(Φ ε u ε ) → πJ * in W −1,p (M ) for any 1 ≤ p < n n−1 as ε → 0. Since we already know from Statement (i) of Theorem C that ⋆J * is a current with the desired properties, it only remains to prove that J(u ε , A ε ) → πJ * in W −1,p (M ) for any 1 ≤ p < n n−1 as ε → 0. To prove this, we observe that, by the gauge-invariance of the Jacobians, we have J(u ε , A ε ) = J(Φ ε u ε , Φ ε · A ε ) for all ε > 0. Thus, for all ε > 0. By definition of J * , we need only to show that To this purpose, we recall that from (1.4) we have for all ε > 0. By (3.16) and the energy estimate (3.17) (or, equivalently, (3.7), since |Φ ε u ε | = |u ε | a.e.), for each ε > 0 it holds for all ε > 0. Then (3.19) follows by using Lemma C.1 exactly as in the last part of Lemma 2.1. The conclusion is now immediate by triangle inequality.
Remark 3.10. In view of (3.12), Corollary 3.9 proves a half of Statement (i) of Theorem A.
The following proposition is the last piece of information we need to combine the above results with Theorem C to deduce the lower bound (11), concluding the proof of Theorem A. Here we will use in a crucial way the fact that the curvatures F Aε satisfy the London equation (8), as alluded in Remark 3.6. (iii) Up to a further (not relabelled) subsequence, {Φ ε ·A ε } converges strongly in W 2,p (M, T * M ), for any 1 ≤ p < n n−1 , to some A * , where A * writes for ψ * ∈ W 3,p (M, Λ 2 T * M ) an exact 2-form and ζ * a harmonic 1-form.
Next, since the space Harm 1 (M ) is finite-dimensional and {ζ εn } is bounded in the L ∞ (M )norm, {ζ εn } is also bounded with respect to the W 2,p (M )-norm. In addition, we can also extract from {ζ εn } a (not relabelled) Cauchy sequence, which is a fortiori a Cauchy sequence in W 2,p (M, T * M ), for any 1 ≤ p < n n−1 , and hence it converges in W 2,p (M, T * M ) to some ζ * . Again, by Hodge-decomposition and strong convergence, ζ * is still a harmonic 1-form.
Thus, up to a not relabelled subsequence, and letting A * := d * ψ * + ζ * , we have Φ ε · A ε → A * in W 2,p (M, T * M ) as ε → 0, for any 1 ≤ p < n n−1 . We now have at disposal everything we need to prove Theorem A.
Proof of Theorem A. (i) As emphasized in Remark 3.6, we can always pass to an optimised sequence associated with {(u ε , A ε )}. Hence, we may assume, for notational convenience, that {(u ε , A ε )} is already optimised. Let {Φ ε } ⊂ W 2,2 (M, S 1 ) be, once again, the family of gauge transformations of Lemma 3.7. By (3.17), we can apply Statement (i) of Theorem C to the sequence {Φ ε u ε }. Let J * be the bounded measure with values in 2-forms associated with {Φ ε u ε } by Statement (i) of Theorem C. The claimed convergence of the gauge-invariant Jacobians J(u ε , A ε ) to J * follows from Corollary 3.9 (and, back to the original sequence, Remark 3.10). Therefore, we still have to prove only the lower bound (11).
To infer (11), we note that, by the gauge-invariance of G ε (u ε , A ε ), it suffices to prove that Once we proved (3.21), the conclusion follows immediately from Statement (i) of Theorem C. Towards the proof of (3.21), we notice that, by interpolation, Proposition 3.11, and the (continuous) Sobolev embedding As remarked, the claimed conclusion now follows from (i) of Theorem C (and, back to the original, not necessarily optimised, sequence {(u ε , A ε )}, inequality (3.11)).
(ii) Given any S * ∈ C, let J * := ⋆S * be the corresponding measure with values in 2-forms J * and denote {u ε } ⊂ W 1,2 (M, E) the recovery sequence given by (ii) of Theorem C. Then the because the inequality holds for E ε (u ε ) by (ii) of Theorem C and because for all ε > 0 we have , where as usual F 0 is the curvature of the reference connection D 0 . The proof is finished. Remark 3.13. There is a local analogue of the Γ-lower inequality for G ε . More precisely, let {(u ε , A ε )} ⊂ W 1,2 (M, E) × W 1,2 (M, T * M ) be a sequence that satisfies (3.7) and let V ⊂ M an arbitrary open set. Up to extraction of a subsequence, assume that J(u ε , A ε ) → πJ * in W −1,p (M ) for any p with 1 ≤ p < n/(n − 1). Then, there holds The proof of (3.23) is completely analogous to the proof of (3.22) above and is based on the local Γ-lower inequality for E ε (Proposition 2.14).
Corollary B is now an almost immediate consequence of Theorem A.
Proof of Corollary B. Let (u min ε , A min ε ) be a minimiser of G ε in W 1,2 (M, E) × W 1,2 (M, T * M ), for any ε > 0. As the class C is non-empty, Theorem A and a comparison argument imply that for some ε-independent constant C. By applying Theorem A again, we find a limit 2-form J * such that, up to extraction of a subsequence, J(u min ε , A min ε ) → πJ * in W −1,p (M ) for any p < n n−1 . Moreover, ⋆J * ∈ C and, by general properties of Γ-convergence, ⋆J * is a chain of minimal mass in C.
As (u min ε , A min ε ) is a minimiser of G ε it follows that |u ε | ≤ 1 (by a truncation argument, along the lines of Lemma 2.1) and that {(u min ε , A min ε )} is an optimised sequence. Then, Proposition 3.11 implies that the curvatures F A min ε converge to a limit F * in W 1,p (M ) for any p < n n−1 . Each F A min ε satisfies the London equation (by the arguments of Proposition 3.2). By passing to the limit as ε → 0 in (3.25), it follows that F * satisfies the London equation It only remains to prove that the rescaled energy densities µ ε := µ ε (u min ε , A min ε ) (defined in (12)) converge to the total variation π |J * |. Due to (3.24), we can extract a subsequence in such a way that µ ε ⇀ µ * , weakly as measures, as ε → 0. Equation (3.23) implies that for any open set V ⊆ M such that µ * (∂V ) = 0. As µ * is a bounded Borel measure, it follows that π |J * | ≤ µ * as measures. However, Statement (ii) in Theorem A implies that Therefore, µ * = π |J * |, as claimed.
for any p < n/(n − 1). Moreover, F * satisfies the London equation (3.26). The proof of this claim follows by the same arguments we used in the proof of Corollary B, word by word. Indeed, the assumption that (u min ε , A min ε ) is a sequence of minimisers is only needed to show that the limit J * is area-minimising in its homology class. As for the rest, the arguments of Corollary B only depend on the fact that F Aε satisfies the London equation (8) and that Proposition 3.11 can be applied. But, according to Remark 3.12, both these facts continue to hold for any sequence of critical points satisfying (3.7).
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Declarations. The authors have no competing interests to declare that are relevant to the content of this article.

A.1 Sobolev spaces of sections and differential forms
We recall below the main definitions and facts concerning several spaces of differential forms and, more broadly, of section of Hermitian vector bundles. We shall do so in a slightly greater generality than strictly needed in this work to make the presentation more transparent and the comparison with the relevant literature easier. The main reference for this appendix is [51], especially Chapters 4,5,9,and 19, where the abstract framework is developed in a much more general context (and using the language of category theory, that we avoid for reader's convenience). A more recent useful reference is [34,Chapter 1]. Both in [51] and in [34] equivalence with other approaches to Sobolev spaces of sections is discussed. (See also [48,Chapter 10], [56,Chapter 1], [65,Appendix B].) Let K be either R or C, and let π : E → M be a K-vector bundle of rank ℓ over a C ∞smooth compact orientable Riemannian manifold M = (M n , g) without boundary. We assume the differentiable structure of M is fixed once and for all.

Regular bundle atlases. Using the compactness of M , it is easily shown that it is always possible to find finite bundle atlases
where N ∈ N, the maps ϕ i : U i → R n are local charts and χ i : π −1 (U i ) =: E| U i → U i × K ℓ local trivialisations, so that: (A1) For all i ∈ {1, 2, . . . , N }, U i is contractible and, moreover, ϕ i (U i ) =: Ω i ⊂ R n is a bounded contractible open set with smooth boundary.
(A2) For all i ∈ {1, 2, . . . , N }, (U i , ϕ i ) can be extended to a smooth chart (V i , ψ i ) contained in the differentiable structure of M , so that U i ⊂ V i and ϕ i = ψ i | U i . This ensures that, for all i, j ∈ {1, 2, . . . , N }, whenever is smooth up to the boundary of ϕ j (U i ∩ U j ) (hence, all its derivatives are bounded).
this comes for free, as the U i are contractible). In addition, each local trivialisation χ i : π −1 (U i ) → U i × K ℓ extends to a smooth map over the corresponding coordinates patches V i associated with U i as in (A2) (up to slightly shrinking V i , if necessary). Consequently, are the charts of E (as an (n + ℓ)-manifold if K = R, and as an (n + 2ℓ) real manifold if K = C) and notice that, under our assumptions, all the derivatives of all coordinate changes χ i • χ j −1 : Ω j × K ℓ → Ω i × K ℓ are bounded. Consequently, denoting pr 2 the projection onto the second factor of a product of the type Ω × K ℓ , with Ω ⊂ R n , the maps pr 2 • χ i • χ j −1 : Ω j × K ℓ → K ℓ are smooth and all their derivatives are bounded.
For convenience, we refer to atlases of M satisfying (A1) and (A2) as regular atlases and to bundle atlases satisfying (A1)-(A3) as regular bundle atlases. We call regular the charts of regular atlases and of regular bundle atlases.
Remark A.1. Since we always cover M by contractible open sets U i , every vector bundle over M , and not only the given bundle π : E → M , trivialises over them.
Remark A.2 (Normal coordinates). As it is well-know (see, e.g., [59, pp. 166-167]), around any point x 0 ∈ M , one can choose normal coordinates so that in the geodesic ball B δ (x 0 ) centered at x 0 , it holds , and consequently Given any atlas of M , by compactness it can be refined so to have a regular atlas in which the local coordinates are normal coordinates.  1, 2, . . . , N ) of u belongs to the usual Sobolev space W m,p (Ω i , K ℓ ). We endow W m,p A (M, E) with the norm

Sobolev spaces of sections of vector bundles
where the inner sum runs over all multi-indexes α of length at most m. Of course, we let . Armed with the above definition and (A1)-(A3), it is not difficult to prove that (cf. e.g., [51,34,56,65]) , and reflexive if p ∈ (1, ∞).  Here, q := p ′ is the Hölder-conjugate exponent of p.
Remark A.4. To simplify notations, we will henceforth drop the subscript A when we denote the norm of a section u ∈ W m,p (M, E). More precisely, we will write u W m,p (M,E) to mean actually that we have fixed a regular bundle atlas A (and a subordinate partition of unity) and we are evaluating u W m,p A (M,E) accordingly to (A.2). Remark A.5. If E → M is a trivial K-bundle of rank ℓ, i.e., if E = M × K ℓ , we can identify W m,p (M, E) and W m,p (M, K ℓ ). Indeed, if u : M → M × K ℓ is a measurable section, thenũ := pr 2 • χ•u : M → K ℓ is a measurable function. Vice versa, ifũ : M → K ℓ is a measurable function, the map x → (x,ũ(x)) gives rise to a section u ∈ Γ(M, E). Moreover, the local representations of u andũ coincide a.e. with respect to the Lebesgue measure. Hence, u ∈ W m,p (M, M × K ℓ ) if and only ifũ ∈ W m,p (M, K ℓ ), and the norms are the same. Therefore, we can identify these two spaces, and we shall do so when convenient even without explicit mention. Sobolev spaces of differential forms Let 0 ≤ k ≤ n be an integer and E = Λ k T * M , the bundle of k-covectors over M . For m ≥ 0 an integer and p ∈ [1, ∞], let the space W m,p (M, Λ k T * M ) be defined according to Definition A.1. Then, the spaces W m,p (M, Λ k T * M ) agree with the Sobolev spaces of differential k-forms over M considered in [46,57,21,38]. For the purpose of exposition, it is convenient to refer this definition of Sobolev spaces of differential k-forms over M to as the classical definition.
Remark A.6. Recall that the bundles Λ k T * M → M are constructed canonically starting from any atlas of M . In particular, any regular atlas {(U i , ϕ i )} N i=1 for M induces a regular bundle atlas A k for Λ k T * M → M for every k ∈ {0, 1, . . . , n}. Thus, for convenience, in this paragraph we use the symbol A to denote the given atlas for M , i.e., , although in the previous paragraph it has been used to denote bundle atlases. for any two measurable maps ω, η : M → Λ k T * M such that the right hand side is welldefined. We say that ω ∈ L 1 (M, Λ k T * M ) has weak exterior differential d ω if there exists Ω ∈ L 1 (M, Λ k+1 T * M ) such that it holds for all smooth test forms η ∈ C ∞ (M, Λ k+1 T * M ). In such case, we set d ω := Ω. Symmetrically, we say that ω ∈ L 1 (M, Λ k T * M ) has weak exterior codifferential d * ω if there exists Ψ ∈ L 1 (M, Λ k−1 T * M ) such that the equation holds for every η ∈ C ∞ (M, Λ k−1 T * M ), and we set d * ω := Ψ. Clearly, when they exists, d ω and d * ω are unique, and they coincide with the classical exterior differential and codifferential of ω if ω is smooth. Thus, for m ≥ 1, the operators are well-defined, linear and continuous. For p ∈ (1, ∞), they are the unique extensions by linearity and density of the classical exterior differential and co-differential. Thus, by density, we have the following "integration by parts" formula: if p, q ∈ (1, ∞) satisfy 1/p + 1/q = 1, then The explicit expressions of d, d * in coordinates will not be needed in this work. Formally, they are the same as in the classical case (see, e.g., [46,Chapter 7]). They can be expressed in terms of the metric g of M on the Levi-Civita connection on TM (which classically induce canonical Riemannian metrics and linear connections over all tensor bundles of M ), see e.g. [56,Chapter 1].
Remark A.7. When defined, d, d * satisfy the same properties as in the classical case. For instance, they are local and commute with restrictions.
It has been firstly shown in [57] that, for M compact and without boundary, the classical definition of W 1,p (M, Λ k T * M ) is equivalent to the following geometrical one [57,Proposition 4.11]: Here, we are using the notation of [57], according to which |∇ω| p := , and ϕ ≡ (x 1 , . . . , x n ) are the local (regular) coordinates on U .
The proof that W 1,p (M, Λ k T * M ) = W 1,p (M, Λ k T * M ) then amounts to prove the reverse continuous embedding. The latter stems on the following L p -version of Gaffney's inequality ([57, Proposition 4.10]).
Proposition A.8 (Gaffney's inequality). Let M be a compact orientable smooth Riemannian manifold without boundary. Then a regular bundle atlas A of M can be found so that the following happens: there exists a positive constant C p , depending on p, n, and A, so that for all ω ∈ W 1,p (M, Λ k T * M ) and all integers 0 ≤ k ≤ n.
Gaffney's-type inequalities hold, for compact manifolds without boundary, also for higher order derivatives. More precisely, we have the following result.  [21,Proposition 4.2.2]). Then the local inequalities can be glued together (using crucially the fact that A is regular) to give (A.5) as in [57,Proposition 4.10]. This concludes the proof if ω is smooth. For general ω ∈ W m,p (M, Λ k T * M ), the result follows by density.
Remark A. 10. In particular, the classical Sobolev spaces W 2,p (M, Λ k T * M ) coincides, when M is as in Proposition A.9, with the space of measurable differential k-forms ω such that for any ω ∈ W 2,p (M, Λ k T * M ) and some constant C that depends only on M , k and p.
Finally, the following fundamental result is proved in [57].
Proposition A.13 (L p -Hodge decomposition, [57,Proposition 6.5]). Let M be a smooth compact oriented Riemannian manifold without boundary and 1 < p < ∞. For any integer 0 ≤ k ≤ n, we have Consequently, any ω ∈ L p (M, Λ k T * M ) can be uniquely written as is a harmonic k-form. In addition, there exists a constant C > 0, depending only on p, k and M , such that there holds for every ω ∈ L p (M, Λ k T * M ), where ϕ, ψ and ξ are as in (A.15).
Remark A.14. The choice of using the L ∞ -norm for the harmonic part of ω in estimate (A.16) is somewhat arbitrary. However, since Harm k (M ) has finite dimension, the L ∞ -norm can be replaced by any other norm (up to enlarging C, if necessary).
For later reference, we point out the following elliptic regularity lemma, immediate consequence of Proposition A.13 and the Open Mapping Theorem.
for some constant C p,j depending only on M , j, k, p.
Proof. Existence and uniqueness readily follow from Proposition A.13. Thus, for any j ≥ 0, holds for all exact k-forms ψ ∈ W 2,p (M, Λ k T * M ). (If ψ is co-exact, d * ψ must be replaced by d ψ in the right hand side of (A. 19).) This fact is used in Section 3.
To conclude this section, we notice that neither Gaffney's inequality nor the L p -Hodge decomposition hold for p = 1, as shown in [5]. However, in the same paper it also proven that Green's operator exists even in this case as a map from measure k-forms into W 1,p (M, Λ k T * M ) for every 1 < p < n n−1 . Importantly, measure k-forms can be regarded as Radon vector measures on M with values in k-forms, as [5,Proposition 2.2] shows. In the language of the present paper, measure k-forms are simply k-currents with finite mass (cf. [5, Definition 2.1] and Appendix B).

A.2 Hermitian line bundles, connections, and weak covariant derivatives of Sobolev sections
A Hermitian metric on a complex line bundle E → M is an assignment, for each x ∈ M , of a positive definite Hermitian form h x : E x × E x → C that is smooth in the sense that, for all sections u 1 , u 2 ∈ C ∞ (M, E), the function x → h x (u 1 (x), u 2 (x)) is smooth. In this case we say that E → M is a Hermitian line bundle. The typical fibre of E → M is of course C. The structure group of a Hermitian line bundle E → M automatically reduces to U(1) [44, pp. 280-1]. Naturally, the metric allows to identify E ′ and E. Associated with a Hermitian metric, there is a canonical scalar product, i.e., its real part, that we denote ·, · . In other words, we set ·, · := 1 2 (h +h).
A (smooth) connection D on a vector bundle E → M is a linear map satisfying Leibniz' rule: For every fixed u ∈ C ∞ (M, E), we can view D u a map taking a vector field on M as argument and giving back a section D u(X) of E → M . We set D X u := D u(X) and call D X u the weak covariant derivative of u with respect to X.
A metric connection on a Hermitian line bundle E → M is a connection D that is compatible with the metric, i.e., satisfying D h ≡ 0. This implies Explicitly, (A.21) means that, for every pair of sections u, v ∈ C ∞ (M, E) and every smooth vector field X ∈ C ∞ (M, TM ), there holds We recall the following important facts: • For every u ∈ C ∞ (M, E) and every X ∈ C ∞ (M, TM ), the value (D X u)(x) of D X u at each x ∈ M depends only on X(x) and the values of u along any smooth curve representing X(x) [44, p. 502]. In fact, D A is a local operator and behaves naturally with respect to restrictions [44, Section 12.1].
• Let U ⊂ M be an open set so that E → M is trivial over U , χ U : E| U → U × C a corresponding local trivialisation, and e U a reference section for E → M over U . Then every u ∈ C ∞ (M, E) writes as u = ue U for some smooth complex-valued function u and we have, with respect to the local trivialisation χ U , . Denoting A U the connection 1-form of D with respect to χ U , then the transformation law holds. From (A.22) it is readily seen that d A U does not depend on the local trivialisation. with canonical isomorphism. Thus, we can identify A with a 1-form with purely imaginary coefficients. However, it is customary to assume instead that A is real valued, writing −iA in place of A. We then rewrite (A.23) as Explicitly, (A.24) means that, for every u ∈ C ∞ (M, E) and every smooth vector field X on M , we have D A, X u = D 0, X u − iA(X)u.
The curvature D 2 A of a connection D A is given by the following formula: for all u ∈ C ∞ (M, E) and all X, Y ∈ C ∞ (M, TM ), One easily checks [44,Section 12.5] that there exists a closed End(E)-valued 2-form F A , called the curvature form of D A , such that As for A, if D A is a metric connection on a Hermitian line bundle, F A is an Ad(E)-valued 2-form taking purely imaginary values in any local trivialisation. Thus, F A is identified with a 2-form on M which is assumed to be real-valued, replacing F A with −iF A in the above formula. Then, denoting F 0 the curvature form of the reference connection, it holds So far, we have dealt with smooth sections and smooth connections. We now extend the previous discussion to Sobolev sections and connections. To this end, we have to define the concept of weak covariant derivative of a non-smooth section u : M → E. For the moment being, we still assume A is a smooth 1-form on M .
The first ingredient we need is the extension, for every integer 0 ≤ k ≤ n, of D A to an operator from C ∞ (M, Λ k T * M ⊗ E) into C ∞ (M, Λ k+1 T * M ⊗ E). This is standardly done by introducing the exterior covariant derivative induced by D A , which we denote d A . For the definition of d A , we address the reader to [44,Section 12.9]. The properties of d A are formally similar to those of D A and they are summarised in [44,Theorem 12.57]. Here we only stress that, obviously, d A coincides with D A on C ∞ (M, E), i.e., if k = 0.
Next, we extend * to an operator from Γ(M, Λ k T * M ⊗ E) to Γ(M, Λ n−k T * M ⊗ E), which we still denote * . To this purpose, it is enough to define the action of * on simple elements of Γ(M, Λ k T * M ⊗ E) by letting for u ∈ Γ(M, E) and ω ∈ Γ(M, Λ k T * M ). The rule (A.27), extended linearly, gives a meaning to * σ for every σ ∈ Γ(M, Λ k T * M ⊗ E) and 0 ≤ k ≤ n integer. Using the scalar product associated with the metric of E and the scalar product of k-forms induced by the metric, we define for every u 1 , u 2 : M → E measurable sections and measurable k-forms ω 1 , ω 2 , and a.e. x ∈ M . From (A.28) we define a corresponding L 2 -product anytime the right hand side of (A.29) exists. Once again, extending (bi)linearly the rule (A.29), we can define the L 2 -product of arbitrary σ 1 , σ 2 ∈ Γ(M, Λ k T * M ⊗ E) by letting anytime the integral at right hand side exists. With all this machinery at disposal, we can define the formal adjoint of d A as the operator which is formally adjoint to D A with respect to the L 2 -product (A. 30). An explicit computation yields (cf., e.g., [43,Section 4.2]) Remark A.17. Although of no use in this work, we record the following fact which will be needed in the forthcoming work [22]. We can associate with D A its formal adjoint D * A , i.e., its adjoint with respect to the L 2 -product ((·, ·)), as follows. Given Remark A.18. The pointwise scalar product ·, · and the L 2 -product ((·, ·)) will be simply denoted ·, · and (·, ·) respectively anytime no ambiguity arises.

B Currents and form-valued distributions
Currents. We recall some standard terminology for currents on smooth manifolds, and refer e.g. to [30,58] for more details. for any k-current S and any smooth (k − 1)-form ω. The boundary operator is sequentially continuous with respect to the weak * convergence of distributions. Given a point x ∈ M and a k-covector ω ∈ Λ k T x M * , we define the comass of ω as (here, |·| is the norm on Λ k T x M induced by the Riemannian metric on M ). The comass is a norm on Λ k T * M , which does not coincide with the norm |·| induced by the Riemannian metric. However, there exists a constant C n,k , depending on n and k only, such that for any ω ∈ Λ k T * x M and any x ∈ M . The equality ω = |ω| holds if and only if ω is simple. For any k-current S, the mass of S is defined as Let E ⊂ M be a k-rectifiable set, oriented by a measurable, simple k-vector field v E : E → Λ k TM that is tangent to E and satisfies |v E | = 1 at H k -almost any point of E. Let θ : E → Z be a measurable function. We denote by E, θ the current carried by E with multiplicity θ, defined by for any smooth k-form τ . In case θ = 1 identically, We write E := E, 1 . A current that can be written in the form (B.5) is called an integer-multiplicity rectifiable k-current. The mass of an integer-multiplicity rectifiable current is given by We denote by R k (M ) the set of integer-multiplicity rectifiable k-currents with finite mass. (with the understanding that inf ∅ = +∞).
Form-valued distributions. We equip the space of smooth k-vector fields, C ∞ (M, Λ k TM ), with the topology induced by the family of C h -seminorms, for all integers h ≥ 1. The space of distributions with values in k-forms is defined as the topological dual of smooth k-vectors, D ′ (M, Λ k T * M ) := (C ∞ (M, Λ k TM )) ′ . The differential and the codifferential extends to operators on form-valued distributions, by duality: The proof of Proposition B.1 is a rather direct application of the definitions above and we omit it for the sake of brevity.
Thus, as each µ k,i is the push-forward of ρ i ν k through ϕ i , and hence for every k = 1, 2, . . . , d, and the claimed conclusion follows immediately.
Proof of Lemma C.1. Recalling the (easy but useful) observation at the end of [5, p. 462], by Nash theorem, M ֒→ R n+r isometrically for some r ∈ N, hence Λ k T * M can be seen as a subbundle of Λ k T * R n+r , and every bounded measure ω with values in k-forms can be regarded as a (Radon) vector measure on M , with values in the Euclidean inner product space Λ k (R n+r ) ′ ∼ = R ( n+r k ) (where the last isomorphism of Euclidean spaces is canonical). Calling V the image of the inclusion I : Λ k T * M ֒→ R ( n+r k ) , V is a finite dimensional Hilbert space space. Then, the conclusion follows from the fact that W 1,p (M, Λ k T * M ) ∼ = W 1,p (M, V) for every p ∈ [1, ∞] (and hence their duals can be identified, too). Indeed, although the underlined isomorphism depends on the isometric embedding J : M ֒→ R n+r , which is generally not unique, any two choices J 1 , J 2 for the embedding give rise to equivalent Banach spaces W 1,p (M, V 1 ), W 1,p (M, V 2 ) because of the compactness of M . Thus, we can apply Lemma C.2 with d = n+r k and reach the conclusion.

C.2 Elliptic regularity
We establish here two results concerning existence, uniqueness and estimates for solutions to London (Lemma C.3) and Poisson (Lemma C.4) equations for differential k-forms and data in W −1,p . Although Lemma C.3 and Lemma C.4 are certainly known to experts, we have not found explicit proofs in the literature. Since they are crucial to our arguments, we provide detailed proofs. for some constant C p depending only on M , k, p.
The proof of Lemma C.3 depends on Gaffney's inequality, see Proposition A.8.
Proof of Lemma C.3. We split the proof into several steps.
Step 1. Let q > 2. We claim the following: for any f ∈ L q (M, Λ k T * M ), the equation (C.1) has a unique solution v ∈ W 2,q (M, Λ k T * M ), which satisfies Step 2. Now, take p ∈ (1, 2). We claim that, for any f ∈ W for some constant C p that depends only on M , k and p. To prove the claim, we will first show existence and uniqueness of a duality solution v ∈ L p (M, Λ k T * M ) and then we will prove that every solution v ∈ L p (M, Λ k T * M ) in the sense of D ′ (M ) is a duality solution.
We notice in first place that p ∈ (1, 2) implies q := p ′ > 2. By Step 1, given h ∈ L q (M, Λ k T * M ), there exists a uniquely determined w h ∈ W 2,q (M, Λ k T * M ) such that w h = (−∆ + Id) −1 h (that it is to say, solving −∆w h + w h = h), which moreover satisfies w h W 2,q ≤ C p h L q , where C p > 0 is constant depending only on p, k and M .
where the constant C p depends only on p, k and M . Consequently, Riesz' theorem implies the existence of a unique v ∈ L p (M, Λ k T * M ) such that i.e., replacing h by the corresponding w = w h ∈ W 2,q (M, Λ k T * M ), ∀w ∈ W 2,q (M, Λ k T * M ), (v, −∆w + w) = f, ⋆w W −2,p ,W 2,q .
Hence, v is the unique duality solution of (C.1) for the given f ∈ W −2,p (M, Λ k T * M ). Since Riesz' theorem also implies v L p = V (L q ) ′ , by (C.9) v satisfies the estimate (C.4).
Conversely, every distributional solution to (C.1) is a duality solution. Indeed, suppose v ∈ L p (M, Λ k T * M ) is a distributional solution to (C.1) for f ∈ W −2,p (M, Λ k T * M ). Then, for every w ∈ C ∞ (M, Λ k T * M ), Step 3. Let f ∈ W −1,p (M, Λ k T * M ) and let v be the unique solution of (C.1). By Step 2 Here is a variant of Lemma C.3, which will also be useful. for some constant C p depending only on M , k, p.
Proof. We proceed in three steps.
Step 1 (Existence and uniqueness of a duality solution). Set q := p ′ . We claim that for any f ∈ W −2,p (M, Λ k T * M ) such that f, ⋆ξ W −2,p ,W 2,q = 0 for every ξ ∈ Harm k (M ) there exists a unique distributional solution v ∈ L p (M, Λ k T * M ) to (C.14), which satisfies the estimate v L p f W −2,p .
Step 2 (Every distributional solution is a duality solution). We argue exactly as in (C.10), with the operator −∆ + Id replaced by ∆G.