A reduced model for domain walls in soft ferromagnetic films at the cross-over from symmetric to asymmetric wall types
aa r X i v : . [ m a t h . A P ] S e p A reduced model for domain walls in soft ferromagnetic filmsat the cross-over from symmetric to asymmetric wall types.
Lukas Döring ∗ Radu Ignat † Felix Otto ∗ January 18, 2018
Abstract
We study the Landau-Lifshitz model for the energy of multi-scale transition layers– called “domain walls” – in soft ferromagnetic films. Domain walls separate domains ofconstant magnetization vectors m ± α ∈ S that differ by an angle α . Assuming trans-lation invariance tangential to the wall, our main result is the rigorous derivation of areduced model for the energy of the optimal transition layer, which in a certain parame-ter regime confirms the experimental, numerical and physical predictions: The minimalenergy splits into a contribution from an asymmetric, divergence-free core which per-forms a partial rotation in S by an angle θ , and a contribution from two symmetric,logarithmically decaying tails, each of which completes the rotation from angle θ to α in S . The angle θ is chosen such that the total energy is minimal. The contribution fromthe symmetric tails is known explicitly, while the contribution from the asymmetric coreis analyzed in [7].Our reduced model is the starting point for the analysis of a bifurcation phenomenonfrom symmetric to asymmetric domain walls. Moreover, it allows for capturing asym-metric domain walls including their extended tails (which were previously inaccessibleto brute-force numerical simulation). Keywords: Γ -convergence, concentration-compactness, transition layer, bifurcation, micromagnet-ics. MSC:
Submitted to:
Journal of the European Mathematical Society ∗ Max Planck Institute for Mathematics in the Sciences, Inselstraße 22, 04103 Leipzig, Germany (email:[email protected], [email protected]) † Institut de Mathématiques de Toulouse, Université Paul Sabatier, 118 Route de Narbonne, 31062Toulouse, France (email: [email protected]) Introduction
We consider the following model: The magnetization is described by a unit-length vectorfield m = ( m , m , m ) : Ω → S , where the two-dimensional domain Ω = R × ( − , corresponds to a cross-section of the sample that is parallel to the x x -plane. The following“boundary conditions at x = ±∞ ” are imposed so that a transition from the angle − α to α ∈ (0 , π ] is generated and a domain wall forms parallel to the x x -plane (see Figure 1): m ( ±∞ , · ) = m ± α := (cos α, ± sin α, , (1)with the convention: f ( ±∞ , · ) = a ± ⇐⇒ Z Ω + | f − a + | dx + Z Ω − | f − a − | dx < ∞ , (2)where Ω + = Ω ∩ { x ≥ } and Ω − = Ω ∩ { x ≤ } . Throughout the paper, we use thevariables x = ( x , x ) ∈ Ω together with the differential operator ∇ = ( ∂ x , ∂ x ) , and wedenote by m ′ = ( m , m ) the projection of m on the x x -plane. Ω x x x Figure 1: The cross-section Ω in a ferromagnetic sample on a mesoscopic level.We focus on the following micromagnetic energy functional depending on a small parame-ter η : E η ( m ) = Z Ω |∇ m | dx + λ ln η Z R | h ( m ) | dx + η Z Ω ( m − cos α ) + m dx, η ∈ (0 , , (3)subject to the boundary conditions (1), where λ > is a fixed constant and h = h ( m ) : R → R stands for the unique L stray-field restricted to the x x -plane that is generated by the We refer to Section 2 for more information on E η and the parameters η and λ . ( ∇ · ( h + m ′ Ω ) = 0 in D ′ ( R ) , ∇ × h = 0 in D ′ ( R ) . (4)The first term of (3) is called the “exchange energy”, favoring a constant magnetization. Thesecond term (called “stray-field energy”) can be written as the ˙ H − ( R ) -norm of the D divergence of m ′ (where m is always extended by outside of Ω ): Z R | h ( m ) | dx = k∇· ( m ′ Ω ) k H − ( R ) := sup (cid:26)Z Ω m ′ · ∇ v dx (cid:12)(cid:12)(cid:12)(cid:12) v ∈ C ∞ c ( R ) , k∇ v k L ( R ) ≤ (cid:27) . The last term in (3) (a combination of material anisotropy and external magnetic field)forces the magnetization to favor the “easy axis” m ± α and serves as confining mechanism forthe tails of the transition layer. We refer to Section 2 for more physical details about thismodel.We are interested in the asymptotic behavior of minimizers m η of E η with the boundarycondition (1) as η ↓ . The main feature of this variational principle is the non-convexconstraint on the magnetization ( | m η | = 1 ) and the non-local structure of the energy (dueto the stray field h ( m η ) ). The competition between the three terms of the energy togetherwith the boundary constraint (1) induces an optimal transition layer that exhibits two lengthscales (cf. Figure 3): • an asymmetric core of size (cid:0) | x | . (cid:1) (up to a logarithmic scale in η ) where themagnetization m η is asymptotically divergence-free (so, generating no stray field) andhence the leading order term in E η is given by the exchange energy; in this region, m η describes a transition path on S between the two directions m ± θ determined by someangle θ . • two symmetric tails of size (cid:0) . | x | . η (cid:1) (up to a logarithmic scale in η ) where m η asymptotically behaves as a symmetric Néel wall: a one-dimensional (i.e., m η = m η ( x ) ) rotation on S := S × { } ⊂ S between the angles θ and α (on the leftand right sides of the core). Here, the formation of the wall profile is driven by thestray-field energy that induces a logarithmic decay of m ,η on these two tails.The constant λ > and the wall angle α play a crucial role in the behavior of a minimizer m η . In fact, for either α ≪ , or α ∈ (0 , π ] arbitrary but λ small, a minimizer is expectedto be asymptotically symmetric (i.e., m η = m η ( x ) ) as η ↓ . However, for sufficiently large λ , there exists a critical wall angle α ∗ where a bifurcation occurs: It becomes favorable tonucleate an asymmetric domain wall in the core of the transition layer.In [10, Section 3.6.4 (E)], Hubert and Schäfer state: Existence and uniqueness of the stray field are a direct consequence of the Riesz representation theoremin the Hilbert space V = n v ∈ L loc ( R ) (cid:12)(cid:12)(cid:12) ∇ v ∈ L , − R B (0 , v dx = 0 o endowed with the norm k∇ v k L : Indeed,by (1), the functional v R Ω (cid:0) m ′ − (cos α, (cid:1) · ∇ v dx is linear continuous on V so that there exists a uniquesolution h = −∇ u with u ∈ V of (4) written in the weak form R R ∇ u · ∇ v dx = R Ω m ′ · ∇ v dx for every v ∈ C ∞ c ( R ) . { E η } η ↓ through the method of Γ -convergence. The limiting reduced model does then show that theminimal energy splits into the separate contributions from the symmetric and asymmetricregions of the transition layer. This makes it possible to infer information on the size of theregions and the conjectured bifurcation from symmetric to asymmetric walls. For details, werefer to Section 1.3. Let α ∈ (0 , π ] and η ∈ (0 , . Observe that for m : Ω → S , finite energy E η ( m ) < ∞ is equivalent to m ∈ ˙ H (Ω , S ) and m ′ ( ±∞ , · ) (2) = (cos α, (which in particular implies | m | ( ±∞ , · ) = sin α , see Lemma 3). In the following we focus on the set of magnetizationsof wall angle α ∈ (0 , π ] with a transition imposed by (1): X α := n m ∈ ˙ H (Ω , S ) (cid:12)(cid:12)(cid:12) m ( ±∞ , · ) = m ± α o . (5)Our main result consists in proving Γ -convergence of { E η } η ↓ , defined on X α ⊂ ˙ H (Ω , S ) ,in the weak ˙ H -topology to the Γ -limit functional E ( m ) = Z Ω |∇ m | dx + 2 π λ (cid:0) cos θ m − cos α (cid:1) , (6)which is defined on a space X ⊂ ˙ H (Ω , S ) :In order to give the definitions of X (see (8)) and the angle θ m associated to m ∈ X (see(7)), we need some preliminary remarks. First, due to the logarithmic penalization of thestray field in (3) as η ↓ , limiting configurations of a family { m η } η ↓ of uniformly boundedenergy E η ( m η ) ≤ C (e.g., minimizers of E η ) are stray-field free. Second, note that for any m ∈ ˙ H (Ω , S ) with ∇ · ( m ′ Ω ) = 0 in D ′ ( R ) (i.e., ∇ · m ′ = 0 in Ω and m = 0 on ∂ Ω )there is a unique constant angle θ m ∈ [0 , π ] such that ¯ m ( x ) := − Z − m ( x , x ) dx = cos θ m for all x ∈ R . (7)Observe that such vector fields have the property m ′ ( ±∞ , · ) = (cos θ m , in the senseof (2) (see (30) and (31) below) and moreover, | m | ( ±∞ , · ) = sin θ m (see Lemma 3 if θ m ∈ (0 , π ) , and Remark 1 below if θ m ∈ { , π } ). We define X as the non-empty (see4ppendix) set of such configurations m that additionally change sign as | x | → ∞ , namely m ( ±∞ , · ) = ± sin θ m in the sense of (2): X := n m ∈ ˙ H (Ω , S ) (cid:12)(cid:12)(cid:12) ∇ · m ′ = 0 in Ω , m = 0 on ∂ Ω , m ( ±∞ , · ) = m ± θ m o . (8)Note, however, that due to vanishing control of the anisotropy energy as η ↓ , a limitingconfiguration m in general satisfies (1) for an angle θ m that differs from α . Remark 1.
Observe that if θ m ∈ { , π } for m ∈ ˙ H (Ω , S ) with ∇ · ( m ′ Ω ) = 0 in D ′ ( R ) – in particular if m ∈ X –, we have m ∈ {± e } : Indeed, since | ¯ m | ≡ in R and | m | = 1 in Ω , we deduce | m | ≡ and m ≡ m ≡ in Ω . We further remark that the first term in the Γ -limit energy (6) accounts for the exchangeenergy of the asymmetric core of a transition layer m η as η ↓ , while the second term in E accounts for the contribution coming from stray field/anisotropy energy through extended(symmetric) tails of the wall configurations at positive η .Our Γ -convergence result is established in three steps. We start with compactness results. Themain difficulty comes from the boundary conditions (1), which are in general not carried overby weak limits of magnetization configurations with uniformly bounded exchange energy.However, since the energy E η is invariant under translations in x -direction, and due tothe constraint (1) in X α , a change of sign in m can be preserved as η ↓ by a suitabletranslation in x . Proposition 1 (Compactness) . Let α ∈ (0 , π ] . The following convergence results hold upto a subsequence and translations in the x -variable :(i) Let { m η } η ↓ ⊂ X α with uniformly bounded energy, i.e., sup η ↓ E η ( m η ) < ∞ . Then m η − ⇀ m weakly in ˙ H (Ω) for some m ∈ X .(ii) Let { m k } k ↑∞ ⊂ X α with uniformly bounded energy E η for η ∈ (0 , fixed, i.e., sup k E η ( m k ) < ∞ . Then m k − ⇀ m weakly in ˙ H (Ω) for some m ∈ X α . Moreover, thecorresponding stray fields { h ( m k ) } k ↑∞ converge weakly in L ( R ) , i.e., h ( m k ) − ⇀ h ( m ) in L ( R ) .(iii) Let { m k } k ↑∞ ⊂ X with uniformly bounded exchange energy, i.e., sup k R Ω |∇ m k | dx < ∞ , such that the angles θ k := θ m k associated to m k in (7) satisfy θ k ∈ [0 , π ] . Then θ k → θ for some angle θ ∈ [0 , π ] and m k − ⇀ m weakly in ˙ H (Ω) for some m ∈ X with θ m = θ (i.e., m ∈ X ∩ X θ ). The main ingredient in Proposition 1 is the following concentration-compactness type lemmarelated to the change of sign at ±∞ : Lemma 1.
Let u k : R → R , k ∈ N , be continuous and satisfy the following conditions: lim sup k ↑∞ Z R | dds u k ( s ) | ds < ∞ , (9) lim sup s ↓−∞ u k ( s ) < and lim inf s ↑∞ u k ( s ) > for every k ∈ N , (10)5 here we denote by dds u k the distributional derivative of the function u k .Then for each k ∈ N , there exists a zero z k of u k and a limit u ∈ ˙ H ( R ) such that u (0) = 0 , u k ( · + z k ) → u locally uniformly in R and weakly in ˙ H ( R ) for a subsequenceand lim sup s ↓−∞ u ( s ) ≤ as well as lim inf s ↑∞ u ( s ) ≥ . (11)The second step consists in proving the following lower bound: Theorem 1 (Lower bound) . Let α ∈ (0 , π ] . For m ∈ X and any family { m η } η ↓ ⊂ X α with m η − ⇀ m in ˙ H (Ω) as η ↓ , the following lower bound holds: lim inf η ↓ E η ( m η ) ≥ E ( m ) . (12)The last step consists in constructing recovery sequences for limiting configurations: Theorem 2 (Upper bound) . For α ∈ (0 , π ] and every m ∈ X there exists a family { m η } η ↓ ⊂ X α with m η → m strongly in ˙ H (Ω) and lim sup η ↓ E η ( m η ) ≤ E ( m ) . (13)As a consequence, one deduces the asymptotic behavior of the minimal energy E η over thespace X α as η ↓ . Corollary 1.
For α ∈ (0 , π ] and θ ∈ [0 , π ] we define E asym ( θ ) = min m ∈ X θ m = θ Z Ω |∇ m | dx and E sym ( α − θ ) = 2 π (cid:0) cos θ − cos α (cid:1) . Then it holds lim η ↓ min m η ∈ X α E η ( m η ) = min m ∈ X E ( m ) = min θ ∈ [0 ,π ] (cid:16) E asym ( θ ) + λ E sym ( α − θ ) (cid:17) . (14) In fact, the optimal angle θ is attained in [0 , π ] . Moreover, every minimizing sequence { m η } η ↓ ⊂ X α of { E η } η ↓ in the sense of E η ( m η ) → min X E is relatively compact inthe strong ˙ H (Ω) -topology, up to translations in x , having as accumulation points in X minimizers of E . One benefit of (14) is splitting the problem of determining the optimal transition layer intotwo more feasible ones: First, the energy of asymmetric walls (i.e. walls of small width)has to be determined (at the expense of an additional constraint on ∇ · m ′ ). Afterwards, aone-dimensional minimization procedure is sufficient to determine the structure of the wallprofile. Direct numerical simulation of (3) has been a difficult endeavor (see [17] and also[10, Section 3.6.4 (E)]). 6 .3 Outlook In the following we briefly discuss an application of our reduced model to the cross-over fromsymmetric to asymmetric Néel wall and point out further interesting (topological) questionsand open problems associated with the energy of asymmetric domain walls.
Bifurcation . The previous result represents the starting point in the analysis of the bifur-cation phenomenon (from symmetric to asymmetric walls) in terms of the wall angle α (seealso [7]). We will prove that there is a supercritical (pitchfork) bifurcation (cf. Figure 2): Thismeans that for small angles α ≪ , the optimal transition layer m η of E η is asymptoticallysymmetric (the symmetric Néel wall); beyond a critical angle α ∗ , the symmetric wall is nolonger stable, whereas the asymmetric wall is. To understand the type of the bifurcation, by(14), we need to compute the asymptotic expansion of the asymmetric energy up to order θ as θ → (since the symmetric part of the energy is quartic for small angles θ, α ≪ , i.e., E sym ( α − θ ) . α ). In fact, we show (see [7]): E asym ( θ ) = 4 πθ + πθ + o ( θ ) as θ ↓ . (15)This allows us to heuristically determine a critical angle α ∗ at which the symmetric Néelwall loses stability and an asymmetric core is generated. Moreover, a new path of stablecritical points with increasing inner wall angle θ branches off of θ = 0 (see Figure 2). Indeed,for small α , combining with (15), the RHS of (14) as function of θ ∈ [0 , α ] has the uniquecritical point θ = 0 if α ≤ α ∗ where the bifurcation angle α ∗ is given by α ∗ = arccos (cid:0) − λ (cid:1) + o (1) , as α → . (Observe that α ∗ ∈ [0 , π ] provided λ ≥ ; therefore, the bifurcation appears only if λ islarge enough.) For α > α ∗ , the symmetric wall becomes unstable under symmetry-breakingperturbations and the optimal splitting angle θ becomes positive; hence, the asymmetric wallbecomes favored by the system. Moreover, the second variation of the RHS of (14) alongthe branch of positive splitting angles is positive so that the bifurcation from symmetric toasymmetric wall is supercritical. Topological degree and vortex singularity . We now discuss topological properties ofstray-field free magnetization configurations: In fact, if m ∈ X satisfies (1) for some angle θ ∈ (0 , π ] , denoting the “extended” boundary of Ω Bdry := ∂ Ω ∪ (cid:18) {±∞} × [ − , (cid:19) ∼ = S , (16)then one can define the following winding number of m on Bdry : due to m = 0 on ∂ Ω as well as m ( ±∞ , · ) = 0 (so, ( m , m ) : Bdry → S ), one obtains (by the homeomorphism(16)) a map ˜ m ∈ H ( S , S ) to which a topological degree can be associated (see, e.g., [4]). Observe that for given α ∈ (0 , π ] the optimal wall angle θ α = argmin ( E asym ( θ ) + λE sym ( α − θ )) ∈ [0 , π ] satisfies the estimate θ α . α . Indeed, by comparison with θ = 0 we have E asym ( θ α ) + 2 πλ (cos θ α − cos α ) ≤ πλ (1 − cos α ) . Omitting E asym ( θ α ) we first obtain θ α → as α ↓ , so that by (15) one deduces that θ α / λ (1 − cos α ) for small α > . From here, the desired estimate follows. table unstable θ αα ∗ Figure 2: Bifurcation diagram for the angle θ of the asymmetric core, depending on theglobal wall angle α .In particular, in the case of smooth ˜ m : S → S , the topological degree (also called windingnumber) of ˜ m is defined as follows: deg( ˜ m ) := 12 π Z S det( ˜ m, ∂ θ ˜ m ) d H where ∂ θ ˜ m is the angular derivative of ˜ m .We will show the following relation between the winding number of m ∈ X on Bdry and topological singularities of ( m , m ) inside Ω : the non-vanishing topological degree of ( m , m ) : Bdry → S generates vortex singularities of ( m , m ) as illustrated in Figure3. By vortex singularity of v := ( m , m ) , we understand a zero of v carrying a non-zerotopological degree. In general, this is implied by the existence of a smooth cycle (i.e., closedcurve) γ ⊂ Ω such that | v | > on γ and deg( v | v | , γ ) = 0 ; the vector field v then vanishes inthe domain bounded by γ . Lemma 2.
Let m ∈ X (i.e. m ∈ ˙ H (Ω , S ) with ∇ · ( m ′ Ω ) = 0 in D ′ ( R ) ) such that (1) holds for some angle θ ∈ (0 , π ] . Suppose that ( m , m ) : Bdry → S has a non-zero windingnumber on Bdry . Then there exists a vortex singularity of ( m , m ) in Ω carrying a non-zerotopological degree. Motivated by Lemma 2, let us introduce the set L θ = { m ∈ X ∩ X θ : deg m = 1 } for a fixed angle θ ∈ (0 , π ] . First of all, we have that L θ = ∅ (see Appendix). Since X = ∪ θ ∈ [0 ,π ] (cid:0) X ∩ X θ (cid:1) , the relation L θ = ∅ obviously implies that X ∩ X θ = ∅ for every θ ∈ (0 , π ) which is essential in our reduced model given by the Γ -convergence program. Anatural question concerns the closure (in the weak ˙ H (Ω) -topology) of the set L θ . This is Naturally, one can address a similar question by imposing an arbitrary winding number n . For the case n = 0 , we analyze this problem in [7] which is typical for asymmetric Néel walls; in particular, for small angles θ , we construct an element m ∈ X ∩ X θ with deg m = 0 and asymptotically minimal energy. Moreover, givenany m ∈ X ∩ X θ with finite energy, one can use a reflection and rescaling argument to define a finite-energymagnetization on Ω with degree (see Remark 5 (iii) ). L θ . Open problem 1.
Is the following infimum inf m ∈L θ Z Ω |∇ m | dx attained for every angle θ ∈ (0 , π ] ? This paper is organized as follows: In Section 2, we explain the relation of (3) to the fullLandau-Lifshitz energy, as well as the physical background of our analysis.In Section 3, we prove the compactness results in Lemma 1 and Proposition 1, which inparticular yield existence of minimizers of E η , E asym ( θ ) and E .Section 4 contains the proofs of the lower and upper bound (Theorems 1 and 2) of our Γ -convergence result and also, the proof of Corollary 1.In the Appendix, finally, we show that the set X ∩ X θ is non-empty for any given angle θ ∈ (0 , π ] . To this end, we construct an admissible configuration in E asym ( θ ) with non-zerotopological degree on the boundary of Ω (i.e., of asymmetric Bloch-wall type). Moreover, weprove Lemma 2. In this section, we denote by ∇ = ( ∂ x , ∂ x , ∂ x ) the full gradient of functions dependingon x = ( x , x , x ) . Recall that the prime ′ denotes the projection on the x x -plane, i.e. ∇ ′ = ( ∂ x , ∂ x ) , x ′ = ( x , x ) . Micromagnetics . Let ω ⊂ R represent a ferromagnetic sample whose magnetization isdescribed by the unit-length vector-field m : ω → S . Assume that the sample exhibitsa uniaxial anisotropy with e = (0 , , as “easy axis”, i.e. favored direction of m . Thewell-accepted micromagnetic model (see e.g. [5, 10]) states that in its ground state themagnetization minimizes the Landau-Lifshitz energy: E D ( m ) = d Z ω |∇ m | dx + Z R | h ( m ) | dx + Q Z ω m + m dx − Z ω h ext · m dx. (17)Here, the exchange length d is a material parameter that determines the strength of theexchange interaction of quantum mechanical origin, relative to the strength of the stray field h = h ( m ) . The stray field is the gradient field h = −∇ u that is (uniquely) generated by thedistributional divergence ∇ · ( m ω ) via Maxwell’s equation ∇ · ( h + m ω ) = 0 in D ′ ( R ) . (18) This question is related to the theory of Ginzburg-Landau minimizers with prescribed degree (see e.g.Berlyand and Mironescu [2]).
Q > is a material constant that measures the relativestrength of the energy contribution coming from misalignment of m with e . The last term,called Zeeman energy, favors alignment of m with an external magnetic field h ext : ω → R . Derivation of our model . We assume the magnetic sample to be a thin film, infinitelyextended in the ( x x ) -plane, i.e. ω = R × ( − t, t ) , where two magnetic domains of almostconstant magnetization m ≈ m ± α have formed for ± x ≫ t . Physically, such a configurationis stabilized by the combination of uniaxial anisotropy and suitably chosen external field h ext = Q cos α e . Moreover, we assume that m and hence, the stray field h = ( h , , h ) are independent of the x -variable so that (17) formally reduces to integrating the energydensity (per unit length in x -direction): E D ( m ) = d Z ω ′ |∇ ′ m | dx ′ + Z R | h ′ | dx ′ + Q Z ω ′ ( m − cos α ) + m dx ′ where ω ′ = R × ( − t, t ) and h ′ = h ′ ( m ) = −∇ ′ u satisfies (4) driven by the D divergenceof m ′ ω ′ . Recall that the prime ′ here denotes a projection onto the coordinate directions ( x , x ) transversal to the wall plane. After non-dimensionalization of length with the filmthickness t , i.e., setting ˜ x ′ = x ′ t , ˜ ω ′ = ω ′ t , ˜ m (˜ x ′ ) = m ( x ′ ) , ˜ u (˜ x ′ ) = u ( x ′ ) t , the above specificenergy (per unit length in x ) is given by ˜ E D ( ˜ m ) = d Z ˜ ω ′ | ˜ ∇ ′ ˜ m | d ˜ x ′ + t Z R | ˜ ∇ ′ ˜ u | d ˜ x ′ + Qt Z ˜ ω ′ ( ˜ m − cos α ) + ˜ m d ˜ x ′ , (19)where the differential operator ˜ ∇ ′ refers to the variables ˜ x ′ = (˜ x , ˜ x ) and ˜ u : R → R is the D stray-field potential given by ˜∆ ′ ˜ u = ˜ ∇ ′ · ( ˜ m ˜ ω ′ ) in D ′ ( R ) . Throughout the section, we omit ˜ and ′ . Symmetric walls . In the regime of very thin films (i.e. for a sufficiently small ratio offilm thickness t to exchange length d , see below for the precise regime), the symmetric Néelwall m is the favorable transition layer: It is characterized by a reflection symmetry w.r.t.the midplane x = 0 , see 3) below. In fact, to leading order in td , it is independent of thethickness variable x , i.e. m = m ( x ) , and in-plane, i.e. m = 0 . The symmetric Néel wall isa two length-scale object with a core of size w core = O ( d t ) and two logarithmically decayingtails w core . | x | . w tail = O ( tQ ) (see e.g. Melcher [15, 16]). It is invariant w.r.t. all thesymmetries of the variational problem (besides translation invariance):1) x → − x , x → − x , m → − m ;2) x → − x , m → − m , m → − m ;3) x → − x , m → − m ;4) Id . A typical, experimentally accessible, soft ferromagnetic material is Permalloy, for which d ≈ nm and Q = 2 . · − . α = π is given by E D ( symmetric Néel wall ) = O ( t w tail w core ) = O ( t t d Q ) (see e.g. [19, 5]). For a symmetric Néel wall of angle α < π , the energy is asymptoticallyquartic in α as it is proportional to (1 − cos α ) (see e.g. [11]). Asymmetric walls . For thicker films, the optimal transition layer has an asymmetric core,where the symmetry 3) is broken (see e.g. [8, 9]). The main feature of this asymmetric coreis that it is approximately stray-field free. Hence to leading order, the asymmetric core isgiven by a smooth transition layer m that satisfies (1) and m : ω → S , ∇ · m ′ = 0 in ω and m = 0 on ∂ω. (20)Observe that ( m , m ) : ∂ω → S since m vanishes on ∂ω , so that one can define a topo-logical degree of ( m , m ) on ∂ω (where ∂ω is the closed “infinite” curve (cid:0) R × {± } (cid:1) ∪ (cid:0) {±∞} × [ − , (cid:1) ). The physical experiments, numerics and constructions predict two typesof asymmetric walls, differing in their symmetries and the degree of ( m , m ) on ∂ω :(i) For small wall angles α , the system prefers the so-called asymmetric Néel wall. Itsmain features are the conservation of symmetries 1) and 4) and a vanishing degreeof ( m , m ) on ∂ω (see Figure 3). Due to symmetry 1), the m component of anasymmetric Néel wall vanishes on a curve that is symmetric with respect to the centerof the wall (by x → − x ). Moreover, the phase of ( m , m ) is not monotone at thesurface | x | = 1 .(ii) For large wall angles α , the system prefers the so-called asymmetric Bloch wall. Thesewalls only have the trivial symmetry 4). Another difference is the non-vanishing topo-logical degree on ∂ω (i.e., deg (cid:0) ( m , m ) , ∂ω (cid:1) = ± ). Therefore, a vortex is nucleatedin the wall core, and the curve of zeros of m is no longer symmetric with respect thecenter of the wall (see Figure 3). Moreover, the phase of ( m , m ) is expected to bemonotone at the surface | x | = 1 .The asymmetric wall has a single length scale w core ∼ t and the specific energy comes fromthe exchange energy (see e.g. [19, 5]). It is of the order E D ( asymmetric wall ) = O ( d ) . For small wall angles, the energy of the optimal asymmetric wall is asymptotically quadraticin α (see [5]). Regime . We focus on the challenging regime of soft materials of thickness t close to theexchange length d (up to a logarithm), where we expect the cross-over in the energy scalingof symmetric walls and asymmetric walls (see [19]): Q ≪ and ln Q ∼ ( td ) . The magnetization was obtained by numerically solving the Euler-Lagrange equation corresponding to E asym ( θ ) . To this end, a Newton method with suitable initial data was employed. x x -1-0.5 0 0.5 1 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 x x Figure 3: Asymmetric Néel wall (on the left) and asymmetric Bloch wall (on the right).Numerics. Rescaling the energy (19) by d and setting η := Q t d ≪ and λ := t d ln η > , then λ = O (1) is a tuning parameter in the system, and the rescaled energy, which is to beminimized, takes the form of energy E η given in (3) under the constraint m : Ω = R × ( − , → S , m ( ±∞ , · ) = m ± α ,h = −∇ u : R → R , ∇ · ( h + m ′ Ω ) = 0 in D ′ ( R ) . Observe that the parameter λ measures the film thickness t relative to the film thickness d ln Q characteristic to the cross-over. The limit η ↓ corresponds to a limit of vanishingstrength of anisotropy, while at the same time the relative film thickness td increases in orderto remain in the critical regime of the cross-over. Other microstructures in micromagnetics . In other asymptotic regimes, different pat-tern formation is observed. Let us briefly mention three other microstructures that wererecently studied: the concertina pattern, the cross-tie wall and a zigzag pattern.
Concertina pattern . In a series of papers ([20, 23] among others) the formation and hysteresisof the concertina pattern in thin, sufficiently elongated ferromagnetic samples were studied.While in this case the transition layers between domains of constant magnetization aresymmetric Néel walls, the program carried out for the concertina (a mixture of theoreticaland numerical analysis, and comparison to experiments) serves as motivation for our workon the energy of domain walls in moderately thin films. Moreover, we hope that our analysisof the wall energy is helpful for studying a different route to the formation of the concertinapattern in not too elongated samples as proposed in [24], see also [6].
Cross-tie wall . An interesting transition layer observed in physical experiments is the cross-tiewall (see [10, Section 3.6.4]). It was rigorously studied in a reduced D model (by assumingvertical invariance of the magnetization) where a forcing term amounts to strong planaranisotropy that dominates the stray-field energy (see [1, 21, 22]). For small wall angles θ ∈ (0 , π ] , the optimal transition layer is given by the symmetric Néel wall; for larger angles12 > π , the domain wall has a two-dimensional profile consisting in a mixture of vortices andNéel walls. The energetic cost of a transition in this D model is proportional to sin θ − θ cos θ ,so it is cubic in θ as θ → . This is due to the scaling of the stray-field energy (because ofthe thickness invariance assumption), which makes this reduced D model seem artificial.In the physics literature, it is known that for the full 3D model and large wall angles thecross-tie wall may also be favored over the asymmetric Bloch wall. We hope that our morerealistic wall-energy density confirms and helps to quantify this issue. A zigzag pattern . In thick films, zigzag walls also occur. This pattern has been studied byMoser [18] in a D model with a uniaxial anisotropy in an external magnetic field perpendic-ular to the “easy axis” (rather similar to our model). In fact, zigzag walls are to be expectedthere; however, this question is still open since the upper bound given for the limiting wallenergy through a zigzag construction does not match the lower bound. Recently, in a reduced D model, Ignat and Moser [13] succeeded to rigorously prove the optimality of the zigzagpattern (for small wall angles). This was due to the improvement of the lower bound basedon an entropy method (coming from scalar conservation laws). Remarkably, the function sin θ − θ cos θ plays an important role for the limiting energy density in that context as wellas for the cross-tie wall. In this section we prove compactness results for sequences { m k } k ↑∞ of magnetizations ofbounded exchange energy. As an application we will derive existence of minimizers of E η (for some fixed η ∈ (0 , ) and E asym ( θ ) subject to a prescribed wall angle θ ∈ (0 , π ) , and show that the optimal angle in E is attained (cf. (14)).All these statements are rather straightforward up to one point: The condition of sign-change ± m ( ±∞ , · ) ≥ can in general not be recovered in the limit as shown in Figure 4. ¯ m ,η x Figure 4: The x -average ¯ m ,η of the m -component. The arrow ←→ denotes that the lengthof the corresponding interval grows to + ∞ as η ↓ . Then the limit ¯ m (as η ↓ ) has thesame sign at + ∞ and −∞ .However, we will show that one can always choose zeros x ,η of ¯ m ,η in such a way that m η ( · + x ,η , · ) has the correct change of sign in the limit η ↓ .13n the sequel we denote by C > a universal, generic constant, whose value may changefrom line to line, unless otherwise stated. We start by proving the D concentration-compactness result stated in Lemma 1. Proof of Lemma 1:
Due to (10), the set Z k := { z ∈ R (cid:12)(cid:12) u k ( z ) = 0 } of zeros of u k is non-empty, and up to a translation in x -direction we may assume u k (0) = 0 for all k ∈ N . Step 1:
For every sequence { z k ∈ Z k } k ↑∞ there exist a subsequence Λ ⊂ N and a limit u : R → R such that u k ( · + z k ) → u locally uniformly for k ↑ ∞ , k ∈ Λ . Moreover, we havethe bound Z R | dds u | ds ≤ lim inf k ↑∞ k ∈ Λ Z R | dds u k | ds < ∞ . Indeed, by Cauchy-Schwarz’s inequality, we have for t = ˜ t that | u k ( t ) − u k (˜ t ) | | t − ˜ t | = (cid:0)R t ˜ t dds u k ds (cid:1) | t − ˜ t | ≤ Z R | dds u k | ds ; thus, by (9), we deduce that { u k ( · + z k ) } k ↑∞ is uniformly Hölder continuous with exponent . In particular, since u k ( z k ) = 0 , we also have that { u k ( · + z k ) } k ↑∞ are locally uniformlybounded. Hence, the Arzelà-Ascoli compactness theorem yields uniform convergence on eachcompact interval [ − n, n ] , n ∈ N , up to a subsequence. By a diagonal argument, one finds asubsequence Λ ⊂ N and a continuous limit u : R → R such that u k ( · + z k ) → u locally uniformly for k ↑ ∞ , k ∈ Λ .Moreover, the L ( R ) -estimate on dds u follows from weak convergence in L of dds u k and weaklower-semicontinuity of the L norm. Step 2:
Inductive construction of zeros.
Assume by contradiction that for every sequence { z k ∈ Z k } k ↑∞ , no accumulation point u (w.r.t. to locally uniform convergence) of the se-quence { u k ( · + z k ) } k ↑∞ satisfies (11). We will show by an iterative construction that one canselect a subsequence of { u k } k ↑∞ such that each term u k has asymptotically infinitely manyzeros (i.e., Z k → ∞ as k ↑ ∞ ) with large distances in-between.More precisely, we prove that for every l ∈ N there exist a limit u l ∈ ˙ H ( R ) and subsequences Λ l ⊂ Λ l − ⊂ . . . ⊂ Λ ⊂ N , such that for all k ∈ Λ l there exists an additional zero z lk ∈ Z k of u k with the properties: min ≤ i = j ≤ l | z ik − z jk | → ∞ and u k ( · + z lk ) → u l locally uniformly, as k ↑ ∞ , k ∈ Λ l . In Step 3, we finally show that this construction implies that u ≡ is one of the accumulationpoints of { u k ( · + z k ) } k ↑∞ for z k ∈ Z k a diagonal sequence of these z lk , i.e., (11) is satisfied,in contradiction to our assumption. 14t level l = 1 , we choose the zero z k = 0 of u k for every k ∈ N . Then by Step 1, there existsa subsequence Λ ⊂ N and a limit u ∈ ˙ H ( R ) such that u k ( · + z k ) → u locally uniformly for k ↑ ∞ , k ∈ Λ . By assumption, u does not satisfy (11). Hence, there exists ε > such that for every s > we can find s > s such that u ( s ) ≤ − ε < or u ( − s ) ≥ ε > . By uniform convergence, we also deduce that for every s > there exists an index k s ∈ Λ such that sup [ − s ,s ] | u k ( · + z k ) − u | ≤ ε for k ≥ k s , k ∈ Λ , which in particular implies that u k ( s + z k ) < or u k ( − s + z k ) > , for k ≥ k s , k ∈ Λ . (21)At level l = 2 , we proceed as follows: By the construction at level l = 1 , for every s := n ∈ N we choose as above s ≥ n and k := k n ∈ Λ (here, { k n } n ↑∞ is to be chosen increasing).We also know that u k satisfies (11) which implies by (21) that u k changes sign at the left of − s + z k or at the right of s + z k . Choose z k ∈ Z k as this new zero of u k . Since z k n = 0 , wehave | z k n − z k n | → ∞ as n ↑ ∞ . Let ˜Λ = { k n | n ∈ N } ⊂ Λ be the sequence of these indices. By Step 1, there exist asubsequence Λ ⊂ ˜Λ and a limit u ∈ ˙ H ( R ) such that u k ( · + z k ) → u locally uniformly for k ↑ ∞ , k ∈ Λ . We now show the general construction, i.e. how one obtains the ( l + 1) th set of zeros fromthe construction after the l th step. Indeed, suppose the functions u , . . . , u l , the sequences Λ l ⊂ . . . ⊂ Λ ⊂ N and the zeros z k , . . . , z lk of u k for every k ∈ Λ l have already beenconstructed. We now construct u l +1 , Λ l +1 and z l +1 k for k ∈ Λ l +1 : By assumption, none ofthe limits u j , ≤ j ≤ l , satisfies (11). Hence, there exists ε l > such that for every s > we can find s , . . . , s l ≥ s with the property: (cid:16) u j ( s j ) ≤ − ε l < or u j ( − s j ) ≥ ε l > (cid:17) for every ≤ j ≤ l. By uniform convergence, we also deduce that for every s > there exists an index k s ∈ Λ l such that for every ≤ j ≤ l and every k ≥ k s with k ∈ Λ l : sup [ − s j ,s j ] | u k ( · + z jk ) − u j | ≤ ε l and min ≤ i = j ≤ l | z ik − z jk | ≥ ≤ j ≤ l s j .
15n particular, for every s := n ∈ N we choose as above s , . . . , s l ≥ n and k := k n ∈ Λ l (again, { k n } n ↑∞ is to be chosen increasing). Then we deduce that for all ≤ j ≤ l and k ∈ Λ l : u k ( s j + z jk ) < or u k ( − s j + z jk ) > , and the l intervals { I j := [ z jk − s j , z jk + s j ] } ≤ j ≤ l are disjoint.Since u k satisfies (11), there exists a new zero z l +1 k ∈ Z k \ S lj =1 I j of u k . Indeed, let usassume (after a rearrangement) that these intervals are ordered I < I < · · · < I l . If thereis no zero to the left of I (i.e., on ( −∞ , z k − s ] ) and in-between these l intervals (i.e., on S l − j =1 [ z jk + s j , z j +1 k − s j +1 ] ), then u k must have a negative sign at the right endpoint of eachinterval I j (i.e., u k ( z jk + s j ) < ) with ≤ j ≤ l . In particular, there must be a zero of u k atthe right of I l , that we call z l +1 k .Set ˜Λ l +1 = { k n | n ∈ N } ⊂ Λ l . Then min ≤ j ≤ l | z jk − z l +1 k | → ∞ as k ↑ ∞ , k ∈ ˜Λ l +1 . Finally, by Step 1, there exist Λ l +1 ⊂ ˜Λ l +1 and u l +1 such that u k ( · + z l +1 k ) → u l +1 locally uniformly in R as k ↑ ∞ , k ∈ Λ l +1 , which finishes the construction at the level l + 1 . Step 3:
Construction of vanishing diagonal sequence.
We prove that the assumption in Step2 (i.e. the assumption that no accumulation point of a sequence of translates of { u k } k ↑∞ satisfies (11)) leads to a contradiction:Consider the construction done in Step 2. The sequence { u l } l ↑∞ is uniformly bounded in ˙ H ( R ) . Hence, as in Step 1, there is a subsequence Λ ⊂ N and a function u such that u l → u locally uniformly for l ↑ ∞ , l ∈ Λ . In the following, we prove that u ≡ on R (in particular(11) is satisfied). Indeed, we first observe that u l (0) → u (0) as l ↑ ∞ , l ∈ Λ ; thus, u (0) = 0 . Let now a > and we want to prove that u ( a ) = 0 . For that, let l ∈ Λ and k ∈ Λ l .Then for ≤ j ≤ l , | u k ( a + z jk ) | a = | u k ( a + z jk ) − u k ( z jk ) | a ≤ Z a + z jk z jk | dds u k | ds. For k = k ( a ) ∈ Λ l sufficiently large, the intervals { [ z jk , a + z jk ] } ≤ j ≤ l are disjoint and we have X ≤ j ≤ l | u k ( a + z jk ) | a ≤ X ≤ j ≤ l Z a + z jk z jk | dds u k | ds ≤ Z R | dds u k | ds. Letting k ↑ ∞ , k ∈ Λ l , it follows X ≤ j ≤ l | u j ( a ) | a ≤ lim sup k ↑∞ Z R | dds u k | ds < ∞ .
16e may now let l ↑ ∞ , l ∈ Λ , and deduce that u l ( a ) → . In particular, this shows u ( a ) = 0 .The same argument adapts to the case a < , so that one concludes u ≡ in R .Therefore, taking a diagonal sequence of the functions constructed in Step 2, one can thenfind a family { u k l ( · + z lk l ) } l ∈ Λ converging (locally uniformly) to the limit function u ≡ thatsatisfies (11) in contradiction to our assumption.The following lemma reduces the problem of finding admissible limits (i.e., satisfying thelimit condition (1)) for a sequence of vector fields { m k : Ω → S } k ↑∞ to shifting the x -average ¯ m ,k of the second component m ,k : Lemma 3.
Let m ∈ ˙ H (Ω , S ) satisfy the limit condition m ′ ( ±∞ , · ) = (cos θ, in the ( m m ) -components in the sense of (2) for some angle θ ∈ (0 , π ) . Then Z Ω (cid:12)(cid:12) | m | − sin θ (cid:12)(cid:12) dx < ∞ . If additionally the x -average ¯ m of m satisfies (11) (i.e. ¯ m changes sign), then we have m ( ±∞ , · ) = m ± θ . Remark 2. (i) Note that the assumption θ
6∈ { , π } is crucial: If we consider m : Ω → S given by m ≡ , m ( x ) = ( cos( π x ) if | x | ≤ , − | x | x if | x | > , m ( x ) = ( sin( π x ) if | x | ≤ , sgn( x ) p − m ( x ) if | x | > , then m ′ ( ±∞ , · ) = (1 , (in the sense of (2) ), and m ∈ ˙ H (Ω , S ) since Z Ω |∇ m | dx = 2 Z R | dds m | − m ds ≤ C + 4 Z ∞ | dds m | − m ds < ∞ , but R Ω m dx = ∞ , so that (2) fails for m .(ii) Under the hypothesis of Lemma 3, in the case θ ∈ { , π } , by Remark 1 one may stillconclude that m ∈ {± e } provided that ∇ · ( m ′ Ω ) = 0 in D ′ ( R ) (i.e. m ∈ X ).Proof of Lemma 3. By m + m + m = 1 = sin θ + cos θ and the triangle inequality wehave: Z Ω | m − sin θ | dx ≤ Z Ω | m − cos θ | + m dx ≤ Z Ω | m − cos θ | + m dx < ∞ , where we used | m − cos θ | = (cid:12)(cid:12) m + cos θ (cid:12)(cid:12) (cid:12)(cid:12) m − cos θ (cid:12)(cid:12) ≤ (cid:12)(cid:12) m − cos θ (cid:12)(cid:12) , ′ ( ±∞ , · ) = (cos θ, and | m | ≤ . Since | m − sin θ | = (cid:12)(cid:12) | m | − sin θ (cid:12)(cid:12) (cid:12)(cid:12) | m | + sin θ (cid:12)(cid:12) ≥ sin θ (cid:12)(cid:12) | m | − sin θ (cid:12)(cid:12) and θ ∈ (0 , π ) , it follows that Z Ω (cid:12)(cid:12) | m | − sin θ (cid:12)(cid:12) dx ≤ θ Z Ω | m − sin θ | dx < ∞ . This proves the first part of the lemma. To establish the second part, we note that due to (cid:12)(cid:12) | m | − | ¯ m | (cid:12)(cid:12) ≤ | m − ¯ m | we have Z R (cid:12)(cid:12) | ¯ m | − sin θ (cid:12)(cid:12) dx = 12 Z Ω (cid:12)(cid:12) | ¯ m | − sin θ (cid:12)(cid:12) dx ≤ Z Ω | m − ¯ m | + (cid:12)(cid:12) | m | − sin θ (cid:12)(cid:12) dx < ∞ , (22)where we used the Poincaré-Wirtinger inequality Z R Z − | m − ¯ m | dx dx ≤ C Z Ω | ∂ x m | dx. (23)Since k| ¯ m |k ˙ H ( R ) = k ¯ m k ˙ H ( R ) ≤ √ k m k ˙ H (Ω) < ∞ , we deduce with help of (22) that | ¯ m | − sin θ ∈ H ( R ) ; in particular, | ¯ m ( s ) | → sin θ > as | s | → ∞ .Under the additional assumption lim inf s ↑∞ ¯ m ( s ) ≥ and lim sup s ↓−∞ ¯ m ( s ) ≤ , we de-duce from | ¯ m ( s ) | → sin θ > as | s | → ∞ that | ¯ m ( s ) | = ¯ m ( s ) and | ¯ m ( − s ) | = − ¯ m ( − s ) if s is sufficiently large, so that (22) translates into Z R − | ¯ m + sin θ | dx + Z R + | ¯ m − sin θ | dx < ∞ . Together with (23), this finally yields Z Ω − | m + sin θ | dx + Z Ω + | m − sin θ | dx ≤ Z Ω | m − ¯ m | dx + 4 Z R − | ¯ m + sin θ | dx + 4 Z R + | ¯ m − sin θ | dx < ∞ . We now prove Proposition 1. In fact, we shall prove it in form of the following propositionthat treats all the cases at once: (i) corresponds to η k ↓ , (ii) corresponds to η k ≡ η ∈ (0 , ,and (iii) corresponds to η k ≡ . Proposition 2.
Suppose that the sequences { θ k } k ↑∞ ⊂ (0 , π ) , { η k } k ↑∞ ⊂ [0 , satisfy θ k → θ and η k → η as k → ∞ with θ ∈ (0 , π ) whenever η ∈ (0 , . (24) Suppose further that the sequence { m k } k ↑∞ ⊂ ˙ H (Ω , S ) satisfies m k ∈ ( X θ k , for η k ∈ (0 , ,X ∩ X θ k , for η k = 0 , (25)18 nd ( E η k ( m k ) , for η k > ,E ( m k ) , for η k = 0 , ) is bounded as k → ∞ . (26) Then there exist zeros x ,k of ¯ m ,k such that after passage to a subsequence, there exists m ∈ ˙ H (Ω , S ) such that m k ( · + x ,k , · ) − ⇀ m weakly in ˙ H (Ω) and weak- ∗ in L ∞ (Ω) , (27) h ( m k ) ( − ⇀ h ( m ) , for η ∈ (0 , , → , for η = 0 , ) in L (Ω) , and m ∈ ( X θ , for η ∈ (0 , ,X ∩ X ˜ θ , for η = 0 , with ˜ θ ∈ [0 , π ] . Proof of Proposition 2.
We divide the proof in several steps:
Step 1:
Compactness of translates of averages { ¯ m ,k } . According to (26), we have that Z R | ddx ¯ m ,k | dx is bounded for k ↑ ∞ ,i.e. (9) for ¯ m ,k . From (25) we obtain Z R − | ¯ m ,k + sin θ k | dx + Z R + | ¯ m ,k − sin θ k | dx < ∞ for each k ∈ N . Since θ k ∈ (0 , π ) , this implies in particular (10) for ¯ m ,k . Hence by Lemma 1, there existzeros x ,k of ¯ m ,k and u ∈ ˙ H ( R ) ∩ C ( R ) s.t. for a subsequence ¯ m ,k ( · + x ,k , · ) − ⇀ u weakly in ˙ H ( R ) and locally uniformly , with u satisfying (11). Step 2:
Convergence of { m k ( · + x ,k , · ) } . Because of (26), by standard weak-compactnessresults, there exists m ∈ ˙ H (Ω) ∩ L ∞ (Ω) s.t. for a subsequence m k ( · + x ,k , · ) − ⇀ m weakly in ˙ H (Ω) and weak- ∗ in L ∞ (Ω) . By Rellich’s compactness result, m k ( · + x ,k , · ) → m in L loc (Ω) and a.e. , One might have that ˜ θ = θ , see Remark 3 (ii).
19o that in particular m k ∈ ˙ H (Ω , S ) yields m ∈ ˙ H (Ω , S ) . We thus may identify u as ¯ m ,i.e. ¯ m ≡ u in R , so that ¯ m satisfies (11).To simplify notation, we identify m k with its translate m k ( · + x ,k , · ) in the sequel of theproof. Step 3: If η ∈ (0 , , we show that m ∈ X θ and compactness of { h ( m k ) } . Indeed, in thiscase, (26) yields in particular (cid:26) Z Ω | m ,k − cos θ k | + m ,k dx (cid:27) k is bounded as k ↑ ∞ , so that Fatou’s lemma and Step 2 lead to: Z Ω | m − cos θ | + m dx < ∞ , that is, m ′ ( ±∞ , · ) = (cos θ, . Since by assumption (24), θ ∈ (0 , π ) , and since ¯ m satisfies(11), Lemma 3 yields m ( ±∞ , · ) = (cos θ, ± sin θ, . Hence, we indeed have m ∈ X θ . Forproving compactness of stray fields, we note that (26) yields in particular (cid:26) Z R | h ( m k ) | (cid:27) k is bounded as k ↑ ∞ . Hence there exists h ∈ L (Ω) s.t. for a subsequence h ( m k ) − ⇀ h weakly in L ( R ) . Passing to the limit in the distributional formulation ∇· (cid:0) h ( m k )+ m ′ k Ω (cid:1) = 0 , ∇× h ( m k ) = 0 to obtain ∇ · (cid:0) h + m ′ Ω (cid:1) = 0 , ∇ × h = 0 in D ′ ( R ) , and using uniqueness of the stray-fieldof m with (1), we learn that h = h ( m ) . Step 4: If η = 0 , we show that h ( m k ) → in L ( R ) and m ∈ X ∩ X ˜ θ for some ˜ θ ∈ [0 , π ] . Indeed, in this case, (26) yields in particular (recall that m k ∈ X yields h ( m k ) ≡ ): Z R | h ( m k ) | dx → , so that passing to the limit in the distributional formulation ∇ · (cid:0) h ( m k ) + m ′ k Ω (cid:1) = 0 welearn that ∇ · ( m ′ Ω ) = 0 in D ′ ( R ) . Since m ∈ ˙ H (Ω) ∩ L ∞ (Ω) , this yields ∇ · m ′ = 0 in Ω , m = 0 on ∂ Ω . (28)On the other hand, ∇ · ( m ′ Ω ) = 0 in D ′ ( R ) implies ddx ¯ m = 0 on R , so that there exists ˜ θ ∈ [0 , π ] with ¯ m = cos ˜ θ on R . (29) Note that h ( m ( · + z , · )) ≡ h ( m )( · + z , · ) by uniqueness of L stray-fields in (4) associated to configu-rations satisfying (1).
20e note that in general, ˜ θ = θ = lim θ ↑∞ θ k . By the Poincaré-Wirtinger inequality in x weobtain from (29): Z Ω | m − cos ˜ θ | dx ≤ C Z Ω | ∂ x m | dx < ∞ . (30)By the Poincaré inequality in x , we obtain from (28): Z Ω m dx ≤ C Z Ω | ∂ x m | dx < ∞ . (31)Hence we have m ′ ( ±∞ , · ) = (cos ˜ θ, . To conclude, we distinguish two cases: Case 1: ˜ θ ∈ (0 , π ) . In this case, we may conclude by Lemma 3 that m ( ±∞ , · ) = (cos ˜ θ, ± sin ˜ θ, as in case of η ∈ (0 , . We thus obtain m ∈ X ∩ X ˜ θ . Case 2: ˜ θ ∈ { , π } . In this case, we apply Remark 1 to conclude from (29) that m is one ofthe constant functions ± e and thus trivially lies in X ∩ X ˜ θ . Remark 3. (i) The assumption θ ∈ (0 , π ) whenever η ∈ (0 , in (24) is due to Remark 2,since in general the condition m ( ±∞ , · ) = ± sin θ fails if θ ∈ { , π } . However, if θ ∈ { , π } , one gets a weaker statement concerning the behavior of ¯ m at ±∞ : Claim . Suppose that the sequences { θ k } k ↑∞ ⊂ (0 , π ) , { η k } k ↑∞ ⊂ (0 , satisfy θ k → θ and η k → η with θ ∈ { , π } and η ∈ (0 , . Consider a sequence { m k } k ↑∞ ⊂ ˙ H (Ω , S ) for which m k ∈ X θ k and { E η k ( m k ) } isbounded. Then there exists m ∈ ˙ H (Ω , S ) such that after passage to a subsequence: m k − ⇀ m weakly in ˙ H (Ω) and weak- ∗ in L ∞ (Ω) ,h ( m k ) − ⇀ h ( m ) weakly in L ( R ) ,E η ( m ) < ∞ , ¯ m ( x ) → as | x | → ∞ . Indeed, we can essentially proceed as in the proof of Steps 1,2 and 3 in Proposition 2.However, note that there is no need to apply Lemma 1; moreover, the application ofLemma 3 (at Step 3) is no longer possible. Instead, note that ¯ m − cos θ, ¯ m ∈ H ( R ) yields lim | x |↑∞ ¯ m ( x ) = cos θ ∈ {± } and lim | x |↑∞ ¯ m ( x ) = 0 . Therefore, | x |↑∞ − Z − | m ( x ) | dx ≥ lim sup | x |↑∞ (cid:12)(cid:12)(cid:12)(cid:0) ¯ m , ¯ m , ¯ m (cid:1) ( x ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:0) , lim sup | x |↑∞ | ¯ m ( x ) | , (cid:1)(cid:12)(cid:12)(cid:12) , i.e., lim | x |↑∞ ¯ m ( x ) = 0 . No translation in x -direction is required here. ii) Note that in the case η = 0 , the angle ˜ θ = θ m associated to the limiting configuration m via (7) in general does not coincide with the limit θ of the sequence θ k . In particular,in the situation of Proposition 1 (i) for θ k ≡ α = θ , the limit angle ˜ θ = θ m describesthe amount of asymmetric rotation in the wall core. Hence, the possibility of having ˜ θ = θ is directly related to observing a non-trivial behavior of the reduced model (14) .However, there are also cases in which θ = lim k θ k coincides with the limit angle ˜ θ , ascan be seen in the statement of Proposition 1 (iii).Proof of Proposition 1: Statements (i) and (ii) are an immediate consequence of Proposi-tion 2 by letting θ k ≡ α .Statement (iii) follows from Remark 1, if there exists a constant subsequence θ k ∈ { , π } .Otherwise, we find a convergent subsequence { θ k } k ↑∞ ⊂ (0 , π ) to which we apply Proposi-tion 2 with η k ≡ . In this latter case, not relabeling the subsequence, it remains to provethat the limit θ := lim k ↑∞ θ k satisfies θ = θ m , i.e., m ∈ X θ . Indeed, exploiting (7) and (27),one obtains cos θ ← cos θ k ≡ ¯ m ,k ( · + x ,k ) → ¯ m ≡ cos θ m as k ↑ ∞ . Since θ, θ m ∈ [0 , π ] , this yields θ = θ m . Due to the compactness statements in Proposition 1, one obtains existence of minimizers for E η , E asym ( θ ) and E . Theorem 3. • For fixed parameters η ∈ (0 , and α ∈ (0 , π ] , there exists a minimizer of E η over theset X α . • For θ ∈ [0 , π ] fixed, there exists a minimizer of E asym ( θ ) over the (non-empty, cf.Appendix) set m ∈ X with θ m = θ . • The Γ -limit energy E admits a minimizer over X . The optimal angle θ in the mini-mization problem (14) is attained.Proof. Observe that the functionals E η and { m R Ω |∇ m | dx } are lower-semicontinuouswith respect to the weak convergence obtained in Proposition 1. Hence, the first two state-ments in Theorem 3 follow immediately by the direct method in the calculus of variations,i.e. by applying the compactness results in Proposition 1 to minimizing sequences.For the third statement, we need an auxiliary lemma that we prove using the existence ofminimizers of E asym ( θ ) we have just shown: Lemma 4.
The map θ ∈ [0 , π ] E asym ( θ ) ∈ R + is lower semicontinuous. roof of Lemma 4. This immediately follows from Proposition 1 (iii) by considering for eachsequence { θ k ∈ [0 , π ] } k ↑∞ , a sequence { m k ∈ X } k ↑∞ of minimizers of E asym ( θ k ) for each k .Now, the third statement in Theorem 3 again follows by the direct method in the calculusof variations, since E is just a continuous perturbation of E asym . Γ -convergence To establish the lower bound (12), one has to estimate the exchange term in E η ( m η ) as wellas stray-field and anisotropy energy from below as η ↓ . If m is the limit of m η (in the weak ˙ H -topology), then the exchange term will be estimated as η ↓ by R Ω |∇ m | dx , while thestray-field and anisotropy energy will be estimated by λE sym ( α − θ m ) , where θ m is associatedto m ∈ X via (7).Let C > always denote a universal, generic constant.W.l.o.g. we may assume E η ( m η ) ≤ C < ∞ for some C > and m η → m in L loc (Ω) anda.e. in Ω as η ↓ . Step 1:
Exchange energy.
We first address estimating the exchange energy from below. Since m η − ⇀ m in ˙ H (Ω) as η ↓ , we obviously have Z Ω |∇ m | dx ≤ lim inf η ↓ Z Ω |∇ m η | dx, (32)by weak lower-semicontinuity of the L norm of {∇ m η } η ↓ . Step 2:
Choice of test function.
Now it remains to estimate both stray-field and anisotropyenergy in E η ( m η ) from below by π λ (cid:0) cos θ m − cos α (cid:1) .Here the idea is to approximate the limit cos θ m − cos α = − Z − − Z − (cid:0) m − cos α (cid:1) dx dx by − R − − R − (cid:0) m ,η − cos α (cid:1) dx dx and to define a suitable test function ζ : R → R , thatcaptures the profile of the tails of a Néel wall (i.e. when | x | ≥ ) and has the property that − Z − − Z − (cid:0) m ,η − cos α (cid:1) dx dx = − Z − − Z − (cid:0) m ′ η − (cos α, (cid:1) · ∇ ζ dx dx . In this way, the stray-field energy will control π λ (cid:0) cos θ m − cos α (cid:1) . Note that the argumenthere is similar to the one used in [19]. 23 emma 5. Let a > and ζ : R → R be the odd piecewise affine function defined by ζ ( x ) := x , < x < , , ≤ x < a, − a ( x − a ) , a ≤ x < a, , a ≤ x (33) (see Figure 5). Let ζ : R → R be given by ζ ( x , x ) = ζ ( x ) on Ω and harmonically extendedto R away from Ω , i.e. ζ satisfies ( ∆ ζ = 0 on R \ ¯Ω ,ζ ( · , ±
1) = ζ on R . (34) Then we have Z R |∇ ζ | dx ≤ π ln a + C (35) for some constant C = O (1) as a ↑ ∞ . Remark 4.
Problem (34) can be solved explicitly via Fourier transform in the x -variable: (cid:0) F x f (cid:1) ( k ) := √ π Z R f ( x ) e − ik x dx , k ∈ R , (36) where f : R → R . In fact, (34) becomes a second-order ODE for F x ( ζ ) in the x -variable.Imposing that ζ ∈ ˙ H ( R \ Ω) , we deduce: F x ( ζ ) ( k , x ) = F x ( ζ ) ( x ) e −| k | ( | x |− , k ∈ R , | x | > . (37) − a − a − −
11 1 a aζ x Figure 5: Test function ζ Proof of Lemma 5.
First we show Z R |∇ ζ | dx = 2 (cid:18)Z R (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ddx (cid:12)(cid:12) ζ (cid:12)(cid:12)(cid:12) dx + Z R (cid:12)(cid:12) ddx ζ (cid:12)(cid:12) dx (cid:19) , (38)where we define Z R (cid:12)(cid:12) | ddx | f (cid:12)(cid:12) dx := Z R | k | |F x f | dk ∈ [0 , ∞ ] , f ∈ L ( R ) . Ω we simply have Z Ω |∇ ζ | dx ζ = ζ in Ω = Z Ω |∇ ζ | dx ζ = ζ ( x ) = 2 Z R (cid:12)(cid:12) ddx ζ (cid:12)(cid:12) dx . Moreover, the contribution from R \ Ω can be computed using (37): Z R \ Ω |∇ ζ | dx = 2 Z R × (1 , ∞ ) |∇ ζ | dx =2 Z ∞ Z R |F x ( ∇ ζ )( k , x ) | dk dx =2 Z ∞ Z R (cid:12)(cid:12) k F x ( ζ )( k , x ) (cid:12)(cid:12) + (cid:12)(cid:12) ∂ x F x ( ζ )( k , x ) (cid:12)(cid:12) dk dx =2 Z ∞ Z R (cid:12)(cid:12) | k | F x ( ζ )( k , x ) (cid:12)(cid:12) + (cid:12)(cid:12) | k |F x ( ζ )( k , x ) (cid:12)(cid:12) dk dx (37) = 2 Z R | k | |F x ( ζ ) | (cid:18) Z ∞ | k | e − | k | ( x − dx | {z } =1 (cid:19) dk =2 Z R (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ddx (cid:12)(cid:12) ζ (cid:12)(cid:12)(cid:12) dx . Therefore (38) is established.To prove (35), one first observes that R R | ddx ζ | dx remains bounded as a ↑ ∞ , so that theleading-order contribution to (38) is given by the homogeneous ˙ H norm of ζ . Recall thatthe ˙ H norm can be expressed as Z R (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ddx (cid:12)(cid:12) ζ (cid:12)(cid:12)(cid:12) dx = min (Z R × R + |∇ ¯ ζ | dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ¯ ζ ∈ ˙ H ( R × R + ) , ¯ ζ ( · ,
0) = ζ ) . (39)Therefore, to estimate (39), we choose an admissible function ¯ ζ : ¯ ζ ( x ) = ¯ ζ ( r, θ ) := ζ ( r ) ϕ ( θ ) , x ∈ R × R + , where ( r, θ ) denote the polar coordinates of x ∈ R × R + and ϕ : [0 , π ] → [ − , is given by ϕ ( θ ) = 1 − π θ, ≤ θ ≤ π. Observe that indeed ¯ ζ ( · ,
0) = ζ in R (since ζ is odd and ϕ (0) = − ϕ ( π ) = 1 ). Therefore,we may estimate Z R (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ddx (cid:12)(cid:12) ζ (cid:12)(cid:12)(cid:12) dx (39) ≤ Z R × R + |∇ ¯ ζ | dx = Z π Z ∞ (cid:16) | ∂∂r ¯ ζ | + | r ∂∂θ ¯ ζ | (cid:17) rdr dθ = Z π ϕ dθ Z ∞ | ddr ζ | rdr | {z } = O (1) + Z π | ddθ ϕ | dθ | {z } = π Z ∞ ζ drr | {z } =ln a + O (1) = 4 π ln a + O (1) , which yields the asserted scaling. 25 tep 3: Stray-field and anisotropy energy.
With the test function constructed in Step 2 wecan establish the relation between λ E sym ( α − θ m ) and stray-field/anisotropy energy. Firstwe use the definition of ζ to rewrite − Z − − Z − (cid:0) m ,η − cos α (cid:1) dx dx = Z − Z − (cid:0) m ,η − cos α (cid:1) ∂ x ζ |{z} =1 dx dx = Z Ω (cid:0) m ,η − cos α (cid:1) ∂ x ζ dx + a Z (( − a, − a ) ∪ ( a, a )) × ( − , (cid:0) m ,η − cos α (cid:1) dx ζ = ζ ( x ) = Z Ω m ′ η · ∇ ζ dx + a Z (( − a, − a ) ∪ ( a, a )) × ( − , (cid:0) m ,η − cos α (cid:1) dx (4) = − Z R h ( m η ) · ∇ ζ dx + a Z (( − a, − a ) ∪ ( a, a )) × ( − , (cid:0) m ,η − cos α (cid:1) dx ≤ (cid:18)Z R | h ( m η ) | dx (cid:19) (cid:18)Z R |∇ ζ | dx (cid:19) + a (4 a ) (cid:18)Z Ω (cid:0) m ,η − cos α (cid:1) dx (cid:19) Lemma ≤ (cid:18)(cid:0) π ln a + O (1) (cid:1) Z R | h ( m η ) | dx (cid:19) + (cid:18) a Z Ω (cid:0) m ,η − cos α (cid:1) dx (cid:19) , as a ↑ ∞ . If we now apply √ α + p β ≤ p α (1 + δ ) + δ − β, which holds for < δ ≤ , α, β ≥ , to α = (cid:0) π ln a + O (1) (cid:1) Z R | h ( m η ) | dx,β = a Z Ω (cid:0) m ,η − cos α (cid:1) dx, we find (cid:18) − Z − − Z − (cid:0) m ,η − cos α (cid:1) dx (cid:19) ≤ (cid:0) π ln a + O (1) (cid:1) (1 + δ ) Z R | h ( m η ) | dx + δ a Z Ω (cid:0) m ,η − cos α (cid:1) dx. Now, for δ ∈ (0 , ] fixed, choose a = a ( η ) such that δ a = λ − δ π η. This implies a ↑ ∞ as η ↓ and ln a = ln η + O (1) as η ↓ . Use √ αβ ≤ δ α + δ β and δ ≤ δ − . λ ln η R R | h ( m η ) | dx ≤ E η ( m η ) is uniformly bounded and thus R R | h ( m η ) | dx → as η ↓ . Together with cos θ m = − R − ¯ m dx ← − R − ¯ m ,η dx , we obtain: (cid:0) cos θ m − cos α (cid:1) = (cid:18) lim inf η ↓ − Z − − Z − (cid:0) m ,η − cos α (cid:1) dx dx (cid:19) ≤ lim inf η ↓ (cid:18) π (1 + δ ) ln a Z R | h ( m η ) | dx + δ a Z Ω (cid:0) m ,η − cos α (cid:1) dx (cid:19) = (1 + δ ) lim inf η ↓ (cid:18) π ln η Z R | h ( m η ) | dx + λ − π η Z Ω (cid:0) m ,η − cos α (cid:1) dx (cid:19) . Letting now δ ↓ , it follows: πλ (cid:0) cos θ m − cos α (cid:1) ≤ lim inf η ↓ (cid:18) λ ln η Z R | h ( m η ) | dx + η Z Ω (cid:0) m ,η − cos α (cid:1) dx (cid:19) . (40) Step 4:
Conclusion.
By combining (32) and (40) one sees E ( m ) ≤ lim inf η ↓ Z Ω |∇ m η | dx + lim inf η ↓ (cid:18) λ ln η Z R | h ( m η ) | dx + η Z Ω (cid:0) m ,η − cos α (cid:1) dx (cid:19) ≤ lim inf η ↓ E η ( m η ) , i.e. the lower bound (12) is proven. For each m ∈ X we construct a recovery sequence { m η } η ↓ ⊂ X α such that m η → m in ˙ H (Ω) and (13) holds. For that, in the general case θ m
6∈ { , α, π } , the basic guideline willbe a decomposition of Ω into several parts (as shown in Figure 6): We consider the regions − a +1 η − ( a + 1) − a a a + 1 a +1 η − Ω T Ω I Ω A Ω I Ω T Néel tails Interp. Asym. wall Interp. Néel tails m η ≈ (cid:16) c o s θ m s i n θ m (cid:17) m η = (cid:16) c o s θ m s i n θ m (cid:17) Figure 6: Construction of recovery sequence Ω A := [ − a, a ] × ( − , , Ω I := (cid:16) [ − ( a + 1) , − a ] ∪ [ a, a + 1] (cid:17) × ( − , , Ω T := (cid:16) [ − a +1 η , − ( a + 1)] ∪ [ a + 1 , a +1 η ] (cid:17) × ( − , , a is a parameter of order ln η ≫ (to be chosen explicitly at Step 1 below). • The (core) region Ω A stands for the asymmetric part of the transition layer m η : Here, m ′ η is of vanishing divergence (so, a stray-field free configuration) with an asymptoticangle transition from − θ m to θ m (as η ↓ ) so that the leading-order term is driven bythe exchange energy. • The (tail) region Ω T corresponds to the symmetric part of the transition layer m η thatmimics the tails of a symmetric Néel wall. Here, the leading order term of the energyis driven by the stray field, the transition angle covering the range [ − α, − θ m ] (at theleft) and [ θ m , α ] (at the right), respectively. • The (intermediate) region Ω I is necessary for the transition between the core of m η andthe tails of a Néel wall. This is because the asymmetric core of m η and the symmetrictails will not fit together perfectly (on Ω A , the angle transition is θ m + o (1) and notexactly θ m ), however, this region only adds energy of order o (1) .So, let m ∈ X . Since θ m ∈ [0 , π ] , we need to treat three different cases: Case 1: θ m
6∈ { , α, π } . We proceed in several steps:
Step 1:
Choice of a . We consider the L ( R ) positive function E : R → R + defined by: E ( x ) := Z − |∇ m ( x , x ) | dx for a.e. x ∈ R . For η ≪ , let b = b ( η ) := ln η (in fact, any choice ln γ η with γ ∈ (1 , would work). Wechoose a = a ( η ) ∈ [ b , b ] such that a and − a are Lebesgue points of E and E ( a ) + E ( − a ) ≤ − Z bb E ( x ) + E ( − x ) dx ≤ b Z Ω |∇ m | dx = C b , (41)with C = 2 R Ω |∇ m | dx < ∞ . In particular, the ˙ H -trace of m on the vertical lines {± a } × ( − , actually belongs to H . Since ¯ m ∓ sin θ m ∈ H ( R ± ) (due to m ( ±∞ , · ) = ± sin θ m ),we also have ¯ m ( ± a ) = ± sin θ m + o (1) as η ↓ . Recall that Sobolev’s embedding theorem yields existence of
C > such that k u k L ∞ ≤ C k dds u k L , for every u ∈ H (cid:0) ( − , (cid:1) , together with k u − ¯ u k L ∞ ≤ C k dds u k L , for every u ∈ H (cid:0) ( − , (cid:1) . r C b ≥ (cid:18)Z − X σ ∈{± } |∇ m ( σa, x ) | dx (cid:19) ≥ (cid:18) X σ ∈{± } Z − | ∂ x m ( σa, x ) | + | ∂ x m ( σa, x ) | + | ∂ x m ( σa, x ) | dx (cid:19) ≥ C X σ ∈{± } (cid:16) k m ( σa, · ) − ¯ m ( σa ) | {z } =cos θ m k L ∞ + k m ( σa, · ) − ¯ m ( σa ) k L ∞ + k m ( σa, · ) k L ∞ (cid:17) . (42)It follows that m ( ± a, x ) = cos θ m + o (1) , m ( ± a, x ) = o (1) and m ( ± a, x ) = ¯ m ( ± a ) + o (1) = ± sin θ m + o (1) uniformly in x ∈ ( − , as η ↓ (since b → ∞ ). Step 2:
Definition of m η . • On Ω A , we choose that m η ( x ) = m ( x ) for every x ∈ Ω A . • On the tail region {| x | ≥ a + 1 } , we choose m η to be the S -valued approximation ofa Néel wall with a transition angle that goes from − α to − θ m (on the left) and from θ m to α (on the right). More precisely, as in [11], let m η : Ω \ (Ω A ∪ Ω I ) → S dependonly on the x -direction and be given by m ,η ( x , x ) := cos α + cos θ m − cos α ln η ln (cid:0) a +1 η | x | (cid:1) , a + 1 ≤ | x | ≤ a +1 η , x ∈ ( − , α, a +1 η ≤ | x | , x ∈ ( − , , m ,η ( x , x ) := sgn( x ) q − m ,η ( x , x ) ,m ,η ( x , x ) := 0 , on { a + 1 ≤ | x |} × ( − , . • On the intermediate region Ω I , i.e. for a < | x | < a + 1 , we define m η by linearinterpolation in m ,η and the phase φ η of ( m ,η , m ,η ) (interpreted as complex number)between Ω T and Ω A . For this, we choose η sufficiently small, such that ± m ( ± a, · ) > sin θ m > in ( − , . (43)Therefore, there exists a unique phase φ η ( ± a, x ) ∈ (0 , π ) of ( m , m )( ± a, x ) ∈ R ≃ C such that ( m + im )( ± a, x ) = q − m ( ± a, x ) e ± iφ η ( ± a,x ) for every x ∈ ( − , . Observe that the function φ η ( ± a, · ) depends on η only through a . Recall that m ,η ( ± ( a +1) , · ) = 0 and ( m ,η + im ,η )( ± ( a + 1) , · ) = e ± iθ m so that we fix φ η ( ± ( a + 1) , · ) := θ m on ( − , . By linear interpolation, we then define m η : Ω I → S and φ η : Ω I → (0 , π ) m ,η ( x ) := (cid:0) a − | x | (cid:1) m ( ± a, x ) , (44) φ η ( x ) := (cid:0) a − | x | (cid:1) φ η ( ± a, x ) + (cid:0) | x | − a (cid:1) θ m , ( m ,η + im ,η )( x ) := q − m ,η ( x ) e ± iφ η ( x ) , (45)whenever a < ± x < a + 1 , x ∈ ( − , . Note that m η ( ±∞ , · ) = m ± α (in the sense of (2)) since m η = m ± α only on the bounded set Ω A ∪ Ω I ∪ Ω T . We will show that m η has H regularity on Ω A , Ω I and Ω T . Moreover, the H -traces of m η on the vertical lines {± a } × ( − , and {± ( a + 1) } × ( − , do agree, sothat finally m η ∈ ˙ H (Ω) , i.e., m η ∈ X α . Step 3:
Exchange energy estimate.
We prove Z Ω |∇ m η | dx ≤ Z Ω |∇ m | dx + o (1) . (46)Indeed, • on Ω A , we have that m η ≡ m so that Z Ω A |∇ m η | dx ≤ Z Ω |∇ m | dx ; (47) • on Ω T , since | m ,η | ≤ max (cid:0) | cos θ m | , cos α (cid:1) =: µ < , we deduce Z − Z a +1 η a +1 |∇ m η | dx = 2 Z a +1 η a +1 | ddx m ,η ( x ) | − m ,η ( x ) dx ≤ (cid:0) cos θ m − cos α ln η (cid:1) Z a +1 η a +1 x − − µ dx ≤ C ( θ m ) b ln η = o (1) as η ↓ , (48)where we used a + 1 ≥ b in the last inequality; • on the intermediate region Ω I , we use the following lemma: Lemma 6.
For a ≤ ± x ≤ a + 1 and x ∈ ( − , , we have ( i ) | ∂ x m ,η ( x ) | ≤ m ( ± a, x ) , ( ii ) | ∂ x m ,η ( x ) | ≤ | ∂ x m ( ± a, x ) | , ( iii ) (cid:12)(cid:12) ∂ x (cid:0) m ,η m ,η (cid:1) ( x ) (cid:12)(cid:12) ≤ m ( ± a, x ) + | φ η ( ± a, x ) − θ m | , ( iv ) (cid:12)(cid:12) ∂ x (cid:0) m ,η m ,η (cid:1) ( x ) (cid:12)(cid:12) ≤ | ∂ x ( m m ) ( ± a, x ) | . (49)30 roof. Inequalities ( i ) and ( ii ) immediately follow from the definition (44) of m ,η on Ω I .To prove the remaining inequalities we use the identity | ∂ x i (cid:0) ρ ( x ) e iϕ ( x ) (cid:1) | = | ∂ x i ρ ( x ) | + | ρ ( x ) ∂ x i ϕ ( x ) | for real-valued functions ρ and ϕ . Therefore, for (iii), using that | m ,η ( x ) | ≤| m ( ± a, x ) | ≤ for η sufficiently small (see (42)), we deduce for x ∈ Ω I : | ∂ x ( m ,η + im ,η )( x ) | = m ,η ( x )1 − m ,η ( x ) | {z } ≤ | ∂ x m ,η ( x ) | + (cid:0) − m ,η ( x ) (cid:1)| {z } ≤ | ∂ x φ η ( x ) | i ) ≤ m ( ± a, x ) + | φ η ( ± a, x ) − θ m | . Similarly, for ( iv ) , using that t t − t is increasing on (0 , , we obtain for the x -derivative of m ,η and m ,η and x ∈ Ω I : | ∂ x ( m ,η + im ,η )( x ) | ≤ m ,η ( x )1 − m ,η ( x ) | ∂ x m ,η ( x ) | + (cid:0) − m ,η ( x ) (cid:1)| {z } ≤ | ∂ x φ η ( x ) | (44) ≤ m ( ± a,x )1 − m ( ± a,x ) | ∂ x m ( ± a, x ) | + 2 (cid:0) − m ( ± a, x ) (cid:1)| {z } ≥ | ∂ x φ η ( ± a, x ) | ≤ (cid:16) m ( ± a,x )1 − m ( ± a,x ) | ∂ x m ( ± a, x ) | + (cid:0) − m ( ± a, x ) (cid:1) | ∂ x φ η ( ± a, x ) | (cid:17) = 2 | ∂ x ( m + im )( ± a, x ) | . Note that for sufficiently small η the function φ η ( ± a, · ) ∈ (0 , π ) is bounded away from and π , such that by Lipschitz continuity of arccos and (42) we have | φ η ( ± a, · ) − θ m | ≤ C ( θ m ) b on ( − , . (50)Therefore, after integrating (49) over Ω I , (50), (41) and (42) show that Z Ω I |∇ m η | dx = Z { a ≤| x |≤ a +1 } Z − |∇ m η | dx dx ≤ C ( θ m ) b = o (1) as η ↓ , (51)which together with (47) and (48) implies (46). Step 4:
Stray-field energy estimate.
We will prove that λ ln η Z R | h ( m η ) | dx ≤ π λ (cid:0) cos θ m − cos α (cid:1) + o (1) . (52)Indeed, we follow the arguments in [11, Proof of Thm. 2(ii)] (see also [12, 14]). First of all,recall that m ,η = 0 on ∂ Ω and that ∇ · m ′ η is supported in the compact set Ω A ∪ Ω I ∪ Ω T .31herefore, the stray field h η = −∇ u η with u η ∈ ˙ H ( R ) satisfies Z R ∇ u η · ∇ v dx = − Z Ω ∇ · m ′ η v dx ∀ v ∈ ˙ H ( R ) , so that by choosing v := u η , we have: Z R |∇ u η | dx = − Z Ω ∇ · m ′ η u η dx = − Z Ω A ∇ · m ′ | {z } =0 u η dx − Z Ω I ∇ · m ′ η u η dx − Z Ω T ∇ · m ′ η u η dx. (53) • On Ω I , since m ,η = 0 on ∂ Ω and ¯ m ,η ( ± a ) = cos θ m as well as m ,η ( ± ( a + 1) , · ) =cos θ m , we have Z (Ω I ) + ∇ · m ′ η dx = Z ( ∂ Ω I ) + m ′ η · ν d H ( x ) = ¯ m ,η ( a + 1) − ¯ m ,η ( a ) = 0 and similarly, R (Ω I ) − ∇ · m ′ η dx = 0 , where ν denotes the outer unit normal to Ω I andwe use the notation (Ω I ) + = Ω I ∩ { x ≥ } , (Ω I ) − = Ω I ∩ { x ≤ } . Therefore wemay subtract the averages ¯ u + η = − Z (Ω I ) + u η dx and ¯ u − η = − Z (Ω I ) − u η dx of u η over the left and right parts of Ω I such that by (51), Cauchy-Schwarz andPoincaré-Wirtinger’s inequality, we deduce: Z Ω I ∇ · m ′ η u η dx = Z (Ω I ) − ∇ · m ′ η (cid:0) u η − ¯ u − η (cid:1) dx + Z (Ω I ) + ∇ · m ′ η (cid:0) u η − ¯ u + η (cid:1) dx ≤ (cid:18)Z Ω I |∇ · m ′ η | dx (cid:19) Z (Ω I ) − | u η − ¯ u − η | dx ! + (cid:18)Z Ω I |∇ · m ′ η | dx (cid:19) Z (Ω I ) + | u η − ¯ u + η | dx ! (51) ≤ (cid:18) Cb (cid:19) (cid:18)Z R |∇ u η | dx (cid:19) = (cid:16) C ln − η (cid:17) (cid:18)Z R |∇ u η | dx (cid:19) . (54) • On Ω T , the stray-field energy can be estimated using the trace characterization (39)and the Cauchy-Schwarz inequality in Fourier space. Indeed, defining m tails ,η : Ω → R
32y setting m tails ,η = m ,η on Ω \ (Ω A ∪ Ω I ) and m tails ,η = cos θ m on Ω A ∪ Ω I , we have (cid:12)(cid:12)(cid:12)Z Ω T ∇ · m ′ η u η dx (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)Z Ω T ddx m ,η u η dx (cid:12)(cid:12)(cid:12) = Z − Z R ddx m tails ,η u η dx dx ≤ Z − Z R | k ||F x m tails ,η ( k ) | |F x u η ( k , x ) | dk dx ≤ Z − Z R | k ||F x m tails ,η ( k ) | dk Z R | k ||F x u η ( k , x ) | dk ! dx (39) ≤ (cid:18)Z R |∇ u η | dx (cid:19) (cid:18) Z R (cid:12)(cid:12)(cid:12) | ddx | m tails ,η (cid:12)(cid:12)(cid:12) dx (cid:19) . (55)By considering the radial extension M ,η ( x ) = m tails ,η ( | x | ) of m tails ,η on R × R + , whichis possible since m tails ,η is even, and using polar coordinates we then can estimate Z R (cid:12)(cid:12)(cid:12) | ddx | m tails ,η (cid:12)(cid:12)(cid:12) dx (39) ≤ Z R × R + |∇ M ,η | dx ≤ π Z a +1 η a +1 | ddx m tails ,η | x dx = π (cid:0) cos θ m − cos α (cid:1) ln η Z a +1 η a +1 1 x dx | {z } =ln η = π (cid:0) cos θ m − cos α (cid:1) ln η . Collecting (53), (54), and (55), it follows that λ ln η Z Ω |∇ u η | dx ≤ λ ln η (cid:18) C ln − η + (cid:16) π (cos θ m − cos α ) ln η (cid:17) (cid:19) ≤ πλ (cid:0) cos θ m − cos α (cid:1) + C ln − η , i.e. (52). Step 5:
Anisotropy energy estimate.
Finally, we prove η Z Ω (cid:0) m ,η − cos α (cid:1) + m ,η dx = o (1) . (56) In fact, this motivates the choice b = ln γ η with γ > . a ∼ b = ln η , on Ω A ∪ Ω I we have η Z a +1 − ( a +1) Z − (cid:0) m ,η − cos α (cid:1) + m ,η dx dx ≤ Cη ln η = o (1) , and on Ω T , m ,η = 0 so that η Z Ω T (cid:0) m ,η − cos α (cid:1) + m ,η dx = 4 η (cid:0) cos θ m − cos α (cid:1) ln η Z a +1 η a +1 ln ( a +1 η x ) dx y = ηx a +1 ≤ Cb (cid:0) cos θ m − cos α (cid:1) ln η Z η ln y dy | {z } = O (1) ≤ C ln − η = o (1) . Moreover, on Ω \ (Ω A ∪ Ω I ∪ Ω T ) we have (cid:0) m ,η − cos α (cid:1) + m ,η = 0 so that (56) holds. Step 6:
Conclusion.
Combining (46), (52) and (56), it follows that E η ( m η ) = Z Ω |∇ m η | dx + λ ln η Z R | h ( m η ) | dx + η Z Ω (cid:0) m ,η − cos α (cid:1) + m ,η dx ≤ Z Ω |∇ m | dx + 2 π λ (cid:0) cos θ m − cos α (cid:1) + o (1) = E ( m ) + o (1) , which is (13). Finally, let us prove that m η → m in ˙ H (Ω) . First, observe that by con-struction, m η ≡ m on Ω A . Therefore, since S η ↓ Ω A = Ω , m η → m in L loc (Ω) . Moreover,(46) implies that { m η } η ↓ is uniformly bounded in ˙ H (Ω) , so that m η − ⇀ m in ˙ H (Ω) . Byweak lower-semicontinuity of k·k L (Ω) and (46), one obtains k∇ m η k L (Ω) → k∇ m k L (Ω) andconcludes that m η → m in ˙ H (Ω) . Case 2: θ m ∈ { , π } . By Remark 1, m is constant, such that its exchange energy does notcontribute to E ( m ) . Thus, we have to construct a sequence m η of asymptotically vanishingexchange energy, whose stray-field and anisotropy energy converge to π λ (cid:0) cos θ m − cos α (cid:1) .The function m η from Case 1 is a good candidate for the second property. However, if θ m ∈ { , π } , it does not belong to H (Ω) , since then − m ,η behaves linearly w.r.t. thedistance to the set { m ,η = 1 } and (48) fails. Therefore, we are obliged to construct atransition region between the two tails where this behavior is corrected.With these considerations, we define m η : Ω → S by m ,η ( x , x ) := cos θ m − (cid:0) cos θ m − cos α (cid:1) ln 2ln η x , | x | ≤ , cos α + (cid:0) cos θ m − cos α (cid:1) ln( ηx )ln η , ≤ | x | ≤ η , cos α, η ≤ | x | , Note that here it is important to have b = ln γ η with γ < . m ,η ( x , x ) := sgn( x ) q − m ,η ( x , x ) ,m ,η ( x , x ) := 0 . Admissibility in X α is obvious and one can then show, using the methods given above, that Z Ω |∇ m η | dx ≤ C ln − η ,λ ln η Z R |∇ u η | dx ≤ π λ (cid:0) cos θ m − cos α (cid:1) + C ln − η ,η Z Ω (cid:0) m ,η − cos α (cid:1) + m ,η dx ≤ C ln − η . The strong convergence m η → m in ˙ H (Ω) also follows as in Step 6 of Case 1 by noting thatthe constructed transition layer m η has the property m η → m = (cid:0) cos θ m , ± sin θ m , (cid:1) a.e. in Ω . Case 3: θ m = α . Since m already has the correct boundary values, we can simply choose: m η := m. Admissibility of m η in X α is clear and we can estimate: Z Ω |∇ m η | dx = Z Ω |∇ m | dx,λ ln η Z R | h ( m η ) | dx = 0 = 2 π λ (cid:0) cos α − cos α (cid:1) ,η Z Ω (cid:0) m ,η − cos α (cid:1) + m ,η dx = o (1) , since m ( ±∞ , · ) = (cid:0) cos α, ± sin α, (cid:1) . (cid:3) Proof of Corollary 1.
The first equality in (14) is a direct consequence of the concept of Γ -convergence. Indeed, we know by Theorem 3 that there exists a minimizer m η ∈ X α of E η for every < η < . By Proposition 1, up to a subsequence and translation in x -direction,we have that m η − ⇀ m in ˙ H (Ω) for some m ∈ X so that Theorem 1 implies lim inf η ↓ min X α E η = lim inf η ↓ E η ( m η ) ≥ E ( m ) ≥ min X E . On the other hand, Theorem 3 also implies existence of a minimizer m ∈ X of E . ByTheorem 2, there exists a family { ˜ m η } η ↓ ⊂ X α such that min X E = E ( m ) ≥ lim sup η ↓ E η ( ˜ m η ) ≥ lim sup η ↓ min X α E η . min X α E η → min X E as η ↓ . For the second equality in (14), note that byTheorem 3 one has min X E = min θ ∈ [0 ,π ] (cid:16) E asym ( θ ) + λ E sym ( α − θ ) (cid:17) . By Lemma 4, the minimum of the RHS is indeed attained. It remains to show that it isachieved for angles θ ∈ [0 , π ] . Indeed, let θ ∈ [0 , π ] be the minimizer of the above RHS and m ∈ X with θ m = θ be the minimizer of E asym ( θ ) . If θ ∈ ( π , π ] , then one considers ˜ m ∈ X given by ˜ m ′ ≡ − m ′ and ˜ m ≡ m , so that θ ˜ m = π − θ m ∈ [0 , π ) . Then ˜ m and m have thesame exchange energy (i.e., E asym ( θ ˜ m ) = E asym ( θ m ) ) and E sym ( α − θ ˜ m ) ≤ E sym ( α − θ m ) which proves (14). Observe that the last inequality is strict whenever α ∈ (0 , π ) , so that forsuch angles α the minimal value of E is achieved only for angles θ ∈ [0 , π ] .Let us now prove the relative compactness in the strong ˙ H -topology of minimizing families { m η } η ↓ ⊂ X α of E η , i.e., which satisfy E η ( m η ) → min X E . By Proposition 1, up to asubsequence and translations in x -direction, we may assume that m η − ⇀ m in ˙ H (Ω) forsome m ∈ X so that Theorem 1 implies min X E = lim η ↓ E η ( m η ) ≥ lim sup η ↓ Z Ω |∇ m η | dx + lim inf η ↓ (cid:18) λ ln η Z R | h ( m η ) | dx + η Z Ω (cid:0) m ,η − cos α (cid:1) dx (cid:19) (32) , (40) ≥ E asym ( θ m ) + λE sym ( α − θ m ) ≥ min X E . Therefore, all above inequalities become equalities, in particular, lim η ↓ R Ω |∇ m η | dx = R Ω |∇ m | dx .Hence, one has m η → m in ˙ H (Ω) , i.e. up to the subsequence taken in Proposition 1 andtranslations the entire family m η converges strongly to m in ˙ H (Ω) where m is a minimizerof E . AppendixA Construction of an asymmetric-Bloch type wall of arbitrarywall angle θ ∈ (0 , π ] In this section we construct a stray-field free domain wall for any given angle θ ∈ (0 , π ] . Inparticular, this shows that the set X ∩ X θ is non-empty, and we may apply the direct methodin the calculus of variations to deduce existence of minimizers of E asym (cf. Theorem 3).The construction we present here is of asymmetric Bloch-wall type in the following sense: Thetrace of m ∈ ˙ H (Ω , S ) on the boundary Bdry := ∂ Ω ∪ (cid:18) {±∞} × [ − , (cid:19) ∼ = S (see (16)) We use that lim sup n ( a n + b n ) ≥ lim sup n a n + lim inf n b n for two bounded sequences ( a n ) and ( b n ) . m = 0 on ∂ Ω as well as m ( ±∞ , · ) = 0 (so, ( m , m ) : Bdry → S ), one obtains (by the homeomorphism (16)) a map ˜ m ∈ H ( S , S ) to which a topological degree can be associated (see, e.g., [4]). Remark 5. (i) Asymmetric Bloch walls as well as the configuration we are about to con-struct do have a non-zero topological degree (e.g. ± ) on Bdry , whereas asymmetricNéel walls have degree on ∂ Ω . We make the following observation (see Lemma 2):the non-vanishing topological degree of ( m , m ) : Bdry → S nucleates at least onevortex singularity of ( m , m ) (carrying a non-zero topological degree) as illustrated inFigure 3.(ii) Note that for the angle θ = 0 , by Remark 1, one has that m ∈ X if and only if m ∈ {± e } , so that m has degree zero on ∂ Ω ; thus, no asymmetric-Bloch type wallexists in this case.(iii) An asymmetric-Néel type configuration ˜ m ∈ X ∩ X θ , i.e. with deg ˜ m = 0 , can be ob-tained from any m ∈ X ∩ X θ using even reflection in ( m , m ) and odd reflection in m across one of the components of ∂ Ω together with a rescaling in x so that ˜ m is definedon Ω . However, starting with m ∈ L θ (introduced at Section 1.3), the reflected config-uration has at least two vortices in ( m , m ) , so that it cannot have minimal energy.In [7], we construct an asymptotically energy minimizing configuration of asymmetric-Néel type for small angles. The degree argument shows that we cannot expect a homotopy between asymmetric Néeland Bloch wall in the class of stray-field free walls. Hence, it is unclear how the neverthelessexpected transition from asymmetric Néel to Bloch wall actually takes place.
Proposition 3.
Given θ ∈ (0 , π ] , there exists a map m : Ω → S with the following proper-ties: • m ∈ ˙ H (Ω) , • m ( x , · ) = m ± θ for all | x | sufficiently large, • ∇ · m ′ = 0 in Ω and m = 0 on ∂ Ω , • deg( m (cid:12)(cid:12) ∂ Ω ) = − .Proof. To construct m , we will search for a stream function ψ : R → R with the followingproperties:(i) ψ ∈ C ( R ) with |∇ ψ | ≤ in R , (ii) ψ ( x ) = − ( x + 1) cos θ for all | x | sufficiently large,(iii) ψ ( · , −
1) = 0 and ψ ( · ,
1) = − θ in R ,(iv) there exists a continuous curve γ , connecting the upper and lower components R ×{ +1 } and R × {− } of ∂ Ω , on which |∇ ψ | = 1 . In fact, we will construct a smooth function ψ . − (cid:0) − cos θ (cid:1) (cid:0) − cos θ (cid:1) L θ ≫ − L θ − Figure 7: Sketch of the level lines of a stream function of an asymmetric domain wall ofBloch type. γ − (cid:0) − cos θ (cid:1) (cid:0) − cos θ (cid:1) − cos θ − − cos θ Figure 8: Enlargement of the area around the vortex, cf. Figure 7.We then define m according to m ′ := ∇ ⊥ ψ, m ( x ) := ( − p − |∇ ψ ( x ) | , if x is to the left of γ, p − |∇ ψ ( x ) | , if x is to the right of γ. (57)Note that by Lemma 7 below, m is Lipschitz continuous. Indeed, we remark that D ( |∇ ψ | ) is globally bounded since |∇ ψ | = cos θ outside of a compact set and apply Lemma 7 to f = 1 − |∇ ψ | ≥ .Figure 7 shows the level lines of ψ , in Figure 8 the region around the vortex is enlarged. − − − ˆ γ Figure 9: Sketch of vortex function s with ellipsoid level sets in the inner part. Step 1:
Construction in the inner part around the vortex.
As a first step in the constructionof ψ we implicitly define a function s ∈ C ∞ ( Q ) , Q = ([ − , × R ) \ { (0 , } , by specifying its level sets (cf. Figure 9). Later, we will define ψ by rescaling, shifting and38 t Figure 10: The semi-major axis t : (0 , ) → R of the ellipses in Figure 9.smoothing the function − s . Consider f : Q × (0 , ∞ ) → R , f (ˆ x, s ) := ((cid:0) ˆ x s (cid:1) − , if s ≥ , ˆ x ∈ Q, (cid:0) st ( s ) (cid:1) (cid:0) ˆ x s (cid:1) + (cid:0) ˆ x s (cid:1) − , if < s < , ˆ x ∈ Q, where t : (0 , ) → R is a smooth function that satisfies the structural condition t ( s ) = s if s ∈ [0 , ] , d ds t ≥ on (0 , ) , (58)and such that t ( s ) vanishes to infinite order at s = , e.g. t ( s ) = e − s if s ∈ [ , ] . We note that the latter implies that f is smooth across s = and thus in the whole domain Q × (0 , ∞ ) . Claim: For every ˆ x ∈ Q there exists a unique solution s = s (ˆ x ) of f (ˆ x, s ) = 0 . We first argue that the solution is unique: Indeed, because of (58) we have in particular dds t ( s ) ≥ so that ∂ s f (ˆ x, s ) < for all (ˆ x, s ) ∈ Q × (0 , ∞ ) , provided we are not in the case of s ≥ and ˆ x = 0 . This case however is not relevant for uniqueness since then f (ˆ x, s ) ≡ − .It follows from the explicit form of f that s (ˆ x ) = ( | ˆ x | , for | ˆ x | ≤ , | ˆ x | , for | ˆ x | ≥ , is a solution.Hence, it remains to show existence of a solution for | ˆ x | < but | ˆ x | > : Indeed, | ˆ x | < implies f (ˆ x, ) < and | ˆ x | > yields f (ˆ x, ) > . Thus, the existence of a solution s = s (ˆ x ) ∈ ( , ) of f (ˆ x, s ) = 0 follows from the intermediate value theorem.39he implicit function theorem yields smoothness of s : Q → (0 , ∞ ) , with ˆ ∇ s (ˆ x ) = − ∇ ˆ x f (ˆ x,s (ˆ x )) ∂ s f (ˆ x,s (ˆ x )) = ( ˆ x t ( s ) , ˆ x s ) ˆ x t ( s ) dtds t ( s ) + ˆ x s s if | ˆ x | ≤ , (59)and ˆ ∇ s (ˆ x ) = ± e if ± ˆ x ≥ . Note that | ˆ ∇ s (ˆ x ) | = ( ˆ x t ( s ) ) t ( s ) + ( ˆ x s ) s ( ˆ x t ( s ) dtds ( s ) t ( s ) + ˆ x s s ) ≤ s (cid:0) ( ˆ x t ( s ) ) + ( ˆ x s ) (cid:1) s ( ˆ x t ( s ) + ˆ x s ) = 1 if | ˆ x | ≤ , since (58) yields t ( s ) ≤ s and dtds ( s ) t ( s ) ≥ s whenever s ∈ (0 , ) .Let us finally remark that the curve ˆ γ ⊂ { } × {| x | ≥ } ∪ (cid:0) B (0 , ) \ B (0 , ) (cid:1) (60)which is indicated in Figure 9, has the property | ˆ ∇ s | = 1 on ˆ γ. Step 2:
Regularization of the vortex at ˆ x = 0 . In this step, we define a function ˆ ψ on [ − , × R that – up to rescaling and recentering – already coincides with the final ψ closeto { x = 0 } . The subsequent steps 3-6 modify ˆ ψ for large ˆ x ∈ R to achieve the boundaryconditions for | x | → ∞ and to make Lemma 7 applicable.In principle, we would like to set ˆ ψ = 1 − s , but since s is not smooth in ˆ x = 0 this wouldgenerate a vortex-type point-singularity at ˆ x = 0 for ˆ ∇ ⊥ ˆ ψ . Instead, let ρ : [0 , ∞ ) → R be asmooth function that satisfies ρ ( s ) = 1 − s if s ≥ , − ≤ dρds ( s ) ≤ if s ≥ , d n ρds n (0) = 0 for all integers n > . Then the function ˆ ψ (ˆ x ) := ρ (cid:0) s (ˆ x ) (cid:1) , ˆ x ∈ Q, is smooth, satisfies | ˆ ∇ ˆ ψ | = | dds ρ | | ˆ ∇ s | ≤ and can be extended to a smooth function ˆ ψ on [ − , × R by setting ˆ ψ (ˆ x = 0) := ρ ( s = 0) . The regularity of ˆ ψ around ˆ x = 0 is due to s (ˆ x ) = | ˆ x | for | ˆ x | ≤ and d n ρds n (0) = 0 for all n > .Note that by definition of ρ and s we still have | ˆ ∇ ˆ ψ | = 1 on ˆ γ, for ˆ γ as in (60). Step 3:
Extending ˆ ψ to R . Here, we use ˆ ψ (defined on [ − , × R ) to define a smoothfunction ˆ ψ on R with the properties ∂ ˆ x ˆ ψ = 0 if | ˆ x | ≥ , | ˆ ∇ ˆ ψ | ≤ on R , | ˆ ∇ ˆ ψ | = 1 on ˆ γ. (61)40 s −
14 14 − − Figure 11: The non-linear change of variables ϕ .Let ϕ : R → [ − , be a smooth odd non-linear change of variables (cf. Figure 11) with ϕ ( s ) = s on (0 , ) , ϕ ( s ) = 1 if s ≥ , < dds ϕ ( s ) ≤ on (0 , . Then we let ˆ ψ (ˆ x ) := ˆ ψ ( ϕ (ˆ x ) , ˆ x ) for ˆ x ∈ R , such that the properties (61) are easily verified. Step 4:
Matching the boundary conditions on ∂ Ω . In this step, we rescale and recenter ˆ ψ according to Figure 7 to achieve the boundary conditions and − θ on the lower andupper components of ∂ Ω , i.e. (iii).More precisely, we want to obtain (63) below. Since ˆ ψ (ˆ x ) = 1 − | ˆ x | for | ˆ x | ≥ , we placethe center of the “regularized vortex” ˆ x = 0 of ˆ ψ at x θ = (cid:0) , − cos θ (cid:1) , and thereby definethe smooth function: ψ ( x ) := (cid:0) − cos θ (cid:1) ˆ ψ (ˆ x ) for x ∈ R , where ˆ x is related to x via x = x θ + (1 − cos θ )ˆ x. (62)Then ψ ( x ) = 1 − cos θ − | x + cos θ | on (cid:8) | x + cos θ | ≥ − cos θ (cid:9) ⊃ ∂ Ω , (63)such that the boundary conditions hold. Moreover, we have |∇ ψ | ≤ in R as well as |∇ ψ | = 1 on the curve γ that ˆ γ induces via the change of variables (62), cf. Figure 7.Note that ψ only depends on x for | x | ≥ − cos θ ) . However, (ii) does not yet hold. Step 5:
Controlling the behavior for | x | ≫ . To allow for an application of Lemma 7 wewant to obtain (ii), in particular bounded second derivatives of f = 1 − |∇ ψ | . To this end,we will first interpolate ψ in x with the boundary data ψ out := − ( x + 1) cos θ for | x | ≫ . In Step 6, we will then interpolate with ψ out in x .We proceed in two steps: First, we employ a regularized max(˜ t, t ) -function to modify ψ outside of Ω to make sure that the slope of ψ agrees with that of ψ out for large | x | . Then,41 ψ − − θ ψ out ψ ψ in asymptote of ψ Figure 12: Sketch of the functions ψ ( x , · ) (for | x | large, fixed) and ψ out , as well as theirinterpolant ψ in .since | ∂ x ψ out | = cos θ < , we can use interpolation with a slowly varying cut-off function todefine a new function ψ in that coincides with ψ out for | x | ≫ and still satisfies |∇ ψ in | ≤ ,cf. Figure 12.Let η : R → [0 , be a smooth, increasing cut-off function with η ≡ on R − , η ≡ on [1 , ∞ ) , and k dds η k ∞ < ∞ . To regularize max(˜ t, t ) = ˜ t + max(0 , t − ˜ t ) = ˜ t + Z t − ˜ t [0 , ∞ ) ( s ) ds we replace [0 , ∞ ) by η , i.e. we define a smooth h : R × R → R via h (˜ t, t ) := ˜ t + Z t − ˜ t η ( s ) ds. (64)Observe that h (˜ t, t ) = ˜ t if ˜ t ≥ t, and h (˜ t, t ) = Z η ( s ) ds − t, if t ≥ ˜ t + 1 . (65)Moreover, ∂ ˜ t h (˜ t, t ) = 1 − η ( t − ˜ t ) and ∂ t h (˜ t, t ) = η ( t − ˜ t ) .Hence, the function ψ : R → R given by ψ ( x ) := ( ψ ( x ) , for x ∈ Ω ,h (cid:0) ψ ( x ) , ψ out ( x ) − (cid:1) , otherwise , is smooth and satisfies ψ (65) ≡ ψ on Ω , ψ − ψ out (65) = Z ηds − if | x | (63) ≥ − cos θ = : M θ , (66) |∇ ψ | ≤ on Ω , ∂ x ψ = 0 for | x | ≥ , |∇ ψ | ≤ (cid:0) − η ( ψ out − − ψ ) (cid:1) |∇ ψ | + η ( ψ out − − ψ ) |∇ ψ out | ≤ on R \ Ω .
42t remains to interpolate ψ and ψ out : For L ≥ M θ + 1 , consider the slowly varying cut-offfunction η L : R + → [0 , given by η L ( t ) := η ( t − M θ L − M θ ) . Then η L ( t ) = 0 if t ≤ , η L ( t ) = 1 if t ≥ L, ( ddt η L ) ≤ C ( θ ) L η L ≤ C ( θ ) L on R + , (67)and we define the smooth function ψ in : R → R by ψ in ( x ) := η L ( | x | ) ψ out ( x ) + (cid:0) − η L ( | x | ) (cid:1) ψ ( x ) , There exists L ( θ ) such that for any L ≥ L ( θ ) we have |∇ ψ in | ≤ on R : Note that |∇ ψ in | = |∇ ψ | ≤ on {| x | ≤ M θ } , while on {| x | ≥ M θ } we have ψ in (66) = ψ out − (2 − R ηds )(1 − η L ( | x | )) , such that due to ∇ ψ out = (0 , − cos θ ) |∇ ψ in | ≤ cos θ + C ddt η L (67) ≤ cos θ + CL ≤ for L sufficiently large . Moreover, ∂ x ψ in = 0 for | x | ≥ − cos θ ) . Step 6:
Interpolation with the boundary conditions at x = ±∞ . In order to obtain (ii) itnow remains to interpolate ψ in with the boundary data ψ out for | x | ≫ .For this, we again consider the cut-off function η L : R + → [0 , with properties (67) anddefine the desired smooth ψ : R → R by ψ ( x ) := η L ( | x | ) ψ out ( x ) + (cid:0) − η L ( | x | ) (cid:1) ψ in ( x ) . Clearly, ψ satisfies the boundary conditions on ∂ Ω as well as ψ = ψ out for | x | ≫ . In thecore region {| x | ≤ M θ } ∩ Ω we have ψ = ψ in = ψ = ψ and therefore |∇ ψ | = 1 on thecurve γ defined in Step 4. Moreover, |∇ ψ | ≤ on {| x | ≤ M θ } ∪ {| x | ≥ L } .For sufficiently large L ≥ L ( θ ) we can also assert |∇ ψ | ≤ on { M θ ≤ | x | ≤ L } : In fact, wehave ∇ ψ = (cid:0) sgn( x )( ψ out − ψ in ) ddt η L , − η L cos θ + (1 − η L ) ∂ x ψ in (cid:1) , where we used that ∂ x ψ in ( x ) = 0 on {| x | ≥ M θ } . Hence, by convexity of z z , η L ∈ [0 , ,and | ∂ x ψ in | ≤ : |∇ ψ | ≤ ( ddt η L ) sup | ψ out − ψ in | | {z } ≤ by def. of ψ in and (64) + (cid:0) (1 − η L ) ∂ x ψ in + η L ( − cos θ ) (cid:1) ≤ C ( ddt η L ) + (1 − η L ) + η L (cos θ ) (67) ≤ − (cid:0) sin θ − CC ( θ ) L (cid:1) η L ≤ if L ≥ L ( θ ) is sufficiently large.43 = R × [ − ,
1] ( m , m ) ∈ S m m Figure 13: ( m , m ) on ∂ Ω . Step 7:
The degree of ( m , m ) . Using η L ( | x | ) = 0 and ψ = ψ (63) = (1 − cos θ ) − | x + cos θ | in a neighbourhood of ∂ Ω , we compute ∇ ψ in ( x ) = ∂ x ψ ( x ) e = − sgn( x ) e on ∂ Ω , and therefore, due to ψ out = ψ in on ∂ Ω ∇ ψ ( x ) = η L ( | x | ) ∇ ψ out ( x ) + (cid:0) − η L ( | x | ) (cid:1) ∇ ψ in ( x )= − (cid:0) η L ( | x | ) cos θ + (1 − η L ( | x | )) sgn( x ) (cid:1) e on ∂ Ω , or in view of the definition of m in (57): m ( x ) = (cid:0) η L ( | x | ) cos θ + (1 − η L ( | x | )) sgn( x ) , sgn( x ) q − m ( x ) , (cid:1) on ∂ Ω . Hence, ( m , m ) , as a map S → S , has degree − on ∂ Ω , cf. Figure 13.In order to prove that the magnetization m defined at (57) belongs to ˙ H (Ω) it is enough tocheck that f = 1 − |∇ ψ | has the property that √ f is Lipschitz in R where ψ is the streamfunction constructed above: Lemma 7.
Let f ∈ C ( R N , R + ) be a non-negative function with D f ∈ L ∞ ( R N ) . Then ∇√ f ∈ L ∞ ( R N ) and we have k∇ p f k ∞ ≤ k D f k ∞ . (68) Proof.
We distinguish two cases:
Case 1: D f ≡ on R N , i.e., f is an affine function. Since by assumption f ≥ in R N ,one has f ≡ const. Thus, the assertion of Lemma 7 becomes trivial. Case 2: k D f k ∞ > . Let x, x ∈ R N . Taylor’s expansion yields for some intermediate ˜ x ∈ R N : ≤ f ( x ) = f ( x ) + ∇ f ( x ) · ( x − x ) + ( x − x ) · D f (˜ x )( x − x ) ≤ | f ( x ) | + ∇ f ( x ) · ( x − x ) + k D f k ∞ | x − x | . (69)44ence, choosing x ∈ R N such that x − x = − ∇ f ( x ) k D f k ∞ , we obtain |∇ f ( x ) | k D f k ∞ ≤ | f ( x ) | , i.e. |∇ p f ( x ) | ≤ √ k D f k ∞ if f ( x ) = 0 . (70)If there exist points at which f vanishes, we apply (70) to f + ε instead of f (with ε > ),and deduce for x, y ∈ R N | p f ( x ) + ε − p f ( y ) + ε | ≤ Z | ( ∇ p f + ε )( tx + (1 − t ) y ) | | x − y | dt (70) ≤ √ k D f k ∞ | x − y | . Letting ε ↓ we obtain | p f ( x ) − p f ( y ) | ≤ √ k D f k ∞ | x − y | for all x, y ∈ R N , such that(68) follows. B Proof of Lemma 2
The relation between the topological degree of ( m , m ) on ∂ Ω and the vortex singularity of ( m , m ) observed in the previous construction is studied in Lemma 2 that we prove in thefollowing: Proof of Lemma 2.
Due to ∇ · m ′ = 0 in D ′ ( R ) we may represent m ′ = ∇ ⊥ ψ for a streamfunction ψ : R → R with ψ ( x , −
1) = 0 , ψ ( x ,
1) = − θ for all x ∈ R . Under the hy-pothesis m ∈ ˙ H (Ω , S ) , one gets that ∇ ψ ∈ ˙ H ∩ L ∞ (Ω) . Since m has non-zero topologicaldegree on Bdry , the set { x ∈ ∂ Ω | m ( x ) < } is non-empty (recall that m ∈ ˙ H / ( ∂ Ω , S ) ).We assume that it has non-empty intersection with R × {− } . (The other case is simi-lar.) Since m = − ∂ x ψ < and ψ = 0 on that subset of R × {− } , one sees that theset { x ∈ Ω | ψ > } is non-empty. In particular, there exists a connected component C of { x ∈ Ω | ψ > } whose boundary intersects R × {− } on a set containing an interval (seeFigure 14).Since ∇ ψ ∈ ˙ H (Ω) , and taking into account the boundary conditions at x = ±∞ , theSobolev embedding theorem on sets { a ≤ | x | ≤ a + 1 } yields ψ ( x , x ) → − ( x + 1) cos θ uniformly in x as | x | ↑ ∞ . Hence, any level set { ψ = ε } for ε > is bounded, and ψ attainsa maximum x in the interior of C ⊂ Ω . Let β = ψ ( x ) > .In the case of a vector field m ∈ C (Ω) , so ψ ∈ C (Ω ⊂ R , R ) , by Sard’s theorem, thereexists a regular value β ∈ (0 , β ) of ψ . In particular, there exists a smooth cycle γ ⊂ C such that ψ ≡ β and |∇ ψ | > on γ . Therefore, ν := ∇ ψ |∇ ψ | : γ → S is a normal vector fieldat γ so that it carries a topological degree equal to . Hence, ( m , m ) = ∇ ⊥ ψ presents a45 = 0 Figure 14: The zero level set of ψ for an asymmetric Bloch wall of angle θ = 0 . (cf. footnotein Fig. 3).vortex singularity inside Ω carrying a non-zero winding number. This argument is still validfor the general case m ∈ ˙ H (Ω) (where ∇ ψ ∈ ˙ H (Ω) ); in fact, Sard’s theorem is valid alsofor ψ ∈ W , loc ( C ) (see e.g., Bourgain-Korobkov-Kristensen [3]) where we recall that ψ > on C and ψ = 0 on ∂C so that almost all β ∈ (0 , β ) is a regular value, i.e., the pre-image ψ − ( β ) is a finite disjoint family of C -cycles and the normal vector field ν on each cycle isabsolutely continuous, in particular, it carries a winding number . Acknowledgements
R.I. gratefully acknowledges the hospitality of Max Planck Institute (Leipzig) where part ofthis work was carried out; he also acknowledges partial support by the ANR projects ANR-08-BLAN-0199-01 and ANR-10-JCJC 0106. L.D. acknowledges support of the InternationalMax Planck Research School. F.O. acknowledges the hospitality of Fondation Hadamard,and both L.D. and F.O. acknowledge the hospitality of the mathematics department of theUniversity of Paris-Sud.
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