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On several types of hysteresis phenomena appearing in porous media

  • In honor of Pavel Krejčí in occasion of his decennial birthday.
  • Received: 13 December 2024 Revised: 29 April 2025 Accepted: 16 June 2025 Published: 19 June 2025
  • This article deals with several hysteresis effects occurring in soils and other porous media. Soils consist of solid particles which are of different size, shape and material behavior depending on the soil type. The pore space is filled with one pore fluid for saturated porous media, with two or more immiscible fluids for partially or unsaturated media. Most of the hysteresis phenomena described can be traced back to the behavior of the pore fluids, e.g., they have to do with different capillary pressures for different degrees of saturation. Hysteresis describes that a quantity or process is not single-valued, that in dependence on the prehistory for the same input value two or more possible output values may occur. The aim of the paper is to show various examples for which hysteresis effects occur and to discuss the reasons – either theoretically or experimentally. Triggered by an actual incident, highlighted in the introduction, in a first paragraph stage/discharge hysteresis of rivers is explained. The most prominent type of hysteresis, hydraulic hysteresis, is treated in the next section. Engineering and mathematical approaches are discussed and results of own laboratory experiments are shown. Further pairs of processes leading to non-unique behavior are adsorption/desorption processes or freezing/thawing processes. But not only the behavior and properties of the pore fluids lead to hysteretic effects. Also the constitutive relations of dry soils exhibit hysteresis. Generally, soils behave differently if first loaded, unloaded and reloaded. It is shown in a further paragraph that the stress-strain curves are hysteretic. The aim of the paper is to show that in porous media where the solid and fluid components interact, accordingly, a variety of hysteresis effects appear and to highlight the different phenomena, show the variety of hysteresis curves and indicate description methods.

    Citation: Bettina Detmann. On several types of hysteresis phenomena appearing in porous media[J]. Mathematics in Engineering, 2025, 7(3): 384-405. doi: 10.3934/mine.2025016

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  • This article deals with several hysteresis effects occurring in soils and other porous media. Soils consist of solid particles which are of different size, shape and material behavior depending on the soil type. The pore space is filled with one pore fluid for saturated porous media, with two or more immiscible fluids for partially or unsaturated media. Most of the hysteresis phenomena described can be traced back to the behavior of the pore fluids, e.g., they have to do with different capillary pressures for different degrees of saturation. Hysteresis describes that a quantity or process is not single-valued, that in dependence on the prehistory for the same input value two or more possible output values may occur. The aim of the paper is to show various examples for which hysteresis effects occur and to discuss the reasons – either theoretically or experimentally. Triggered by an actual incident, highlighted in the introduction, in a first paragraph stage/discharge hysteresis of rivers is explained. The most prominent type of hysteresis, hydraulic hysteresis, is treated in the next section. Engineering and mathematical approaches are discussed and results of own laboratory experiments are shown. Further pairs of processes leading to non-unique behavior are adsorption/desorption processes or freezing/thawing processes. But not only the behavior and properties of the pore fluids lead to hysteretic effects. Also the constitutive relations of dry soils exhibit hysteresis. Generally, soils behave differently if first loaded, unloaded and reloaded. It is shown in a further paragraph that the stress-strain curves are hysteretic. The aim of the paper is to show that in porous media where the solid and fluid components interact, accordingly, a variety of hysteresis effects appear and to highlight the different phenomena, show the variety of hysteresis curves and indicate description methods.



    Lattice systems are discrete variational models, whose energy depends on a spin field defined in a lattice. In frustrated lattice systems, spins cannot find an orientation that simultaneously minimizes the nearest-neighbor (NN) and the next-nearest-neighbor (NNN) interactions. Such interactions are said to be ferromagnetic or antiferromagnetic if they favour alignment or anti-alignment (we address the reader to [13] for a complete dissertation).

    Three-dimensional frustrated magnets generally exist in the magnetic diamond and pyrochlore lattices (see [14]) and edge-sharing chains of cuprates provide a natural example of frustrated lattice systems (see [16]). Furthermore, jarosites are the prototype for a spin-frustrated magnetic structure, because these materials are composed exclusively of kagomé layers (see [20]).

    A different frustration mechanism can also be caused by magnetic anisotropy, as it is common in spin ices (see [17]). Magnetic anisotropy refers to the dependence of the magnetization of a material on the direction of the applied magnetic field, which acts as a potential barrier (we address the reader to [23] for a comprehensive overview of magnetism, including a chapter on magnetic anisotropy and the energy barrier). The interplay between the two frustration mechanisms may result in very complicated Hamiltonians (see [22]). Most recently, the physics community attempts to find new fundamental effects such as the magnetization plateaus and the magnetization jumps which represent a genuine macroscopic quantum effect. For example, kagomé staircases have been of particular interest because of the concurrent presence of both highly frustrated lattice and strong quantum fluctuations (see [24]).

    In this paper we study a frustrated lattice spin system whose spins take values on the unit sphere of R3. More precisely, a spin of the system u is a vectorial function whose codomain is the union of two fixed disjoint circles, S1 and S2, of the unit sphere, which have the same radius R and are identified by two versors, v1 and v2, Figure 1. We set the problem in one and two dimensions: in the one-dimensional case (Section 3) spin fields are parametrized over the points of the discrete set [0,1]λnZ and satisfy a periodic boundary condition; in the two-dimensional case (Section 4) they are parametrized over the points of the discrete set ΩλnZ2, where ΩR2 is an open bounded regular domain. In both cases {λn}nN is a vanishing sequence of lattice spacings. In the first setting, the energy of a given spin of the system u:λni[0,1]λnZuiS1S2 is

    En(u)=En(u)+Pn(u),
    Figure 1.  S1 and S2 circles of anisotropy transitions.

    with

    En(u)=i[0,1]λnZλn[αuiui+1+uiui+2]andPn(u)=λnkn|DA(u)|(I),

    where α(0,+) is the frustration parameter of the system that rules the NN and NNN interactions and {kn}nN is a divergent sequence of positive numbers. The term A(u) indicates the spins' magnetization direction (the so-called magnetic anisotropy) in the two circles. If the number of magnetic anisotropy transitions, i.e., the number of the jumps between the two circles, is finite, A(u) is a BV function and |DA(u)|(I) counts them. According to physical considerations, we require that the energy Pn gives a penalizing contribution to the total energy.

    It is easy to see that while the first term of the energy En is ferromagnetic and favors the alignment of neighboring spins, the second one, being antiferromagnetic, frustrates it as it favors antipodal next-to-nearest neighboring spins. A more refined analysis, contained in Proposition 3.5 and Remark 3.6, shows that, for n sufficiently large, the ground states of the system take values on one of the two circles and for α4 are ferromagnetic (the spins are made up of aligned vectors), while for 0<α4 they are helimagnetic (the spins consists in rotating vectors with a constant angle ϕ=±arccos(α/4)). The property of the latter case is known in literature as chirality simmetry: the two possible choices for the angle correspond to either clockwise or counterclockwise spin rotations, or in other words to a positive or a negative chirality.

    In this paper, we address a system close to the ferromagnet/helimagnet transition point (see [15]), that is when α is close to 4 from below. We also require that λnkn is close to some positive value (that can be also infinite). This assumption is reasonable, since from a physical point of view the change of the spin's polarization involves a larger amount of energy. Our aim is to provide a careful description of the admissible states and compute their associated energy. In particular, we find the correct scalings to detect the following two phenomena: the spins' magnetic anistropy transitions and chirality transitions that break the rigid simmetry of minimal configurations.

    In [12], the authors studied a one-dimensional ferromagnetic/antiferromagnetic frustrated spin system with nearest and next-to-nearest interactions close to the helimagnet/ferromagnet transition point as the number of particles diverges. In that case, spin fields take values in the unit circle. The proposed model is different from that one, where no anisotropy functional Pn was introduced. In [12] the presence of a periodic boundary condition allowed manipulating En in such a way that it can be recast as a discrete version of a Modica-Mortola type energy, whose Γ-convergence is well-known in literature (see [18] and [19]). Indeed, expanding the functional at the first order, under a suitable scaling, spin fields can make a chirality transition on a scale of order λnδn, when λnδn approaches to a finite nonnegative value, as n+ (otherwise no chirality transitions emerge).

    To set up our problem, we let the ferromagnetic interaction parameter α depend on n and be close to 4 from below, that is, we substitute α by αn=4(1δn) for some positive vanishing sequence {δn}nN. As in [12], the Γ-limit of En (with respect to the weak convergence in L) does not provide a detailed description of the phenomena (as a consequence of Theorem 3.12) and suggests that, in order to get further information on the ground states of the system, we need to consider higher order Γ-limits (see [6] and [7]).

    The two phenomena can be detected at different orders. At the first order we are led to normalize the energy En of the system and study the asymptotic behavior of (a rescaling of) the new functional Gn defined by

    Gn=EnminEn.

    Rescaling Gn by λn, we prove that magnetic anisotropy transitions can be captured when λnkn is close to any positive finite value, for n large enough (see Theorem 3.16). At the scale value λn, the energy spent for spin's magnetic anisotropy transitions is equal to the minimal energetic value corresponding to the sum of all the interactions in proximity of the transition points. In Figure 2 it can be seen an occurrence of the phenomenon that we are analyzing.

    Figure 2.  Magnetic anisotropy transitions.

    Chirality transitions can be detected at the next order by means of a technical decomposition of the energy Gn. The idea behind the construction in Subsection 3.6 is to split the problem set in the sphere into finitely many problems set in one of the two circles each. We associate each spin field u with a unique and finite partition of [0,1] containing intervals Ij such that u|Ij takes values only in one circle. We note that the intervals Ij depend on n because u is defined on the lattice [0,1]λnZ. We modify such restrictions u|Ij in such a way that they still satisfy a similar periodic boundary condition on Ij, denoting them as ˜uIj. In Lemma 3.13 we decompose the functional Gn as follows:

    Gn(u)=jMMn(˜uIj)+j(Rn)j(u)+(Rn)M(u)(u)+Rn(u).

    The energy MMn is of discrete Modica-Mortola type and collects the pairwise interactions of spins' vectors pointing to the same circle; the functionals (Rn)j and (Rn)M(u) gather the interactions between consecutive spins' vectors that point to different circles. Rn is a correction addend. The first sum and the other addend in the right-hand side of the previous formula need to be rescaled in different ways, the first sum being a higher order term. Thus, at the second order we deal with the energy

    Gn(u)=Gn(u)j(Rn)j(u)(Rn)M(u)(u)Rn(u)=jMMn(˜uIj).

    In Theorem 3.18 we apply the Γ-convergence result contained in [12] to each functional MMn, rescaled by λnδ3/2n. It turns out that different scenarios may occur, depending on the value of limnλn/δn:=l[0,+]. If l=+, chirality transitions are forbidden. Otherwise a spin field can make a chirality transition on a lenght-scale λn/δn. In particular, if l>0, it may have diffuse and regular macroscopic (on an order one scale) chirality transitions in each Sj whose limit energy is finite on H1(Ij) (provided some boundary conditions are taken into account); if l=0, chirality transitions on a mesoscopic scale are allowed. In this case, the continuum limit energy is finite on BV(Ij) and counts the number of jumps of the chirality of the spin field.

    Systems defined in planar structures are much more difficult to study, due to the higher dimensional setting (see [1,4,5,10,11]). We address here the two-dimensional analogue of the frustrated spin chain studied in the first part of the paper. The energy of a given spin of the system u:(i,j)ΩλnZ2ui,jS1S2 is

    En(u;Ω)=En(u;Ω)+Pn(u;Ω),

    where

    En(u;Ω)=α(i,j)λ2n(ui,jui+1,j+ui,jui,j+1)+(i,j)λ2n(ui,jui+2,j+ui,jui,j+2)

    and

    Pn(u;Ω):=λnkn|DA(u)|(Ω).

    We assume that the functional Pn(;Ω) is bounded. The number α>0 is the frustration parameter of the system and {kn}nN is a divergent sequence of positive numbers. The term |DA(u)|(Ω) is related to magnetic anistropy transitions. In the two-dimensional setting, they occur on the edges of the lattice ΩλnZ2 and the natural number |DA(u)|(Ω)λn|v1v2| is an upper bound on the spins' transitions from a circle to the other in Ω.

    Motivated by the variational analysis of the one-dimensional problem, we assume that the frustation parameter depend on n and is close to the helimagnet/ferromagnet transition point as the number of particles diverges, i.e., αn4. In view of detecting spins' chirality transitions, which cannot be captured by means of the Γ-limit of the energy at the zero order, we are interested in the functional defined by

    Hn(u;Ω):=12λnδ32n12λ2n(i,j)[|ui+2,jαn2ui+1,j+ui,j|2+|ui,j+2αn2ui,j+1+ui,j|2],

    which is the two-dimensional analogue of Gn, up to additive constants.

    In [10] the authors studied a similar frustrated spin chain whose spin fields take values in the unit circle of R2. In [10,Theorem 2.1] they proved the emergence of spins' chirality transitions by means of the Γ-convergence of the functional Hn with respect to the local L1-convergence of two chirality parameters.

    In view of applying their result in our setting, we employ an idea that recalls the construction carried out in the one-dimensional problem. We restrict every spin u to connected open sets Cs that partition Ω in such a way that u|Cs takes values only in one circle. In order to avoid more complicated notation, we do not impose boundary conditions on Ω and we state the result by means of a local convergence. We note that the sets Cs depend on n because u is defined on the lattice ΩλnZ2.

    We decompose

    Hn(u;Ω)=s[Hn(u;Cs)+(Rn)Cs(u)],

    where Hn collects the interactions of spins' vectors pointing to the same circle and (Rn)Cs gathers the interactions between spins' vectors that point to different circles.

    While in the one-dimensional setting the partition associated with a spin contains intervals, which guaranty the compactness results stated, in this case the sets Cs could be very wild, as the spacing of the lattice shrinks. Therefore, we require as additional regularity condition for the components Cs, that is the BVG regularity. Its definition can be found in [21] and is recalled in Definition 4.1.

    With this regularity assumption, we can apply the Γ-convergence result proved in [9] to each addend of the functional

    Gn(h;Ω)=Hn(h,Ω)s(Rn)Cs(h)=sHn(h;Cs),

    as it is shown in Theorem 4.5, that is the main result of Section 4. It turns out that chirality transitions are possible and they can take place both in the vertical and horizontal slices of Cs.

    Given xR, we denote by x the integer part of x. For a set K we denote by co(K) the convex hull of K, by #K the number of its elements and χK its characteristic function. We write vw for the Euclidean scalar product of the vectors v,wR3 and by S2 the unit sphere of R3. For all vR3 we denote by πv the Euclidean projection on v and by πv the projection on the orthogonal complement of v. If A is a subset of the Euclidean space we denote by ¯A its closure respect the Euclidean topology. We denote by C a generic constant that may vary from line to line in the same formula and between formulas. Relevant dependencies on parameters and special constants will be suitably emphasized using parentheses or subscripts.

    If IR is an interval and all wBV(I;R3), we denote by DwMb(I;R3) the distributional differential of w, and by |Dw|Mb(I) the total variation measure of Dw. We say that a sequence {un}nN converges weakly in BV(I;R3) to a function uBV(I;R3) if and only if

    unuin L1(I;R3)  and supnN|Dun|(I)<+,

    (see [3,Definition 3.11 and Proposition 3.13]). We denote it by unBVu.

    Fixing v1,v2S2 and R(0,1), we define the set

    Si:={wS2:|πvi(w)|=R,w,vi>0},for i{1,2}. (2.1)

    It is easy to observe that the set Si is a circle centered in vi1R2 and it can be easily verified that for R<RMax:=1v1v22 the sets S1 and S2 are disjoint. Throughout the paper we assume that R(0,RMax).

    If S is an open set of RN and C is a collection of open subsets of S, we say that C is an open partition of S if C does not contain empty sets and

    ¯S=CC¯C,C1C2=,C1,C2C.

    Given two vectors w=(w1,w2),¯w=(¯w1,¯w2) of R2, we define the function

    χ[w,¯w]:=sign(w1¯w2w2¯w1),

    with the convention that sign(0)=1.

    We let I=(0,1) and we consider a sequence {λn}nNR+ that vanishes as n+. It represents a sequence of spacings of the lattice ¯IλnZ.

    We introduce the class of functions valued in S1S2 which are piecewise constant on the edges of the lattice ¯IλnZ and satisfy a periodic boundary condition:

    PCλn:={u:¯IS1S2:u(t)=u(λni) for tλn[i+[0,1)]¯I and λni¯IλnZ,u0u1=u1λn1u1λn}. (3.1)

    We will identify a piecewise constant function u:¯IS1S2 with the function defined on the points of the lattice given by λni¯IλnZui:=u(λni). Conversely, given values uiS1S2 for λni¯IλnZ, we define u:¯IS1S2 by u(t):=ui for tλn[i+[0,1)].

    There exists a natural projection map A:L(I;S1S2)L(I;{v1,v2}) defined as follows:

    A(u)(t)={v1if u(t)S1,v2if u(t)S2,tI. (3.2)

    For each spin u, the function A(u) indicates the spins' magnetization direction and its jumps correspond the the spins' magnetic anisotropy transitions. In general, A can be defined analogously on L(I;K1K2), if K1 and K2 are two disjoint subsets of R3 containing, respectively, S1 and S2. In this case, we remark that if a spin field uL(I;K1K2) switches from K1 to K2 a finite number of times, i.e., A(u)BV(I;{v1,v2}) and so |DA(u)|(I)<+, the interval I can be partitioned in finitely many regions where the function u takes values only in one of the two sets K1 and K2. In other words, there exist M(u)N and a collection of open intervals, {Ij}j{1,,M(u)}, such that

    {Ij}j{1,,M(u)} is an open partition of I, (3.3)
    either u(Ij)K1 or u(Ij)K2, for any j{1,,M(u)}, (3.4)
    u(Ij)×u(Ij+1)(K1×K2)(K2×K1),for any j{1,,M(u)1}. (3.5)

    The last two properties imply that this partition is unique. We observe that, if uL(I;S1S2) and A(u)BV(I;{v1,v2}) (or, in particular, if uPCλn), then

    M(u)=|DA(u)|(I)|v1v2|+1.

    The following definition will be useful throughout the section.

    Definition 3.1. Let uL(I;S1S2) be such that A(u)BV(I;{v1,v2}). We say that Cn(u)={Ij|j{1,,M(u)}} is the open partition associated with u if M(u)=|DA(u)|(I)|v1v2|+1 and the collection of open intervals {Ij}j{1,,M(u)} satisfies (3.3), (3.4) and (3.5).

    In this subsection we recall some classical properties of the Lebesgue space L(I;K), where KR3 is a compact set. The statements and the proofs are fully analogous if the setting is a N-dimensional Euclidean space.

    Proposition 3.2. Let {fn}nNL(I;K). Then, up to subsequences, fnfL(I;co(K)) in the weak topology of L(I;R3). Moreover, for all uL(I;co(K)) there exists a sequence {un}nNL(I;K) of piecewise constant functions such that unu.

    Proof. Since the set K is bounded then, up to a subsequence, there exists fL(I;R3) such that fnf. Now we prove that f(t)co(K) for almost every tI. For every ξco(K) there exist an affine function hξ:RNR and α<0 such that

    hξ(ξ)>0>α>hξ(x),xco(K).

    By the weak convergence of {fn}nN we have that for any measurable set AI

    Ahξ(f(t))dt=limn+Ahξ(fn(t))dt|A|α<0.

    Hence, by the arbitrariness of A, we obtain

    hξ(f(t))<0, for a.e. tI. (3.6)

    Recalling that

    co(K)=jN{yR3:hξj(y)<0,ξjQNco(K)},

    by formula (3.6) we obtain

    f(t)co(K), for a.e. tI.

    Now we prove the second statement of the proposition. Let uL(I;co(K)). There exists a sequence {un}nNL(I;co(K)) such that un=mj=1ajχIj, where ajco(K) and IjI is an interval, for any j{1,,m}, and un converges to u in L1(I;R3). Hence, unu. Therefore, without loss of generality, we may prove the statement for u=aco(K).

    We define the following function:

    h(t):={a1if t(0,λ),a2if t[λ,1),

    where a=λa1+(1λ)a2 with a1,a2K and for some λ[0,1]. Then the sequence un(t):=h(nt) converges to u in the weak topology of L by Riemann-Lebesgue's lemma.

    Corollary 3.3. The closure of the set L(I;K) with respect to the weak topology of L(I;R3) is the set L(I;co(K)).

    Proof. Since the space L1(I;R3) is separable, every bounded subset of L(I;R3) is metrizable with respect to the weak topology of L(I;R3). Hence the set L(I;K) is metrizable. Therefore, by Proposition 3.2, we have that the set L(I;co(K)) is the weak closure of the set L(I;K).

    In this subsection we cite an abstract Γ-convergence result proved in [2] that will be applied in Subsection 3.5. For this purpose, we introduce the following notation. Let KRN be a compact set and for all ξZ let fξ:R2NR be a function such that

    (H1) fξ(x,y)=fξ(y,x),

    (H2) for all ξZ, fξ(x,y)=+ if (x,y)K2,

    (H3) for all ξZ there exists Cξ0 such that

    sup(x,y)K2|fξ(x,y)|CξandξZCξ<+.

    For any nN we define the functional space

    Dn(I;RN):={u:RRN:u is constant in λn(i+[0,1)) for all λni¯IλnZ}.

    With the notation already used, we denote the value of the function u in the interval λn(i+[0,1)) by ui for all λni¯IλnZ. We introduce the sequence of functionals Fn:L(I;RN)(,+] defined as follows:

    Fn(u):={ξZiRξn(I)λnfξ(ui,ui+ξ) for uDn(I;RN),+ for uL(I;RN)Dn(I;RN),

    where Rξn(I):={λni¯IλnZ:λni+ξ¯IλnZ}, for ξZ. For any open and bounded set AR and for every u:ZRN, we define the discrete average of u in A as

    (u)1,A:=1#(ZA)iZAui.

    Theorem 3.4 (See [2]). Let {fξ}ξZ be a family of functions that satisfies H1, H2, H3. Then the sequence Fn Γ-converges, as n+, with respect to the weak topology of L(I;RN), to

    F(u):={Ifhom(u(t))dtforuL(I;co(K)),+foruL(I;RN)L(I;co(K)),

    where fhom:RNR is given by the following homogenization formula

    fhom(z)=limρ0limk+1kinf{ξZβRξ1((0,k))fξ(u(β),u(β+ξ))|us.t.(u)1,(0,k)¯B(z,ρ)}.

    This subsection is devoted to the mathematical formulation of the model and the characterization of its ground states.

    Let α>0 be a fixed parameter and let {kn}nNR+ be a divergent sequence of positive numbers. Denoting

    In(I):=(¯IλnZ){λn(1λn1),λn1λn},

    and in general if J=(a,b)I we define

    In(J):=(¯JλnZ){λn(bλn1),λnbλn}.

    We define the energy of the system as the sum of two addends. The first addend is a bulk scaled energy of a frustrated F-AF spin chain, En:PCλn(,+), having the following form:

    En(u):=λniIn(I)[αuiui+1+uiui+2].

    The second addend of the energy, Pn:PCλn[0,+), is a term of confinement in S1S2 and is defined as follows:

    Pn(u):=λnkn|DA(u)|(I),

    where A is the function defined in formula (3.2). We consider the family of energies En:PCλn(,+) defined by

    En(u)=En(u)+Pn(u).

    Furthermore, we define the functional Hn:PCλn[0,+) by

    Hn(u):=12λniIn(I)|ui+2α2ui+1+ui|2.

    If uPCλn, since |ui|=1 for all λni¯IλnZ, thanks to the boundary condition contained in the definition of PCλn (see (3.1)), we compute:

    Hn(u)=12λniIn(I)(2+α24+2uiui+2αuiui+1αui+1ui+2)=12iIn(I)λn[α(uiui+1+ui+1ui+2)+2uiui+2]+λn(1+α28)#In(I)=En(u)+λn(1+α28)#In(I). (3.7)

    Thus we gain a new expression for En:

    En=Hn+Pnλn(1+α28)#In(I). (3.8)

    Thanks to this decomposition, we characterize the ground states of En.

    Proposition 3.5 (Characterization of the ground states of En). Let 0<α4. Then, for nN sufficiently large, it holds

    minuPCλnEn(u)=λn#In(I)[R2(1+α28)+(α1)(1R2)].

    Furthermore, a minimizer un of En over PCλn takes values only in one circle S, with {1,2}, and satisfies

    πvuinπvui+1n=R2α4andπvuinπvui+2n=R2(α281),iIn(I).

    Proof. Let us postpone the proof of the following equality:

    minuPCλnu(I)S1oru(I)S2En(u)=λn#In(I)[R2(1+α28)+(α1)(1R2)] (3.9)

    after the next claim. We claim that, for n sufficiently large, if uPCλn is a minimizer of En, then u(I)S1 or u(I)S2. We may assume that the open partition associated with u is {I1,I2}, i.e., M=2, and u(I1)S1, u(I2)S2. The general case MN can be proved similarly. We have that

    En(u)=En(u)+λnknc=λniIn(I1)In(I2)[αuiui+1+uiui+2]+λniIn(I)(In(I1)In(I2))][αuiui+1+uiui+2]+λnknc, (3.10)

    where c:=|v1v2|. We observe that #[In(I)(In(I1)In(I2))]2. We define

    L:=min{αuv+uwαvw+vz:u,vS1,w,zS2}

    and we observe that

    LiIn(I)(In(I1)In(I2))[αuiui+1+uiui+2].

    Therefore by the formula (3.10) we have that

    minuPCλnu(I1)S1En(u)+minuPCλnu(I2)S2En(u)+λn(L+knc)En(u). (3.11)

    In order to prove the claim we are left to show that, for nN sufficiently large,

    minuPCλnu(I)S1oru(I)S2En(u)<minuPCλnu(I1)S1En(u)+minuPCλnu(I2)S2En(u)+λn(L+knc), (3.12)

    which is equivalent to prove that

    λn#In(I)[R2(1+α28)+(α1)(1R2)]<λn(#In(I1)+#In(I2))[R2(1+α28)+(α1)(1R2)]+λn(L+knc),

    where we used formula (3.9). Since #In(I)#In(I1)#In(I2)2, R1 and α4, we have that, for n sufficiently large,

    λn(#In(I)#In(I1)#In(I2))[R2(1+α28)+(α1)(1R2)]12λn<λn(L+knc),

    because kn+. We have proved the validity of (3.12). Thus, combining (3.12) and (3.11), we get

    minuPCλnEn(u)=minuPCλnu(I)S1oru(I)S2En(u).

    We prove that

    minuPCλnu(I)S1oru(I)S2En(u)=λn#In(I)[R2(1+α28)+(α1)(1R2)].

    We fix {1,2} and consider uPCλn such that u(I)S. By geometric and trigonometric identities we deduce that

    uiui+1=1R2+πuiπui+1,

    where πui:=πvui. Of course an analogous statement holds for uiui+2. Thus

    En(u)=iIn(I)λn[απuiπui+1+πuiπui+2](α1)(1R2)λn#In(I)=˜En(u)(α1)(1R2)λn#In(I), (3.13)

    where we have defined

    ˜En(u):=iIn(I)λn[απuiπui+1+πuiπui+2].

    Now we are led to minimize ˜En. We find its minimum by following the same argument in [12]. With an easy computation similar to the one in (3.7), we remark that

    ˜En(u)=12λniIn(I)|πui+2α2πui+1+πui|2R2(1+α28)λn#In(I)=˜Hn(u)R2(1+α28)λn#In(I), (3.14)

    where

    ˜Hn(u):=12λniIn(I)|πui+2α2πui+1+πui|2.

    We fix ϕ[π2,π2] so that cosϕ=α4. We may assume for simplicity of notation that v=en. Let

    ui:=(Rcos(ϕi),Rsin(ϕi),1R2)S,i¯IλnZ,

    so that πui=(Rcos(ϕi),Rsin(ϕi),0). By trigonometric identities, we have that

    πui+πui+2=2cos(ϕ)πui+1=α2πui+1,iIn(I).

    Remarking that ˜Hn(u)=0, we combine the previous identity with (3.14) to get that

    minuPCλnu(I)S1oru(I)S2˜En(u)=R2(1+α28)λn#In(I).

    The computation of the minimum follows from (3.13).

    Now we consider a minimizer uPCλn of En. For n sufficently large, it must hold that u(I)S, for some {1,2}, and

    ˜En(u)=R2(1+α28)λn#In(I),

    thus implying that ˜Hn(u)=0. It follows that

    πui+1=2α(πui+πui+2),iIn(I).

    Squaring the modulus of both sides in the previous equality, we infer

    πuiπui+2=R2(α281).

    Hence

    πuiπui+1=2α(πuiπui+πuiπui+2)=2α(R2+πuiπui+2)=R2α4,

    which concludes the proof.

    From now on we assume that n is sufficiently large to satisfy the thesis of the above proposition.

    Remark 3.6. The case α>4 is trivial and the ground states of En are all ferromagnetic, i.e., ui=¯u, for all i¯IλnZ and for some ¯uS1S2. Indeed, denoting by E(α=4)n the energy of formula (3.8) for α=4, we have that

    En(u)=E(α=4)n(u)λn(α4)iIn(I)uiui+1,

    for all uPCλn. By the above proposition, the energy E(α=4)n is minimized on ferromagnetic states, which trivially also holds true for the second term in the above sum. The minimal value of En is

    minuPCλnEn(u)=λn(α1)#In(I).

    In this subsection we study the Γ-convergence of En at the zero order. With a slight abuse of notation, we extend the energies En, Pn, En and Hn to the space L(I;co(S1)co(S2)), setting their value as + in L(I;co(S1)co(S2))PCλn. With a slight abuse of notation, we extend the projection map A to the space L(I;co(S1)co(S2)) by setting

    A(u)(t)={v1if u(t)co(S1),v2if u(t)co(S2),

    for uL(I;co(S1)co(S2)). Furthermore we define

    D:={uL(I;co(S1)co(S2)):A(u)BV(I;co(S1)co(S2))}=A1(BV(I;co(S1)co(S2))). (3.15)

    It is natural to extend Definition 3.1 to any spin field uD. The following notion of convergence will be used.

    Definition 3.7. Let {un}nNL(I;co(S1)co(S2)) and uD. We say that un D-converges to u (we write unDu) if and only if unu in the weak topology of L(I;R3) and A(un) converges to A(u) weakly in BV(I;{v1,v2}).

    Remark 3.8. We observe that the notion of convergence introduced in the previous definition is induced by the smallest topology on D containing the set

    {A:A is an open set of the weak topology of L(I;co(S1)co(S2))or A=A1(U), where U is an open set of the weak topology of BV(I;co(S1)co(S2))}.

    For further details about the weak topology of a BV space we address the reader to [3,Remark 3.12].

    We prove the following proposition, which relies on the properties contained in Subsection 3.2 and will be useful in this subsection.

    Proposition 3.9. Let {un}nNL(I;S1S2) be such that A(un)BV(I;{v1,v2}), for any nN, and let Cn(un)={Inj|j{1,,M(un)}} be the open partition associated with un. We assume that

    supnNM(un)<+. (3.16)

    Then there exists uD such that, up to subsequences, unDu.

    Proof. By Proposition 3.2 it follows that, up to a subsequence, unuL(I;co(S1S2)). Thanks to (3.16), up to the extraction of a subsequence, M=M(un) is independent of nN. Up to subsequence, InjIj in the Hausdorff sense, for some intervals Ij and for any j{1,,M}. Note that some Ij could be empty. Let us fix j{1,,M}. For all ε>0 there exists n0N such that

    (Ij)ε={tIj:dist(t,Ij)>ε}Injnn0.

    We define the following two sets:

    A1={nn0:un(t)S1 for a.e. t(Ij)ε},A2={nn0:un(t)S2 for a.e. t(Ij)ε}.

    One of the following three alternatives may occur:

    1.#A1=,#A2<;2.#A1<,#A2=;3.#A1=,#A2=.

    In the first case we have that unL((Ij)ε;S1) for all nn0, up to finitely many indices of the sequence. Thus, by Proposition 3.2, unuL((Ij)ε;co(S1)) and hence, by the arbitrariness of ε>0, we obtain that uL(Ij;co(S1)). The second case is fully analogous to the first case. If we repeat the above argument for all j{1,,M}, we deduce that unu.

    Finally, we get the thesis by remarking that

    limn+I|A(un)A(u)|dt=limn+Mj=1Ij|A(un)A(u)|dt=0.

    The third alternative leads to a contradiction. Indeed, if it holds true, we can find two subsequences {n(1)k}kN and {n(2)k}kN such that un(1)kL((Ij)ε;S1) and un(2)kL((Ij)ε;S2), for all kN. By Proposition 3.2, there exist u1L((Ij)ε;co(S1)) and u2L((Ij)ε;co(S2)) such that un(1)ku1 and un(2)ku2. On the other hand, applying again Proposition 3.2, we infer that unuL(I;co(S1S2)). Then, by the uniqueness of the limit in the weak topology, we infer that u1(t)=u2(t)=u(t) for almost every t(Ij)ε, which is a contradiction since co(S1)co(S2)=.

    Firstly, we study the Γ-convergence of En. The following theorem relies on a straightforward application of Theorem 3.4.

    Theorem 3.10. The sequence En Γ-converges to the functional

    E(u):={Ifhom(u(t))dtifuL(I;co(S1S2)),+otherwise,

    with respect to the weak topology of L(I;R3), where fhom:co(S1S2)R is defined by

    fhom(z)=limρ0limk+1kinf{k2i=1[αuiui+1+uiui+2]|us.t.(u)1,(0,k)¯B(z,ρ)}. (3.17)

    Proof. The result immediately follows by applying Theorem 3.4 to

    fξ(u,v)={α2uvif |ξ|=1,12uvif |ξ|=2,0otherwise,

    where u,vK:=S1S2, extended to + outside K.

    Remark 3.11. The function fhom defined in (3.17) does not depend on the parameter λn. Therefore, in the theorem above the Γ-limit does not depend on the choice of λn.

    Furthermore, an analogous statement of Theorem 3.10 above can be obtained if the functional En is defined only in L(I;S) for some {1,2} (see [12,Theorem 3.4]). Its Γ-limit has the same form and it is finite on L(I;co(S)).

    The following theorem is the main result of this subsection.

    Theorem 3.12 (Zero order Γ-convergence of En). Assume that there exists limn+λnkn=:η(0,+]. Then the following Γ-convergence and compactness results hold true.

    (i) If η(0,+), then En Γ-converges to the functional

    E(u)={Ifhom(u(t))dt+η|DA(u)|(I)ifuD,+ifuL(I;co(S1)co(S2))D,

    with respect to the D-convergence of Definition 3.7, where fhom and D are defined in (3.17) and (3.15) respectively. Moreover if {un}nNL(I;co(S1)co(S2)) satisfies

    supnNEn(un)<+,

    then, up to a subsequence, unDuD.

    (ii) If η=+, then En Γ-converges to the functional

    E(u):={Ifhom(u(t))dtif uL(I;co(S1))L(I;co(S2)),+if uL(I;co(S1)co(S2))(L(I;co(S1))L(I;co(S2))),

    with respect to the weak topology of L(I;R3), where fhom is defined in (3.17). Moreover if {un}nNL(I;co(S1)co(S2)) satisfies

    supnNEn(un)<+

    then, up to a subsequence, unu for some uL(I;co(S1))L(I;co(S2)).

    Proof. We first deal with case (ⅰ). We start by proving the compactness result. Let {un}nNL(I;co(S1)co(S2)) be such that

    supnNEn(un)<¯C, (3.18)

    for some ¯C>0. Thus we have that {un}nNPCλn. Let us consider the open partition Cn(un)={(Ij)n|j{1,,M(un)}} associated with un, where M(un)1=|DA(un)|(I)|v1v2|N. By formula (3.8) and by the definition of A, we compute

    En(un)=Hn(un)+Pn(un)λn(1+α28)#In(I)Pn(un)λn(1+α28)#In(I)=knλn|DA(un)|(I)λn(1+α28)#In(I)=knλn(M(un)1)|v1v2|λn(1+α28)#In(I)C(α)+knλn(M(un)1)|v1v2|, (3.19)

    for some constant C=C(α)>0, where the last inequality is obtained by observing that λn#In(I)=λn1λnλn1, as n+, and thus it is bounded. Therefore by formulae (3.18) and (3.19) we obtain that

    supnNM(un)<C(η,¯C,α,|v1v2|).

    Hence, the sequence {un}nN satisfies the hypotheses of Proposition 3.9 and so we deduce the existence of uD such that, up to a subsequence, unDu.

    Now we prove the liminf inequality. Let {un}nNL(I;co(S1)co(S2)) be such that unDuD. It is not restrictive to assume that {un}nNPCλn. By the liminf inequality of Theorem 3.10 we have

    lim infn+En(un)Ifhom(u(t))dt. (3.20)

    On the other hand, by the lower semicontinuity of the total variation respect the weak convergence in BV(I;{v1,v2}), we have

    lim infn+Pn(un)=lim infn+knλn|DA(un)|(I)η|DA(u)|(I). (3.21)

    Hence by formulae (3.20) and (3.21) we obtain

    lim infn+En(un)lim infn+En(un)+lim infn+Pn(un)Ifhom(t)dt+η|DA(u)|.

    We finally prove the limsup inequality. Let uL(I;co(S1)co(S2)). We may assume that uD. Since A(u)BV(I;co(S1)co(S2)), it is not restrictive to suppose that the number of jumps of u from one circle to the other is one, i.e., |DA(u)|(I)=|v1v2|. Furthermore, by the same density argument exploited in Proposition 3.2 and the locality of the construction, we may assume that

    u(t)={a1if t[0,12],a2if t(12,1],

    where a1co(S1) and a2co(S2). Let {vjn}nNL(I;Sj) be the recovery sequence for the constant function aj obtained by the Γ-convergence result in Remark 3.11 with 2λn as the spacing of the lattice, i.e., vjnaj and

    fhom(aj)=limn+En(vjn)=limn+2λn12λn2i=0[α(vjn)i(vjn)i+1+(vjn)i(vjn)i+2]. (3.22)

    We define

    un(t)={v1n(2t)if t[0,12],v2n(2t1)if t(12,1].

    Remarking that, for all nN,

    A(un)(t)=A(u)(t)={v1if t[0,12],v2if t(12,1],

    we deduce that unDu. We compute

    En(un)=1212λn2i=02λn[α(v1n)i(v1n)i+1+(v1n)i(v1n)i+2]+1212λn2i=02λn[α(v2n)i(v2n)i+1+(v2n)i(v2n)i+2]+12λni=12λn1λn[αuinui+1n+uinui+2n]. (3.23)

    We observe that

    |12λni=12λn1λn[αuinui+1n+uinui+2n]|C(α)λn0, (3.24)

    as n+. By formulae (3.22), (3.23), (3.24), we obtain that

    limn+En(un)=fhom(a1)+fhom(a2)2=Ifhom(u(t))dt. (3.25)

    Since |DA(un)|(I)=|DA(u)|(I)=|v1v2| we get

    limn+Pn(un)=limn+λnkn|v1v2|=η|v1v2|. (3.26)

    Combining (3.25) and (3.26), we deduce the limsup inequality.

    Now we deal with case (ⅱ). Firstly, we prove the compactness result. Let {un}nNL(I;co(S1)co(S2)) be such that

    supnNEn(un)<¯C,

    for some constant ¯C>0. Thus we have that {un}nNPCλn. With the same compactness argument used in the previous case, we deduce the existence of uD such that unDu. In particular unu. By the lower semicontinuity of the total variation respect the weak convergence in BV(I;{v1,v2}), remarking that EnC(α), for some positive constant C(α), we get

    0=lim infn+¯Cλnknlim infn+1λnkn[En(un)+λnkn|DA(un)|(I)]lim infn+(C(α)λnkn+|DA(un)|(I))|DA(u)|(I),

    hence uL(I;co(S1))L(I;co(S2)).

    Let us prove the liminf inequality. Let {un}nNL(I;co(S1)co(S2)) be such that unuL(I;co(S1)co(S2)) and suppose that

    lim infn+En(un)<+.

    Up to the extraction of a subsequence, we may assume that the previous lower limit is actually a limit. By compactness, we infer that unuL(I;co(S1))L(I;co(S2)). Hence, by Theorem 3.10, we obtain

    lim infn+En(un)lim infn+En(un)Ifhom(u(t))dt.

    We finally prove the limsup inequality. Let uL(I;co(S1)), the case uL(I;co(S2)) being fully analogous. The recovery sequence obtained from Remark 3.11, {un}nNL(I;S1), satisfies the limsup inequality.

    In this subsection and in the following one we study the system when it is close to the helimagnet/ferromagnet transition point as the number of particles diverges. In what follows we let α=αn and we assume that αn4, as n+, and that n is sufficiently large so that Proposition 3.5 holds true.

    Once again, with a slight abuse of notation, we extend the energies En, Pn and En to the space L(I;R3), setting their value as + in L(I;R3)PCλn. Similarly, we extend A from L(I;co(S1)co(S2)) to L(I;R3).

    The main result of this subsection, Theorem 3.16, concerns the phenomenon of magnetic anisotropy transitions. Having in mind Proposition 3.5 and (3.9), we define the functional

    Gn:=EnminwPCλnEn(w)=Enλn#In(I)[R2(1+α2n8)+(αn1)(1R2)].

    At this point we need to introduce modified spin fields in order to understand better the asymptotic behaviour of the energy Gn. Let uPCλn and let Cn(u)={Ij|j{1,,M(u)}} be the open partition associated with u, with Ij=(tj,tj+1), for j{1,,M(u)1}, and IM(u)=(tM(u),1). We set tM(u)+1:=λn1λn. Since u is piecewise constant on the edges of the lattice [0,1]λnZ, we have that t1=0 and t2,,tM(u)+1 are multiples of λn, so that tjλnN, for any j{2,,M(u)+1}.

    We define the auxiliary spin ˜uIj:¯IjS1S2 by

    ˜uIj(t)={u(t)if t[tj,tj+1),wjif t=tj+1,

    and we set ˜uIM(u)(t)=wM(u) for t(tM(u)+1,1], where wjS1S2 is a vector such that the following boundary condition is satisfied in ¯Ij:

    utj+1λn1wj=utjλnutjλn+1. (3.27)

    We prove the following decomposition lemma.

    Lemma 3.13 (Decomposition of Gn). Let uPCλn and let Cn(u)={Ij|j{1,,M(u)}} be the open partition associated with u. We have

    Gn(u)=M(u)j=1MMn(˜uIj)+M(u)1j=1(Rn)j(u)+(Rn)M(u)(u)+Rn(u), (3.28)

    where, for all j{1,,M(u)},

    MMn(˜uIj):=λniIn(Ij)(αn˜uiIj˜ui+1Ij+˜uiIj˜ui+2Ij)+λnR2(1+α2n8)#In(Ij)+λn(αn1)(1R2)(#In(I)M(u)+1)M(u),

    and, for all j{1,,M(u)1},

    (Rn)j(u):=λn(αnutj+1λn1utj+1λn+utj+1λn1utj+1λn+1+utj+1λn2utj+1λnutj+1λn2wj),
    (Rn)M(u)(u):=λn(utM(u)+1λn2utM(u)+1λnutM(u)+1λn2wM(u)),
    Rn(u):=λnR2(1+α2n8)(M(u)1)+λn(αn1)(1R2)(M(u)1).

    Proof. Remarking that

    In(Ij)={tjλn,tjλn+1,,tj+1λn2},j{1,,M(u)},

    we may write

    Gn(u)+minwPCλnEn(w)=λnM(u)1j=1iIn(Ij)(αnuiui+1+uiui+2)+M(u)1j=1(αnutj+1λn1utj+1λn+utj+1λn1utj+1λn+1)+iIn(IM(u))(αnuiui+1+uiui+2).

    After adding and subtracting the terms utj+1λn2wj, for any j{1,,M(u)}, we interchange utj+1λn2wj and utj+1λn2utj+1λn in the first and the third sums, for any j{1,,M(u)}, obtaining

    Gn(u)+minwPCλnEn(w)=λn[M(u)j=1iIn(Ij)(αn˜uiIj˜ui+1Ij+˜uiIj˜ui+2Ij)+M(u)1j=1(αnutj+1λn1utj+1λn+utj+1λn1utj+1λn+1+utj+1λn2utj+1λnutj+1λn2wj)+(utM(u)+1λn2utM(u)+1λnutM(u)+1λn2wM(u))]=M(u)j=1MMn(˜uIj)+M(u)1j=1(Rn)j(u)+(Rn)M(u)(u)λnR2(1+α2n8)M(u)j=1#In(Ij)λn(αn1)(1R2)(#In(I)M(u)+1).

    We conclude the proof by computing

    minwPCλnEn(w)λnR2(1+α2n8)M(u)j=1#In(Ij)λn(αn1)(1R2)(#In(I)M(u)+1)=λnR2(1+α2n8)[#In(I)M(u)j=1#In(Ij)]+λn(αn1)(1R2)(M(u)1)=λnR2(1+α2n8)(M(u)1)+λn(αn1)(1R2)(M(u)1)=Rn(u),

    where we used

    Mj=1#In(Ij)=#In(I)M(u)+1. (3.29)

    Remark 3.14. In the decomposition (3.28) of Gn(u) the functional MMn(˜uIj) represents the energy of the j-th modified spin field ˜uIj, which is localized in one circle. The remainders for such modifications, (Rn)j(u) and (Rn)M(u)(u), consist of the interactions between spins with values in two neighboring intervals, Ij and Ij+1. Furthermore, they contain an additional term linked to the boundary condition (3.27). The term Rn(u) contains a corrective addend.

    Remark 3.15. Following the same computations done in (3.7), we infer that MMn(˜uIj)0, for all j{1,,M(u)} and uPCλn.

    The next theorem shows that the correct scaling of the energy to capture spin fields' magnetic anisotropy transitions is λn. To this end, for MN, we set

    RM:=inf{lim infn+1λn[M1j=1(Rn)j(un)+(Rn)M(un)+Rn(un)]|{un}nNPCλn such that A(un)BVvBV(I;{v1,v2}), with M=|Dv|(I)|v1v2|+1N}.

    Theorem 3.16 (First order Γ-convergence of En). Assume that there exists limn+λnkn=:η(0,+). Then the following compactness and Γ-convergence results hold true:

    (i) (Compactness) If for {un}nNL(I;R3) there exists a constant C>0 independent of n such that

    supnNGn(un)λnCandsupnNPn(un)C, (3.30)

    then, up to subsequences, A(un)BVvBV(I;{v1,v2}).

    (ii) (liminf inequality) For all vBV(I;{v1,v2}) and {un}nNPCλn such that A(un)BVv and (3.30) holds for some constant C>0, then

    lim infn+Gn(un)λnRM,

    where M=|Dv|(I)|v1v2|+1N.

    (iii) (limsup inequality) For all vBV(I;{v1,v2}) there exists {un}nNPCλn such that A(un)BVv, (3.30) holds for some constant C>0 and

    limn+Gn(un)λn=RM,

    where M=|Dv|(I)|v1v2|+1N.

    Proof. We start by proving (ⅰ). Let {un}nNL(I;R3) be such that (3.30) holds true. It follows that {un}nNPCλn. Since η(0,+), by the second inequality of formula (3.30), we deduce that the sequence {|DA(un)|(I)}nN is bounded and so the sequence {A(un)}nN is bounded in the space BV(I;{v1,v2}). Thus, up to a subsequence, {A(un)}nN converges to a function vBV(I;{v1,v2}) weakly in BV(I;{v1,v2}).

    We prove (ⅱ). Let vBV(I;{v1,v2}) and {un}nNPCλn be such that A(un)BVv and (3.30) holds. By assumption, {|DA(un)|(I)}nN is bounded. Let Cn(un)={(Ij)n|j{1,,M(un)}} be the open partition associated with un. Up to subsequences, we may assume that M=M(un) is independent of n. By Lemma 3.13, Remark 3.15 and the definition of RM we have

    lim infn+Gn(un)λnlim infn+Mj=1MMn(˜un(Ij)n)λn+lim infn+[M1j=1(Rn)j(un)λn+(Rn)M(un)+Rn(un)λn]lim infn+[M1j=1(Rn)j(un)λn+(Rn)M(un)+Rn(un)λn]RM.

    We finally prove (ⅲ). Let vBV(I;{v1,v2}). It is not restrictive to assume that v=v1χ[0,12]+v2χ(12,1] and thus we can choose uPCλn such that A(u)=v. By the definition of RM and by [12,Theorem 4.2], we gain the existence of {un}nN such that A(un)BVA(u), unχ[0,12]S1,unχ(12,1]S2 and the following formulae are satisfied:

    limn+1λn[M1j=1(Rn)j(un)+(Rn)M(un)+Rn(un)]=RM,
    (1αn4)32MMn(unχ[0,12])λn<C,(1αn4)32MMn(unχ(12,1])λn<C.

    Therefore

    limn+Gn(un)λn=RM.

    We let α=αn:=4(1δn), where {δn}nN is a positive vanishing sequence.

    At the second order we split the global functional on the 2-dimensional sphere into finitely many functionals localized in circles, where we repeat the analysis lead in [12]. For each circle S we define a convenient order parameter.

    Let uPCλn. According to the notation introduced in Subsection 3.6, for j{1,,M(u)} and i{tjλn,tjλn+1,,tj+1λn1}, we consider the pair (˜uiIj,˜ui+1Ij) of vectors that take values in S, for some =j{1,2}. We associate each pair with the corresponding oriented angle θiIj[π,π) with vertex in the center of the circle S given by

    θiIj:=χ[πv(˜uiIj),πv(˜ui+1Ij)]arccos(πv(˜uiIj)πv(˜ui+1Ij)).

    We set

    wiIj:=84αnsinθiIj2=2δnsinθiIj2

    and

    w(t)=wiIj{for} tλn{i+[0,1)},i{tjλn,,tj+1λn1},j{1,,M(u)}.

    We extend w(t)=wtj+1λn1IM(u), for t[tM(u)+1,1], so that w is well-defined in the whole interval ¯I. Note that we can define a map Tn by setting

    Tn(u):=(w,A(u)),uPCλn,

    and we denote ~PCλn:=Tn(PCλn). We observe that if h=Tn(u)=Tn(v) then u(t) and v(t) belong to the same circle, for any t¯I, and u,v differ by a constant rotation. Furthermore, Gn(u)=Gn(v) and Pn(u)=Pn(v). The same identity holds for the functionals defined in Lemma 3.13. Therefore, with a slight abuse of notation, we now set

    Gn(h):={Gn(u)if h~PCλn,+otherwise,
    Pn(h):={Pn(u)if h~PCλn,+otherwise,
    MMn(h|Ij):={MMn(˜uIj)if h~PCλn,+otherwise,
    (Rn)j(h):={(Rn)j(u)if h~PCλn,+otherwise,
    (Rn)M(u)(h):={(Rn)M(u)(u)if h~PCλn,+otherwise,
    Rn(h):={Rn(u)if h~PCλn,+otherwise,

    for j{1,,M(h)}, where hL1(I;R×{v1,v2}), h=Tn(u) and M(h):=M(u).

    We want to study the convergence of the functional

    Gn(h)={Gn(h)M(h)1j=1(Rn)j(h)+(Rn)M(h)(h)+Rn(h)if h~PCλn,+otherwise,=M(h)j=1MMn(˜h|Ij)

    for hL1(I;R×{v1,v2}). In order to establish the related result, we need a notion of convergence.

    Definition 3.17. Let {hn}nN~PCλn and hL1(I;R×{v1,v2}). We say that hn θ-converges to h (we write hnθh) if and only if the following conditions are satisfied:

    ● there exist {un}nNPCλn and a positive constant C such that if Cn(un)={(Ij)n|j{1,,M(un)}} is the open partition associated with un, then

    hn=Tn(un) and Pn(hn)<C,

    M(un)MN as n+,

    (Ij)nIj in the Hausdorff sense, as n+, for any j{1,,M}.

    hnχ(Ij)nhχIj in L1(I;R×{v1,v2}), for all j{1,,M}.

    We point out that the intervals Ij of the previous definition may be also empty.

    The next theorem shows that the correct scaling of the energy to capture spin fields' chirality transitions is 2λnδ32n.

    Theorem 3.18 (Second order Γ-convergence of En). Assume that there exist limn+λnkn=:η(0,+) and l:=limn+λn(2δn)12[0,+].

    Then the following statements are true:

    (i) (Compactness) If for {hn}nNL1(I;R×{v1,v2}) there exists a constant C>0 such that

    supnNGn(hn)2λnδ32nCandsupnNPn(hn)C, (3.31)

    then, up to a subsequence, hnθh, where

    if l=0, hBV(I;{1,1}×{v1,v2});

    if l(0,+), h|IjH1|per|(Ij;R×{v1,v2}) for all j{1,,M(h)};

    if l=+, h is piecewise constant with values in R×{v1,v2}.

    The space H1|per|((a,b);R×{v1,v2}) is equal to

    {hH1((a,b);R×{v1,v2}):|w(a)|=|w(b)|whereh=(w,A(u))}.

    (ii) (liminf inequality)

    If l=0, for all h=(w,A(u))BV(I;{1,1}×{v1,v2}) and for all {hn}nN~PCλn such that hnθh and (3.31) holds true for some constant C>0, then

    lim infn+Gn(hn)2λnδ32n43R2M(h)j=1|Dw|(Ij).

    If l(0,+), for all h=(w,A(u))L1(I;R×{v1,v2}) such that h|IjH1|per|(Ij;R×{v1,v2}), for every j{1,,M(h)}, and for all {hn}nN~PCλn such that hnθh and (3.31) holds true for some constant C>0, then

    lim infn+Gn(hn)2λnδ32nM(h)j=1R2[1lIj(w2(x)1)2dx+lIj(w(x))2dx].

    If l=+, for all piecewise constant functions h:IR×{v1,v2} and for all {hn}nN~PCλn such that hnθh, and (3.31) holds true for some constant C>0, then

    lim infn+Gn(hn)2λnδ32n0.

    (iii) (limsup inequality)

    If l=0, for all h=(w,A(u))BV(I;{1,1}×{v1,v2}) there exists {hn}nN~PCλn such that hnθh, (3.31) holds true for some constant C>0 and

    limn+Gn(hn)2λnδ32n=43R2Mj=1|Dw|(Ij).

    If l(0,+), for all h=(w,A(u))L1(I;R×{v1,v2}) such that h|IjH1|per|(Ij;R×{v1,v2}) for all j{1,,M(h)}, there exists {hn}nN~PCλn such that hnθh, (3.31) holds true for some constant C>0 and

    limn+Gn(hn)2λnδ32n=Mj=1R2[1lIj(w2(x)1)2dx+lIj(w(x))2dx].

    If l=+, for all piecewise constant functions h:IR×{v1,v2} there exists {hn}nN~PCλn such that hnθh, (3.31) holds true for some constant C>0 and

    limn+Gn(hn)2λnδ32n=0.

    Proof. We prove the statement only in the case l=0, the other cases being fully analogous. We start by proving (ⅰ). Let {hn}nNL1(I;R×{v1,v2}) be such that (3.31) holds true for some constant C>0. By formula and Remark 3.15, we infer that

    MMn(hn|Inj)λnδ32nC,for all j{1,,M(hn)} and nN.

    It is easy to see that, up to subsequences, M=M(hn) is independent of nN and the interval (Ij)nIj=(tj1,tj), in the Hausdorff sense, for every j{1,,M(hn)} (it may happen that Ij=, for some j). In the following computations we drop for simplicity the dependence on n writing Ij in place of (Ij)n.

    Reasoning as in Proposition 3.5, thanks to (3.29), we compute

    Gn(hn)=Mj=1λniIn(Ij)(αnπ˜uinIjπ˜ui+1nIj+π˜uinIjπ˜ui+2nIj)+λnR2(1+α2n8)Mj=1#In(Ij)+λn(αn1)(1R2)(#In(I)M+1)λn(αn1)(1R2)Mj=1#In(Ij)=Mj=1λniIn(Ij)(αnπ˜uinIjπ˜ui+1nIj+π˜uinIjπ˜ui+2nIj)+λnR2(1+α2n8)(#In(I)M+1),

    where we set π˜uinIj:=πv˜uinIj, with =j{1,2} such that ˜uinIjS.

    By the definition of ˜uinIj and geometric and trigonometric identities, we observe that

    R2π˜uinIjπ˜ui+1nIj=2R2sin2(θiIj2),
    R2π˜uinIjπ˜ui+2nIj=R2[1cos(θiIj+θi+1Ij)],

    where, for simplicity of notation, we have dropped the dependence on n of the angles θiIj. Taking into account the previous formulae, we gain

    Gn(hn)=λnMj=1iIn(Ij){αn[R2π˜uinIjπ˜ui+1nIj][R2π˜uinIjπ˜ui+2nIj]}+λnR2(1+α2n8)(#In(I)M+1)+λnR2(1αn)Mj=1#In(Ij)=λnR2Mj=1iIn(Ij){2αnsin2(θiIj2)[1cos(θiIj+θi+1Ij)]}+λnR2(2αn+α2n8)(#In(I)M+1). (3.32)

    The proof can be carried out as in [12,Theorem 4.2]. For reader's convenience we give here its sketch. By trigonometric identities, it holds

    \begin{equation*} 8\sin^2\bigg( \frac{\theta^i_{I_j}}{2} \bigg)-2\sin^2(\theta^i_{I_j}) = 8\sin^4\bigg( \frac{\theta^i_{I_j}}{2} \bigg). \end{equation*}

    Moreover, taking into account the boundary condition (3.27), we can find a vanishing sequence \{\gamma_n\}_{n\in \mathbb{N}}\subset \mathbb{R} such that

    \begin{align*} & \sum\limits_{i\in\mathcal{I}^n(I_j)}\big[2\sin^2(\theta^i_{I_j})-1+\cos(\theta^i_{I_j}+\theta^{i+1}_{I_j})\big]\\ & \geq 2(1-\gamma_n)\sum\limits_{i\in\mathcal{I}^n(I_j)}\bigg(\sin\bigg( \frac{\theta^{i+1}_{I_j}}{2} \bigg)-\sin\bigg( \frac{\theta^i_{I_j}}{2} \bigg)\bigg)^2. \end{align*}

    We insert the previous two formulae in (3.32) and compute

    \begin{align*} & \mathcal{G}_n(h_n)\\ & = \lambda_n R^2 \sum\limits_{j = 1}^M\sum\limits_{i\in\mathcal{I}^n(I_j)}\Bigg\{8\sin^2\bigg( \frac{\theta^i_{I_j}}{2} \bigg)-(8-2\alpha_n)\sin^2\bigg( \frac{\theta^i_{I_j}}{2} \bigg)+2\bigg(1-\frac{\alpha_n}{4}\bigg)^2\Bigg\}\\ & -\lambda_n R^2 \sum\limits_{j = 1}^M\sum\limits_{i\in\mathcal{I}^n(I_j)}\big[1-\cos(\theta^i_{I_j}+\theta^{i+1}_{I_j})\big]\\ & -2\lambda_n R^2\bigg(1-\frac{\alpha_n}{4}\bigg)^2(\#\mathcal{I}^n(I)-M+1) + \lambda_n R^2\left(2-\alpha_n+\frac{\alpha_n^2}{8}\right)(\#\mathcal{I}^n(I)-M+1)\\ & = \lambda_n R^2 \sum\limits_{j = 1}^M\sum\limits_{i\in\mathcal{I}^n(I_j)}\Bigg\{8\sin^2\bigg( \frac{\theta^i_{I_j}}{2} \bigg)-2\sin^2(\theta^i_{I_j})-(8-2\alpha_n)\sin^2\bigg( \frac{\theta^i_{I_j}}{2} \bigg)+2\bigg(1-\frac{\alpha_n}{4}\bigg)^2\Bigg\}\\ & +\lambda_n R^2 \sum\limits_{j = 1}^M\sum\limits_{i\in\mathcal{I}^n(I_j)}\big[2\sin^2(\theta^i_{I_j})-1+\cos(\theta^i_{I_j}+\theta^{i+1}_{I_j})\big]\\ & = 8\lambda_n R^2\sum\limits_{j = 1}^M\sum\limits_{i\in\mathcal{I}^n(I_j)}\bigg[\sin^2\bigg( \frac{\theta^i_{I_j}}{2} \bigg)-\frac{1}{2}\bigg(1-\frac{\alpha_n}{4}\bigg)\bigg]^2\\ & +\lambda_n R^2 \sum\limits_{j = 1}^M\sum\limits_{i\in\mathcal{I}^n(I_j)}\big[2\sin^2(\theta^i_{I_j})-1+\cos(\theta^i_{I_j}+\theta^{i+1}_{I_j})\big]\\ & \geq \lambda_n R^2\sum\limits_{j = 1}^M\sum\limits_{i\in\mathcal{I}^n(I_j)}\Bigg\{8\bigg[\sin^2\bigg( \frac{\theta^i_{I_j}}{2} \bigg)-\frac{1}{2}\bigg(1-\frac{\alpha_n}{4}\bigg)\bigg]^2+2(1-\gamma_n)\bigg[\sin\bigg( \frac{\theta^{i+1}_{I_j}}{2} \bigg)-\sin\bigg( \frac{\theta^i_{I_j}}{2} \bigg)\bigg]^2\Bigg\}. \end{align*}

    Dividing by \sqrt{2}\lambda_n\delta_n^\frac{3}{2} and recalling that \alpha_n = 4(1-\delta_n) , we infer that

    \begin{equation} \frac{\mathcal{G}_n(h_n)}{\sqrt{2}\lambda_n\delta_n^{\frac{3}{2}}}\geq R^2\Bigg\{ \frac{\sqrt{2}\delta_n^{\frac{1}{2}}}{\lambda_n}\sum\limits_{i\in\mathcal{I}^n(I_j)}\lambda_n\big[(w_{nI_j}^i)^2-1\big]^2+\frac{\lambda_n}{\sqrt{2}\delta_n^{\frac{1}{2}}}(1-\gamma_n)\sum\limits_{i\in\mathcal{I}^n(I_j)}\bigg(\frac{w_{nI_j}^{i+1}-w_{nI_j}^i}{\lambda_n}\bigg)^2\Bigg\}. \end{equation} (3.33)

    If \varepsilon > 0 is sufficiently small such that I_j^\varepsilon: = ({t_j}+\varepsilon, {t_{j+1}}-\varepsilon) \subset (I_j)_n , for all n\in \mathbb{N} , then

    \begin{equation*} MM_{n}(w_{n| I_j^\varepsilon}) \leq \lambda_n \delta_n^{\frac{3}{2}} C \end{equation*}

    and (3.33) holds with I_j^\varepsilon in place of I_j , for any j\in\{1, \dots, M\} . Therefore, applying [12,Theorem 2.2 and Remark 2.3] (see also [8]), \{ w_{n} \chi_{I_j^\varepsilon} \}_{n \in \mathbb{N}} converges, up to subsequences, to w \in BV(I_j) in \mathrm{L}^1 . Thus we deduce the existence of h \in BV(I; \{-1, 1\}\times\{ v_1, v_2\}) such that h_n: = (w_n, \mathcal{A}(u_n)) \overset{{\theta}}{\longrightarrow} h .

    Now we prove (ⅱ). Let h = (w, \mathcal{A}(u))\in BV(I; \{-1, 1\}\times\{v_1, v_2\}) and \{ h_n\}_{n \in \mathbb{N}}\subset \widetilde{PC}_{\lambda_n} be such that h_n \overset{{\theta}}{\longrightarrow} h and (3.31) holds true for some constant C > 0 . Up to a subsequence, M = M(h_n) is independent of n . Moreover, denoting I_j = (t_j, t_{j+1}) , for \varepsilon > 0 sufficiently small, it holds that I_j^\varepsilon: = (t_j+ \varepsilon, t_{j+1}-\varepsilon) \subset (I_j)_n , for all j\in\{1, \dots, M(h_n)\} and n\in \mathbb{N} . By the definition of \mathcal{G}_n , we have

    \begin{equation*} \liminf\limits_{n \rightarrow + \infty} \frac{{\mathcal{G}_n}(h_n)}{\sqrt{2}\lambda_n\delta_n^{\frac{3}{2}}} = \liminf\limits_{n \rightarrow + \infty} \frac{ \sum\limits_{j = 1}^{M} MM_n(h_{n|I_j})}{\sqrt{2}\lambda_n\delta_n^{\frac{3}{2}}} \geq {\frac{4}{3}R^2} \sum\limits_{j = 1}^{M}\left| Dw \right|(I_j^\varepsilon ), \end{equation*}

    where in the last step we have used the liminf inequality of [12,Theorem 4.2]. Letting \varepsilon\rightarrow0 , we obtain the liminf inequality.

    We finally prove (ⅲ). Let h = (w, \mathcal{A}(u)) \in BV(I; \{-1, 1\}\times\{ v_1, \, v_2\}) . We can find M > 0 and an open partition of I made by the intervals \mathcal{C} = \{I_j\}_{j\in\{1, \dots, M\}} such that h_{|I_j} = ({w_{|I_j}, \overline{v}_j)\in BV(I_j; \{-1, 1\})\times\{v_1, \, v_2\})} . Thanks to the limsup inequality proved in [12,Theorem 4.2], for all j \in \{1, \dots, M\} there exists a sequence \{(z_j)_n\}_{n \in \mathbb{N}}\subset\mathrm{L}^1(I_j; \mathbb{R}) , such that (z_j)_n\rightarrow {w_{|I_j}} in \mathrm{L}^1(I_j; \mathbb{R}) and

    \begin{equation} \quad \lim\limits_{n \rightarrow + \infty} \frac{MM_n(h_{n|I_j})}{\sqrt{2} \lambda_n \delta_n^{\frac{3}{2}}} = {\frac{4}{3}R^2} \vert D{w}| (I_j), \end{equation} (3.34)

    where h_{n|I_j}: = ((z_j)_n, \overline{v}_j) . By the definition of \mathcal{G}_n and (3.34) we gain

    \begin{equation*} \lim\limits_{n \rightarrow + \infty} \frac{{\mathcal{G}_n}(h_n)}{\sqrt{2}\lambda_n\delta_n^{\frac{3}{2}}} = \lim\limits_{n \rightarrow + \infty} \frac{ \sum\limits_{j = 1}^{M} MM_n(h_{n_{|I_j}})}{\sqrt{2}\lambda_n\delta_n^{\frac{3}{2}}} = {\frac{4}{3}R^2}\sum\limits_{j = 1}^{M}\left| D{w} \right|(I_j), \end{equation*}

    that is the thesis.

    In this section we analyze the problem in the two-dimensional case. Therefore we need to introduce proper notation and new definitions.

    Let \{\lambda_n\}_{n\in \mathbb{N}}\subset \mathbb{R}^+ be a vanishing sequence of positive lattice spacings. Given i, j\in \mathbb{Z} , we denote by Q_{\lambda_n}(i, j): = (\lambda_n i, \lambda_n j)+[0, \lambda_n)^2 the half-open square with left-bottom corner in (\lambda_n i, \lambda_n j) . For a given set S , we introduce the class of spin fields with values in S which are piecewise constant on the squares of the lattice \lambda_n \mathbb{Z}^2 :

    \begin{equation*} \mathcal{PC}_{\lambda_n}( \mathbb{R}^2;S): = \{ {u}\colon \mathbb{R}^2\rightarrow S\,:\, {u}(x) = {u}(\lambda_n i,\lambda_n j) \text{ for }x\in Q_{\lambda_n}(i,j)\}. \end{equation*}

    We will identify a function {u}\in \mathcal{PC}_{\lambda_n}(\mathbb{R}^2;S) with the function defined on the points of the lattice \lambda_n \mathbb{Z}^2 given by (i, j)\mapsto {u}^{i, j}: = {u}(\lambda_n i, \lambda_n j) , for i, j\in \mathbb{Z} . Conversely, given values {u}^{i, j}\in S for i, j\in \mathbb{Z} , we define {u}\in \mathcal{PC}_{\lambda_n}(\mathbb{R}^2;S) by setting {u}(x): = {u}^{i, j} , for x\in Q_{\lambda_n}(i, j) .

    Furthermore, we define the projection function \mathcal{A}\colon \mathcal{PC}_{\lambda_n}(\mathbb{R}^2;S_1\cup S_2)\rightarrow \mathrm{L}^\infty(\mathbb{R}^2;\{v_1, v_2\}) by setting

    \begin{equation*} { \mathcal{A}(u)(x) = \begin{cases} v_1\quad&\text{if }u(x)\in S_1,\\ v_2 &\text{if }u(x)\in S_2, \end{cases} \quad \forall x\in \mathbb{R}^2.} \end{equation*}

    In this paper we will make use of the notion of BVG regularity. BVG domains and BVG functions have been introduced in [21] (see also [9,Section 3]).

    Definition 4.1. Let I\subset \mathbb{R} be an open set. We define the space of BVG functions by

    \begin{equation*} BVG(I): = \{\phi\in W^{1,\infty}(I)\,:\, \nabla\phi\in BV(I)\}. \end{equation*}

    A bounded connected open set \Omega\subset \mathbb{R}^2 is called a BVG domain if \Omega can be described locally at its boundary as the epigraph of a BVG function with respect to a suitable choice of the axes, i.e., if for every x\in \partial\Omega there exist a neighborhood U_x\subset \mathbb{R}^2 , a function \psi_x\in BVG(\mathbb{R}) and an isometry R_x\colon \mathbb{R}^2\rightarrow \mathbb{R}^2 satisfying

    \begin{equation*} R_x(\Omega\cap U_x) = \{(y_1,y_2)\in \mathbb{R}^2\,:\,y_1 > \psi_x(y_2)\}\cap R_x(U_x). \end{equation*}

    We remark that smooth domains and polygons are BVG domains and BVG domains are Lipschitz domains.

    As in the one-dimensional case we observe that, if u\in\mathcal{PC}_{\lambda_n}(\mathbb{R}^2;S_1\cup S_2) , then a bounded connected open set \Omega\subset \mathbb{R}^2 can be uniquely partitioned in regions where the spin field u takes values only in one of the two circles. In other words, there exist M(u)\in \mathbb{N} and a collection of connected open sets, \{C_s\}_{s\in\{1, \dots, M(u)\}} , such that

    \begin{equation} \{C_s\}_{s\in\{1,\dots,M(u)\}}\text{ is an open partition of }\Omega, \end{equation} (4.1)
    \begin{equation} {\text{either } u(C_s)\subset S_1\text{ or } u(C_s)\subset S_2 ,\text{ for any $s\in\{1,\dots,M(u)$\}}}, \end{equation} (4.2)
    \begin{align} & \text{if }u(C_{s_1})\times u(C_{s_2})\subset S_\ell\times S_\ell,\text{ for some $s_1,s_2\in\{1,\dots,M(u)\}$ and $\ell\in\{1,2\}$,} \end{align} (4.3)
    \begin{align} & \text{then }\overline{C}_{s_1}\cap \overline{C}_{s_2}\text{ has at most a finite number of points.} \end{align}

    The last two properties imply that this partition is unique. We remark that the sets C_s are squares or union of squares. In particular, (4.3) ensures that u maps two confining sets of the open partition in different circles, if their intersection contain edges of squares.

    The following definition will be useful throughout the section.

    Definition 4.2. Let u\in\mathcal{PC}_{\lambda_n}(\mathbb{R}^2;S_1\cup S_2) and \Omega\subset \mathbb{R}^2 be a bounded connected open set. We say that \mathcal{C}_n(u) = \{ C_s\, |s\in\{1, \dots, M(u)\} \} is the open partition of \Omega associated with u if M(u)\in \mathbb{N} and the collection \{C_s\}_{s\in\{1, \dots, M(u)\}} of open connected sets satisfies (4.1), (4.2) and (4.3). If \Omega is a BVG domain, we call \mathcal{C}_n(u) the open BVG partition of \Omega associated with u if C_s is also a BVG domain, for all s\in\{1, \dots, M(u)\} .

    Our model is an energy on discrete spin fields defined on square lattices inside a given domain \Omega\subset \mathbb{R}^2 belonging to the following class:

    \begin{equation*} \mathfrak{A}_0: = \{\Omega\subset \mathbb{R}^2\,:\,\Omega \text{ is a {simply connected $BVG$ domain}}\}{.} \end{equation*}

    To define the energies in our model, we introduce the set of indices

    \begin{equation*} \mathcal{I}^n(\Omega): = \{(i,j)\in \mathbb{Z}^2\,:\,\overline{Q}_{\lambda_n}(i,j), \overline{Q}_{\lambda_n}(i+1,j), \overline{Q}_{\lambda_n}(i,j+1)\subset\Omega \}, \end{equation*}

    for \Omega\in \mathfrak{A}_0 . Let \alpha_n: = 4(1-\delta_n) , where \{\delta_n\}\subset \mathbb{R}^+ is a vanishing sequence, and let \{ k_n\}_{n\in \mathbb{N}}\subset \mathbb{R}^+ be a divergent sequence. In the following we shall assume that \varepsilon_n: = \frac{\lambda_n}{\sqrt{\delta_n}}\rightarrow0 and \lambda_n k_n\rightarrow \eta\in(0, +\infty) , as n\rightarrow +\infty .

    We consider the functionals H_n, P_n\colon \mathrm{L}^\infty(\mathbb{R}^2;S_1\cup S_2)\times \mathfrak{A}_0\rightarrow [0, +\infty] defined by

    \begin{equation*} H_n(u;\Omega): = \frac{1}{\sqrt{2}\lambda_n\delta_n^{\frac{3}{2}}}\frac{1}{2}\lambda_n^2\sum\limits_{(i,j)\in\mathcal{I}^n(\Omega)}\left[\left| u^{i+2,j}-\frac{\alpha_n}{2}u^{i+1,j}+u^{i,j}\right|^2+\left| u^{i,j+2}-\frac{\alpha_n}{2}u^{i,j+1}+u^{i,j}\right|^2\right], \end{equation*}
    \begin{equation} P_n(u;\Omega): = \lambda_nk_n|D\mathcal{A}(u)|(\Omega), \end{equation}

    for u\in\mathcal{PC}_{\lambda_n}(\mathbb{R}^2;S_1\cup S_2) and extended to +\infty elsewhere.

    Similarly to the analysis at the first and second order in the one-dimensional case, we split the functional H_n as follows:

    \begin{equation*} H_n(u;\Omega) = \sum\limits_{s = 1}^{M(u)}\Big[\mathcal{H}_n(u;C_s)+(R_n)_{C_s}(u)\Big], \end{equation*}

    where

    \begin{equation*} \mathcal{H}_n(u;C_s): = H_n(u;C_s)+\frac{1}{\sqrt{2}\lambda_n\delta_n^{\frac{3}{2}}}\cdot 2\lambda_n^2(\alpha_n-1)(1-R^2)\#\mathcal{I}^n(C_s), \end{equation*}
    \begin{align*} (R_n)_{C_s}(u): = & \frac{1}{\sqrt{2}\lambda_n\delta_n^{\frac{3}{2}}}\frac{1}{2}\lambda_n^2\sum\limits_{(i,j)\in(C_s\cap\mathcal{I}^n(\Omega))\setminus\mathcal{I}^n(C_s)}\bigg[\left| u^{i+2,j}-\frac{\alpha_n}{2}u^{i+1,j}+u^{i,j}\right|^2\\ &+\left| u^{i,j+2}-\frac{\alpha_n}{2}u^{i,j+1}+u^{i,j}\right|^2\bigg] {-\frac{1}{\sqrt{2}\lambda_n\delta_n^{\frac{3}{2}}}\cdot 2\lambda_n^2(\alpha_n-1)(1-R^2)\#\mathcal{I}^n(C_s),} \end{align*}

    for any s\in\{1, \dots, M(u)\} . The functionals (R_n)_{C_s} collect the remainders associated with the decomposition of the energy in the open partition \mathcal{C}_n(u) = \{ C_s\, |s\in\{1, \dots, M(u)\} \} . They consist of the interactions between spin field's vectors located in different circles.

    In this subsection we introduce the chirality order parameter associated with a spin field. Let u\in\mathcal{PC}_{\lambda_n}(\mathbb{R}^2;S_1\cup S_2) and let \mathcal{C}_n(u) = \{ C_s\, |s\in\{1, \dots, M(u)\} \} be the partition associated with u . For (i, j)\in\mathcal{I}^n(C_s) , we consider the pairs (u^{i, j}, u^{i+1, j}) and (u^{i, j}, u^{i, j+1}) of vectors that take values in S_\ell , for some \ell = \ell_s\in\{1, 2\} . We define the horizontal and vertical oriented angles between two adjacent spin vectors by

    \begin{equation*} \widetilde{\theta}^{i,j}_{C_s}: = \chi [\pi_{v_\ell^\bot}(u^{i,j}), \pi_{v_\ell^\bot}(u^{i+1,j})] \arccos(\pi_{v_\ell^\bot}(u^{i,j})\cdot \pi_{v_\ell^\bot}(u^{i+1,j}))\in[-\pi,\pi), \end{equation*}
    \begin{equation*} \check{\theta}^{i,j}_{C_s}: = \chi [\pi_{v_\ell^\bot}(u^{i,j}), \pi_{v_\ell^\bot}(u^{i,j+1})]\arccos(\pi_{v_\ell^\bot}(u^{i,j})\cdot \pi_{v_\ell^\bot}(u^{i,j+1}))\in [-\pi,\pi). \end{equation*}

    We define the order parameter ({(w, z)}, \mathcal{A}(u))\in\mathcal{PC}_{\lambda_n}(\mathbb{R}^2; \mathbb{R}^2)\times{\mathrm{L}^\infty(\Omega; \{v_1, v_2\})} (we will write (w, z, \mathcal{A}(u)) for simplicity) by setting

    \begin{equation*} w^{i,j}: = \begin{cases} \sqrt{\frac{2}{\delta_n}}\sin\frac{\widetilde{\theta}^{i,j}_{C_s}}{2}\quad&\text{if }(i,j)\in\mathcal{I}^n(C_s)\text{ for some }s\in\{1,\dots,M(u)\},\\ 0 & \text{otherwise}, \end{cases} \end{equation*}
    \begin{equation*} z^{i,j}: = \begin{cases} \sqrt{\frac{2}{\delta_n}}\sin\frac{\check{\theta}^{i,j}_{C_s}}{2}\quad&\text{if }(i,j)\in\mathcal{I}^n(C_s)\text{ for some }s\in\{1,\dots,M(u)\},\\ 0 & \text{otherwise}. \end{cases} \end{equation*}

    It is convenient to introduce the transformation T_n\colon\mathcal{PC}_{\lambda_n}(\mathbb{R}^2;S_1\cup S_2)\rightarrow \mathcal{PC}_{\lambda_n}(\mathbb{R}^2; \mathbb{R}^2)\times\mathrm{L}^\infty(\Omega; \{v_1, v_2\}) given by

    \begin{equation*} T_n(u): = \big(w,z,\mathcal{A}(u)). \end{equation*}

    With a slight abuse of notation we define the functional H_n\colon\mathrm{L}^1_{loc}(\mathbb{R}^2; \mathbb{R}^2\times\{v_1, v_2\})\times \mathfrak{A}_0\rightarrow [0, +\infty) by setting

    \begin{equation} H_n(h;\Omega) = \begin{cases} H_n(u;\Omega) \quad&\text{if }T_n(u) = h\text{ for some }u\in\mathcal{PC}_{\lambda_n}( \mathbb{R}^2;S_1\cup S_2),\\ +\infty&\text{otherwise}. \end{cases} \end{equation} (4.4)

    Notice that H_n does not depend on the particular choice of u , since it is rotation-invariant. The same notation can be adopted for P_n , (R_n)_{C_s} and \mathcal{H}_n .

    We study the convergence of the functional

    \begin{align*} G_n(h;\Omega) & : = \begin{cases} H_n(h,\Omega)-\sum\limits_{s = 1}^{M(h)} (R_n)_{C_s}(h)\quad&\text{if }T_n(u) = h\text{ for some }u\in\mathcal{PC}_{\lambda_n}( \mathbb{R}^2;S_1\cup S_2),\\ +\infty &\text{otherwise} \end{cases}\\ & = \sum\limits_{s = 1}^{M(h)}\mathcal{H}_n(h;C_s). \end{align*}

    where M(h): = M(u) . Hence, we introduce the functional G\colon\mathrm{L}^1_{loc}(\mathbb{R}^2; \mathbb{R}^2\times\{v_1, v_2\})\times \mathfrak{A}_0\rightarrow [0, +\infty) by setting

    \begin{equation*} {G}(h;\Omega): = \begin{cases} \frac{4}{3}{R^2}\sum\limits_{s = 1}^{M(h)}(|D_1 w|(C_s)+|D_2 z|(C_s))\quad &\text{if }h = (w,z,\alpha)\in\text{Dom}({G};\Omega),\\ +\infty &\text{otherwise}, \end{cases} \end{equation*}

    where

    \begin{align*} \text{Dom}({G};\Omega): = & \bigg\{(w,z,\alpha)\in\mathrm{L}^1_{loc}( \mathbb{R}^2; \mathbb{R}^2\times\{v_1,v_2\}) \,:\,\exists\{C_s\}_{\substack{s\in\{1,\dots,M\}}}\text{ open partition of }\Omega\text{ s.t. }\\ & (w_{|C_s},z_{|C_s},\alpha_{|C_s})\in BV(C_s;\{-1,1\}^2\times\{v_{\ell_s}\}),\,\text{for some } \ell_s\in\{1,2\},\\ & \text{curl}(w_{|C_s},z_{|C_s}) = 0\text{ in } \mathcal{D}'(C_s; \mathbb{R}^2) \bigg\}. \end{align*}

    For h\in\mathrm{Dom}(G; \Omega) we say that the collection \{C_s\}_{s\in\{1, \dots, M\}} existing in virtue of the definition of \mathrm{Dom}(G; \Omega) is the open partition associated with h .

    We have denoted by \mathcal{D}'(C_s; \mathbb{R}^2) the space of distributions and by \mathrm{curl} the distribution curl defined by

    \begin{equation*} \langle ( \text{curl}(T))_{h,k}, \xi \rangle: = - \langle T^k, \partial_h \xi \rangle+ \langle T^h, \partial_k \xi \rangle, \quad {\forall} \xi \in C_c^{\infty}(C_s),\,\forall T \in \mathcal{D}'(C_s; \mathbb{R}^2), \end{equation*}

    for any h, k\in\{1, 2\} .

    The following notion of convergence will be used.

    Definition 4.3. Let \{h_n\}_{n\in \mathbb{N}}\subset \mathrm{L}^1_{loc}(\mathbb{R}^2; \mathbb{R}^2\times\{v_1, v_2\}) . We say that h_n {\Theta} -converges to h\in\mathrm{L}^1_{loc}(\mathbb{R}^2; \mathbb{R}^2\times\{v_1, v_2\}) (we write h_n \overset{{\Theta}}{\longrightarrow} h ) if the following conditions are satisfied:

    ● there exist \{u_n\}_{n\in \mathbb{N}}\subset\mathcal{PC}_{\lambda_n}(\mathbb{R}^2;S_1\cup S_2) , a positive constant C such that

    h_n = T_n(u_n) and P_n(u_n; \Omega) < C ,

    M(u_n)\rightarrow M\in \mathbb{N} as n\rightarrow +\infty ,

    (C_s)_n\rightarrow C_s in the Hausdorff sense, as n\rightarrow +\infty , for any s\in\{1, \dots, M\} ,

    where \mathcal{C}_n(u_n) = \{(C_{s})_n|\, s\in\{1, \dots, M(h_n)\}\} is the open partition associated with u_n .

    {h_n}\chi_{(C_s)_n}\rightarrow h\chi_{C_s} in \mathrm{L}^1_{loc}(\mathbb{R}^2; \mathbb{R}^2\times\{v_1, v_2\}) , for any s\in\{1, \dots, M\} .

    As in formula (4.4) we define P_n(h; \Omega): = P_n(u; \Omega) for h = T_n(u) with u \in \mathcal{PC}_{\lambda_n}(\mathbb{R}^2;S_1\cup S_2) .

    We remark that in general it is not possible to prove a compactness result for a sequence \{h_n = T_n(u_n)\}_{n\in \mathbb{N}}\subset T_n(\mathcal{PC}_{\lambda_n}(\mathbb{R}^2;S_1\cup S_2)) satisfying only the following natural conditions:

    \begin{equation*} {\sup\limits_{n\in \mathbb{N}}G_n(h_n;\Omega) < C\quad\text{and}\quad \sup\limits_{n\in \mathbb{N}}P_n(h_n;\Omega) < C}. \end{equation*}

    Indeed, it could happen that the region \{\mathcal{A}(u_n) = v_1\} has an increasing number of holes vanishing in the limit so that \{M(u_n)\}_{n\in \mathbb{N}} is divergent. Neither the Hausdorff convergence of the sets of the open partition is ensured.

    In the following proposition we show that, if strong and technical conditions hold, then \{h_n\}_{n\in \mathbb{N}} converges, up to subsequences, with respect to the \Theta -convergence.

    Proposition 4.4. Let \{h_n = T_n(u_n)\}_{n\in \mathbb{N}}\subset T_n(\mathcal{PC}_{\lambda_n}(\mathbb{R}^2;S_1\cup S_2)) be a sequence such that

    \begin{equation} {\sup\limits_{n\in \mathbb{N}}G_n(h_n;\Omega) < C\quad\;{and}\;\quad \sup\limits_{n\in \mathbb{N}}P_n(h_n;\Omega) < C}, \end{equation} (4.5)

    for some constant C > 0 . Furthermore, we assume that the open partition associated with u_n , \mathcal{C}_n(u_n) = \{(C_{s})_n|\, s\in\{1, \dots, M(u_n)\}\} , is such that

    \begin{equation*} M(u_n)\rightarrow M\in \mathbb{N}\quad {{as}}\;n\rightarrow +\infty, \end{equation*}
    \begin{equation*} (C_s)_n\rightarrow C_s \quad{{in \;the\; Hausdorff \;sense, \;as}}\;n\rightarrow +\infty,\,\forall s\in\{1,\dots,M\}. \end{equation*}

    Then there exists h\in \mathit{\text{Dom}}({G}; \Omega) such that, up to a subsequence, h_n \overset{{\Theta}}{\longrightarrow} h .

    Proof. Let \{h_n = (w_n, z_n, \mathcal{A}(u_n))\}_{n\in \mathbb{N}}\subset T_n(\mathcal{PC}_{\lambda_n}(\mathbb{R}^2;S_1\cup S_2)) be a sequence satisfying (4.5). Since {u_n}_{|C_s}\in S_\ell , for some \ell = \ell_s\in\{1, 2\} , then, by geometric and trigonometric identities, we deduce that

    \begin{equation*} {{u^{i,j}}} \cdot {{u^{i+1,j}}} = 1-R^2+{{\pi u^{i,j}}} \cdot {{\pi u^{i+1,j}}}, \end{equation*}
    \begin{equation*} {{u^{i,j}}} \cdot {{u^{i,j+1}}} = 1-R^2+{{\pi u^{i,j}}} \cdot {{\pi u^{i,j+1}}}, \end{equation*}

    where \pi u^{i, j}: = \pi_{v_\ell^\bot} u^i . Thus we may write

    \begin{equation*} G_n(h_n;\Omega) = \sum\limits_{s = 1}^M \widetilde{H}_n(u_n;C_s), \end{equation*}

    where

    \begin{align*} &\widetilde{H}_n(h_n;C_s)\\ &: = \frac{1}{\sqrt{2}\lambda_n\delta_n^{\frac{3}{2}}}\frac{1}{2}\lambda_n^2\sum\limits_{(i,j)\in\mathcal{I}^n(C_s)}\left[\left| \pi u_n^{i+2,j}-\frac{\alpha_n}{2}\pi u_n^{i+1,j}+\pi u_n^{i,j}\right|^2+\left| \pi u_n^{i,j+2}-\frac{\alpha_n}{2}\pi u_n^{i,j+1}+\pi u_n^{i,j}\right|^2\right]. \end{align*}

    Fixing \varepsilon > 0 sufficiently small, we have that for all n\in \mathbb{N} , up to a subsequence, (C_s)_\varepsilon: = \{x\in C_s\, :\, \text{dist}(x, \partial C_s) > \varepsilon\}\subset (C_s)_n and u_{n_{|(C_s)_\varepsilon}} takes values only in one circle. We infer that

    \begin{equation*} {\sum\limits_{s = 1}^M \widetilde{H}_n(h_n;(C_s)_\varepsilon)}\leq {G}_n(h_n;\Omega) < C, \end{equation*}

    which of course implies that {\widetilde{H}}_n(h_n; (C_s)_\varepsilon) < C , for all s\in\{1, \dots, M\} . We are in position to apply [9,Theorem 2.1 ⅰ) and Remark 2.2] to deduce the existence of (w_{(C_s)_\varepsilon}, z_{(C_s)_\varepsilon})\in BV((C_s)_\varepsilon; \{-1, 1\}^2) such that, up to subsequences, (w_n, z_n)\rightarrow (w_{(C_s)_\varepsilon}, z_{(C_s)_\varepsilon}) in \mathrm{L}^1_{loc}((C_s)_\varepsilon; \mathbb{R}^2) and curl (w_{(C_s)_\varepsilon}, z_{(C_s)_\varepsilon}) = 0 in \mathcal{D}'((C_s)_\varepsilon; \mathbb{R}^2) . The couples (w_{(C_s)_\varepsilon}, z_{(C_s)_\varepsilon}) can be extended to 0 in C_s\setminus(C_s)_\varepsilon . We preliminary observe that

    \begin{equation} (w_{(C_s)_{\varepsilon_2}},z_{(C_s)_{\varepsilon_2}}) = (w_{(C_s)_{\varepsilon_1}},z_{(C_s)_{\varepsilon_1}}) \quad\text{a.e. on }(C_s)_{\varepsilon_2}, \end{equation} (4.6)

    for any 0 < \varepsilon_1 < \varepsilon_2 . Indeed, since (C_s)_{\varepsilon_2}\subset(C_s)_{\varepsilon_1} , we have that

    \begin{equation*} (w_n,z_n)\rightarrow (w_{(C_s)_{\varepsilon_1}},z_{(C_s)_{\varepsilon_1}})\quad\text{in}\mathrm{L}^1_{loc}((C_s)_{\varepsilon_2}; \mathbb{R}^2). \end{equation*}

    The uniqueness of the limit in the \mathrm{L}^1_{loc} -topology implies (4.6). We now define the couples (w_{C_s}, z_{C_s})\colon C_s\rightarrow \mathbb{R}^2 by

    \begin{equation*} (w_{C_s},z_{C_s}): = \lim\limits_{\varepsilon\rightarrow 0^+}(w_{(C_s)_\varepsilon},z_{(C_s)_\varepsilon}). \end{equation*}

    The definition is well-posed; indeed, since by (4.6),

    \begin{equation*} \lim\limits_{\varepsilon'\rightarrow 0^+}(w_{(C_s)_{\varepsilon'}},z_{(C_s)_{\varepsilon'}}) = (w_{(C_s)_{\frac{1}{n}}},z_{(C_s)_{\frac{1}{n}}}) \quad\text{a.e. in }(C_s)_{\frac{1}{n}}, \end{equation*}

    for all n\in \mathbb{N} , then

    \begin{align*} &\left|\left\{ x\in C_s\,:\,\nexists\lim\limits_{\varepsilon'\rightarrow 0^+}(w_{(C_s)_{\varepsilon'}}(x),z_{(C_s)_{\varepsilon'}}(x)) \right\}\right|\\ & = \left|\bigcup\limits_{n = 1}^{+\infty}\left\{ x\in(C_s)_{\frac{1}{n}}\,:\,\nexists\lim\limits_{\varepsilon'\rightarrow 0^+}(w_{(C_s)_{\varepsilon'}}(x),z_{(C_s)_{\varepsilon'}}(x)) \right\}\right| = 0. \end{align*}

    Furthermore we define (w, z)\colon\Omega\rightarrow \mathbb{R}^2 by setting

    \begin{equation} (w,z)(x) = (w_{C_s},z_{C_s})(x), \end{equation}

    for a.e. x\in\Omega with x\in C_s , for some s\in\{1, \dots, M\} . Of course (w_{|C_s}, z_{|C_s}) = (w_{C_s}, z_{C_s})\in BV(C_s; \{-1, 1\}^2) , as it is the limit of BV functions. In order to show the \mathrm{L}^1_{loc} -convergence, we fix A\subset\subset C_s . Since dist (A, \partial C_s) > 0 , there exists \varepsilon > 0 such that A\subset\subset (C_s)_\varepsilon . We obtain:

    \begin{equation*} \left\lVert{(w_n,z_n)-(w_{C_s},z_{C_s})}\right\rVert_{\mathrm{L}^1(A; \mathbb{R}^2)} = \left\lVert{(w_n,z_n)-(w_{(C_s)_\varepsilon},z_{(C_s)_\varepsilon})}\right\rVert_{\mathrm{L}^1(A; \mathbb{R}^2)}, \end{equation*}

    which vanishes as n\rightarrow +\infty , up to subsequences. This leads to the convergence

    \begin{equation*} (w_n,z_n)\rightarrow ( w_{C_s},z_{C_s}) \quad\text{in }L^1_{loc}(C_s; \mathbb{R}^2). \end{equation*}

    Finally, we prove that \mathrm{curl}(w_{C_s}, z_{C_s}) = 0 in \mathcal{D}'(C_s; \mathbb{R}^2) . If \xi\in C_c^\infty(C_s) , then \mathrm{supp}\xi\subset(C_s)_\varepsilon for some \varepsilon > 0 and so

    \begin{align*} \langle \text{curl}(w_{C_s},z_{C_s}),\xi \rangle & = -\int_{(C_s)_\varepsilon}w_{(C_s)_\varepsilon} \partial_2\xi\,dx+\int_{(C_s)_\varepsilon}z_{(C_s)_\varepsilon} \partial_1\xi\,dx\\ & = \langle \text{curl}(w_{(C_s)_\varepsilon},z_{(C_s)_\varepsilon}),\xi \rangle = 0. \end{align*}

    Now we state the main theorem of this section. The regularity assumption on \Omega and on the open partition of h in the statement ⅱ) are required in order to apply [9,Theorem 2.1 ⅲ)] locally. As explained in [9] a simply connected BVG domain guaranties an extension property for BVG functions, which is needed to construct a recovery sequence for h . On the contrary, the proof of the liminf inequality ⅰ) actually works without assuming this kind of regularity (see [9,Remark 2.2]).

    Theorem 4.5. Let \Omega\in \mathfrak{A}_0 . Then the following statements hold true:

    i) (liminf inequality) Let \{h_n\}_{n\in \mathbb{N}}\subset\mathrm{L}^1_{loc}(\mathbb{R}^2; \mathbb{R}^2\times\{v_1, v_2\}) and h\in\mathrm{L}^1_{loc}(\mathbb{R}^2; \mathbb{R}^2\times\{v_1, v_2\}) . Assume that {\sup_{n\in \mathbb{N}}}P_n(h_n; \Omega) < C for some constant C > 0 and h_n \overset{{\Theta}}{\longrightarrow} h . Then

    \begin{equation*} {G}(h;\Omega)\leq\liminf\limits_{n\rightarrow +\infty}{G}_n(h_n;\Omega). \end{equation*}

    ii) (limsup inequality) Let h\in\mathrm{Dom}(G; \Omega) be such that its open partition consists of BVG domains. Then there exists a sequence \{h_n\}_{n\in \mathbb{N}}\subset\mathrm{L}^1_{loc}(\mathbb{R}^2; \mathbb{R}^2\times\{v_1, v_2\}) such that h_n \overset{{\Theta}}{\longrightarrow} h and

    \begin{equation*} \limsup\limits_{n\rightarrow +\infty}{G}_n(h_n;\Omega)\leq{G}(h;\Omega). \end{equation*}

    Proof. We start by proving ⅰ). Let \{h_n\}_{n\in \mathbb{N}}\subset\mathrm{L}^1_{loc}(\mathbb{R}^2; \mathbb{R}^2\times\{v_1, v_2\}) and h\in\mathrm{L}^1_{loc}(\mathbb{R}^2; \mathbb{R}^2\times\{v_1, v_2\}) be such that {\sup_{n\in \mathbb{N}}}P_n(h_n; \Omega) < C and h_n \overset{{\Theta}}{\longrightarrow} h . Up to subsequences, we may assume that the lower limit in the right hand side of the liminf inequality is actually a limit. Furthermore we may assume that it is finite, the inequality being otherwise trivial. In particular, we have

    \begin{equation*} {\sup\limits_{n\in \mathbb{N}}G_n}(h_n;\Omega) < C, \end{equation*}

    possibly with a larger C . By the definition of {\Theta} -convergence, h_n = (w_n, z_n, \mathcal{A}(u_n)) = T_n(u_n) for some u_n\in\mathcal{PC}_{\lambda_n}(\mathbb{R}^2;S_1\cup S_2) . Up to subsequences, M = M(h_n) is independent of n and we may assume, for \varepsilon > 0 sufficiently small, that (C_s)_\varepsilon\subset (C_s)_n and u_{n_{|(C_s)_\varepsilon}} takes values only on one circle S_\ell , for all n\in \mathbb{N} . Reasoning as in ⅰ), we infer

    \begin{equation*} {{G}_n(h_n;\Omega)\geq\sum\limits_{s = 1}^M \widetilde{H}_n(h_n;(C_s)_\varepsilon)}, \end{equation*}

    Since {h_n}\rightarrow h in \mathrm{L}^1((C_s)_\varepsilon; \mathbb{R}^2\times\{v_\ell\}) , as n\rightarrow +\infty , we are in position to apply [9,Theorem 2.1 ⅱ) and Remark 2.2] so that, passing to the lower limit, we get

    \begin{align*} \liminf\limits_{n\rightarrow +\infty}{G}_n(h_n;\Omega) &\geq \sum\limits_{s = 1}^M\liminf\limits_{n\rightarrow +\infty}{\widetilde{H}}_n(h_n;(C_s)_\varepsilon) \geq \sum\limits_{s = 1}^M \frac{4}{3}R^2[|D_1 w|((C_s)_\varepsilon)+|D_2 z|((C_s)_\varepsilon)], \end{align*}

    where h = (w, z, \alpha) . Letting \varepsilon\rightarrow 0^+ we get the thesis.

    Let us prove ⅱ). Let h\in\text{Dom}(G; \Omega) . This implies that h = (w, z, \alpha)\in\mathrm{L}^1_{loc}(\mathbb{R}^2; \mathbb{R}^2\times\{v_1, v_2\}) and the existence of an open partition of \Omega , \mathcal{C} = \{C_{s}|\, s\in\{1, \dots, M\}\} consisting of BVG domains such that, for some \ell = \ell_s\in\{1, 2\} ,

    \begin{equation*} (w_{|C_s},z_{|C_s},\alpha_{|C_s})\in BV(C_s;\{-1,1\}^2\times\{v_\ell\})\quad\text{and}\quad \text{curl}(w_{|C_s},z_{|C_s}) = 0\text{ in } \mathcal{D}'(C_s; \mathbb{R}^2). \end{equation*}

    Applying [9,Theorem 2.1 ⅲ)] to any (w_{|C_s}, z_{|C_s}) , we get the existence of a sequence {\{((w_n)_{C_s}, (z_n)_{C_s})\}_{n\in \mathbb{N}}\subset}\mathrm{L}^1_{loc}(\mathbb{R}^2; \mathbb{R}^2) such that {((w_n)_{C_s}, (z_n)_{C_s})}\rightarrow (w_{|C_s}, z_{|C_s}) in \mathrm{L}^1(C_s; \mathbb{R}^2) and

    \begin{equation*} \limsup\limits_{n\rightarrow +\infty}{\mathcal{H}}_n({(w_n)_{C_s},(z_n)_{C_s}},v_\ell)\leq \frac{4}{3}{R^2}(|D_1 w|(C_s)+|D_2 z|(C_s)) \end{equation*}

    Defining (w_n, z_n, \alpha_n)\colon \mathbb{R}^2\rightarrow \mathbb{R}^2\times\{v_1, v_2\} by

    \begin{equation*} (w_n,z_n,\alpha_n)(x): = {((w_n)_{C_s}(x),(z_n)_{C_s}(x),v_\ell)}, \end{equation*}

    if x\in\Omega such that x\in C_s for some s\in\{1, \dots, M\} , and arbitrarily extended outside \Omega , and summing on s\in\{1, \dots, M\} the previous inequality we obtain the thesis.

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    The authors wish to thank the reviewer for numerous suggestions that improved the paper. The authors warmly thank Prof. Marco Cicalese for the insightful discussions. L. Lamberti wishes to acknowledge the hospitality of the Faculty of Mathematics of the Technical University of Munich, where part of this research was carried out. The authors are members of the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). A. Kubin was supported by the DFG Collaborative Research Center TRR 109 "Discretization in Geometry and Dynamics". L. Lamberti was supported partially by the DFG Collaborative Research Center TRR 109 "Discretization in Geometry and Dynamics" and by COST Action 18232 MAT-DYN-NET, supported by COST (European Cooperation in Science and Technology).

    The authors declare no conflict of interest.



    [1] B. Albers, Coupling of adsorption and diffusion in porous and granular materials. A 1D example of the boundary value problem, Arch. Appl. Mech., 70 (2000), 519–531. https://doi.org/10.1007/s004190000082 doi: 10.1007/s004190000082
    [2] B. Albers, Makroskopische Beschreibung von Adsorptions-Diffusions-Vorgängen in porösen Körpern, Ph.D. Thesis, TU Berlin, Logos-Verlag, 2000.
    [3] B. Albers, Modeling and numerical analysis of wave propagation in saturated and partially saturated porous media, Vol. 48, In: Veröffentlichungen des Grundbauinstitutes der Technischen Universität Berlin, Habilitation thesis, Shaker Verlag, Aachen, 2010.
    [4] B. Albers, Modeling the hysteretic behavior of the capillary pressure in partially saturated porous media: a review, Acta Mech., 225 (2014), 2163–2189. https://doi.org/10.1007/s00707-014-1122-4 doi: 10.1007/s00707-014-1122-4
    [5] B. Albers, On modeling three-component porous media incorporating hysteresis, In: E. Onate, J. Oliver, A. Huerta, Proceedings of the 11th World Congress on Computational Mechanics (WCCM XI), 2014, 3240–3251.
    [6] B. Albers, Main drying and wetting curves of soils: on measurements, prediction and influence on wave propagation, Eng. Trans., 63 (2015), 5–34. https://doi.org/10.24423/engtrans.286.2015 doi: 10.24423/engtrans.286.2015
    [7] B. Albers, On the influence of the hysteretic behavior of the capillary pressure on the wave propagation in partially saturated soils, J. Phys.: Conf. Ser., 727 (2016), 012001. https://doi.org/10.1088/1742-6596/727/1/012001 doi: 10.1088/1742-6596/727/1/012001
    [8] B. Albers, P. Krejčí, Hysteresis in unsaturated porous media–two models for wave propagation and engineering applications, In: B. Albers, M. Kuczma, Continuous media with microstructure 2, Springer, Cham, 2016,217–229. https://doi.org/10.1007/978-3-319-28241-1_15
    [9] B. Albers, P. Krejčí, Unsaturated porous media flow with thermomechanical interaction, Math. Methods Appl. Sci., 39 (2016), 2220–2238. https://doi.org/10.1002/mma.3635 doi: 10.1002/mma.3635
    [10] A. Basile, G. Ciollaro, A. Coppola, Hysteresis in soil water characteristics as a key to interpreting comparisons of laboratory and field measured hydraulic properties, Water Resour. Res., 39 (2003), 1355. https://doi.org/10.1029/2003WR002432 doi: 10.1029/2003WR002432
    [11] J. Bear, Y. Bachmat, Introduction to modeling of transport phenomena in porous media, Springer Dordrecht, 1990. https://doi.org/10.1007/978-94-009-1926-6
    [12] J. Bear, A. Verruijt, Modeling flow in the unsaturated zone, In: Modeling groundwater flow and pollution, Theory and Applications of Transport in Porous Media, Springer, Dordrecht, 2 (1987), 123–152. https://doi.org/10.1007/978-94-009-3379-8_5
    [13] G. Bertotti, I. D. Mayergoyz, The science of hysteresis: 3-volume set, Academic Press, 2005.
    [14] G. Bonan, Predictor–Corrector solution for the \varphi-based Richards equation, accessed: 2023/08/18. Available form: https://zmoon.github.io/bonanmodeling/08/01.html.
    [15] R. H. Brooks, A. T. Corey, Hydraulic properties of porous media, Colorado State University ProQuest Dissertations & Theses, 1965.
    [16] S. Brunauer, L. S. Deming, W. E. Deming, E. Teller, On a theory of the van der Waals adsorption of gases, J. Am. Chem. Soc., 62 (1940), 1723–1732. https://doi.org/10.1021/ja01864a025 doi: 10.1021/ja01864a025
    [17] S. Brunauer, P. H. Emmet, E. Teller, Adsorption of gases in multimolecular layers, J. Am. Chem. Soc., 60 (1938), 309–319. https://doi.org/10.1021/ja01269a023 doi: 10.1021/ja01269a023
    [18] I. Chatzis, F. A. L. Dullien, Dynamic immiscible displacement mechanisms in pore doublets: theory versus experiment, J. Colloid Interf. Sci., 91 (1983), 199–222. https://doi.org/10.1016/0021-9797(83)90326-0 doi: 10.1016/0021-9797(83)90326-0
    [19] H. Chen, K. Chen, M. Yang, A new hysteresis model of the water retention curve based on pore expansion and contraction, Comput. Geotech., 121 (2020), 103482. https://doi.org/10.1016/j.compgeo.2020.103482 doi: 10.1016/j.compgeo.2020.103482
    [20] O. Coussy, Poromechanics, John Wiley & Sons, 2004.
    [21] O. Coussy, L. Dormieux, E. Detourney, From mixture theory to Biot's approach for porous media, Int. J. Solids Struct., 35 (1998), 4619–4635. https://doi.org/10.1016/S0020-7683(98)00087-0 doi: 10.1016/S0020-7683(98)00087-0
    [22] B. Detmann, Capillary rise and infiltration in sand – phenomena, 1D tests and analysis, submitted for publication, 2024.
    [23] B. Detmann, P. Krejčí, A multicomponent flow model in deformable porous media, Math. Methods Appl. Sci., 42 (2019), 1894–1906. https://doi.org/10.1002/mma.5482 doi: 10.1002/mma.5482
    [24] A. S. Dias, M. Pirone, M. V. Nicotera, G. Urciuoli, Hydraulic hysteresis of natural pyroclastic soils in partially saturated conditions: experimental investigation and modelling, Acta Geotech., 17 (2022), 837–855. https://doi.org/10.1007/s11440-021-01273-y doi: 10.1007/s11440-021-01273-y
    [25] J. A. Ewing, X. Experimental researches in magnetism, Phil. Trans. R. Soc., 176 (1885), 523–640. https://doi.org/10.1098/rstl.1885.0010 doi: 10.1098/rstl.1885.0010
    [26] M. J. Fayer, C. S. Simmons, Modified soil water retention functions for all matric suctions, Water Resour. Res., 31 (1995), 1233–1238. https://doi.org/10.1029/95WR00173 doi: 10.1029/95WR00173
    [27] D. Flynn, Modelling the flow of water through multiphase porous media with the Preisach model, Ph.D. Thesis, University College Cork, 2008.
    [28] D. Flynn, H. McNamara, P. O'Kane, A. Pokrovskii, Chapter 7 – Application of the Preisach model to soil-moisture hysteresis, In: G. Bertotti, I. D. Mayergoyz, The science of hysteresis, III (2005), 689–744. https://doi.org/10.1016/B978-012480874-4/50025-7 doi: 10.1016/B978-012480874-4/50025-7
    [29] D. G. Fredlund, H. Rahardjo, M. D. Fredlund, Unsaturated soil mechanics in engineering practice, John Wiley & Sons, 2012. https://doi.org/10.1002/9781118280492
    [30] H. Freundlich, Kapillarchemie, Akademische Verlagsgeselschaft, Leipzig, 1923.
    [31] D. Gallipoli, A hysteretic soil-water retention model accounting for cyclic variations of suction and void ratio, Géotechnique, 62 (2012), 605–616. https://doi.org/10.1680/geot.11.P.007 doi: 10.1680/geot.11.P.007
    [32] R. W. Gillham, A. Klute, D. F. Heermann, Hydraulic properties of a porous medium: Measurement and empirical representation, Soil Sci. Soc. Amer. J., 40 (1976), 203–207. https://doi.org/10.2136/sssaj1976.03615995004000020008x doi: 10.2136/sssaj1976.03615995004000020008x
    [33] R. W. Gillham, A. Klute, D. F. Heermann, Measurement and numerical simulation of hysteretic flow in a heterogeneous porous medium, Soil Sci. Soc. Amer. J., 43 (1979), 1061–1067. https://doi.org/10.2136/sssaj1979.03615995004300060001x doi: 10.2136/sssaj1979.03615995004300060001x
    [34] W. H. Graf, Fluvial hydraulics: flow and transport processes in channels of simple geometry, John Wiley & Sons, New York, 1998.
    [35] W. H. Graf, Z. Qu, Flood hydrographs in open channels, Proceedings of the Institution of Civil Engineers–Water Management, 157 (2004), 45–52. https://doi.org/10.1680/wama.2004.157.1.45 doi: 10.1680/wama.2004.157.1.45
    [36] S. J. Gregg, K. S. W. Sing, Adsorption, surface area and porosity, Academic Press, London, 1982.
    [37] S. Guglielmi, M. Pirone, A. S. Dias, F. Cotecchia, G. Urciuoli, Thermohydraulic numerical modeling of slope-vegetation-atmosphere interaction: case study of the pyroclastic slope cover at Monte Faito, Italy, J. Geotechn. Geoenviron. Eng., 149 (2023), 05023005. https://doi.org/10.1061/JGGEFK.GTENG-11240 doi: 10.1061/JGGEFK.GTENG-11240
    [38] R. Haverkamp, P. Reggiani, P. J. Ross, J. Y. Parlange, Soil water hysteresis prediction model based on theory and geometric scaling, In: P. A. C. Raats, D. Smiles, A. Warrick, Environmental mechanics, water, mass and energy transfer in the biosphere, American Geophysical Union, 129 (2002), 213–246. https://doi.org/10.1029/129GM19
    [39] P. P. Jansen, L. van Bendegom, J. van den Berg, M. de Vries, A. Zanen, Principles of river engineering: the non-tidal alluvial river, Water Resources Engineering Series, Pitman, 1979.
    [40] D. B. Jaynes, Comparison of soil-water hysteresis models, J. Hydrol., 75 (1984), 287–299. https://doi.org/10.1016/0022-1694(84)90054-4 doi: 10.1016/0022-1694(84)90054-4
    [41] B. E. Jones, A method of correcting river discharge for a changing stage, Technical report, US Geological Survey, 1916.
    [42] D. Kolymbas, Introduction to hypoplasticity: advances in geotechnical engineering and tunnelling, CRC Press, 2000.
    [43] R. W. R. Koopmans, R. D. Miller, Soil freezing and soil water characteristic curves, Soil Sci. Soc. Amer. J., 30 (1966), 680–685. https://doi.org/10.2136/sssaj1966.03615995003000060011x doi: 10.2136/sssaj1966.03615995003000060011x
    [44] I. Langmuir, The adsorption of gases on plane surfaces of glass, mica and platinum, J. Am. Chem. Soc., 40 (1918), 1361–1403. https://doi.org/10.1021/ja02242a004 doi: 10.1021/ja02242a004
    [45] V. H. Le, R. Glasenapp, F. Rackwitz, Cyclic hysteretic behavior and development of the secant shear modulus of sand under drained and undrained conditions, Int. J. Geomech., 24 (2024), 04024126. https://doi.org/10.1061/IJGNAI.GMENG-9380 doi: 10.1061/IJGNAI.GMENG-9380
    [46] R. J. Lenhard, J. C. Parker, J. J. Kaluarachchi, Comparing simulated and experimental hysteretic two-phase transient fluid flow phenomena, Water Resour. Res., 27 (1991), 2113–2124. https://doi.org/10.1029/91WR01272 doi: 10.1029/91WR01272
    [47] A. C. Liakopoulos, Theoretical approach to the solution of the infiltration problem, International Association of Scientific Hydrology. Bulletin, 11 (1966), 69–110. https://doi.org/10.1080/02626666609493444 doi: 10.1080/02626666609493444
    [48] N. Lu, W. J. Likos, Unsaturated soil mechanics, Wiley, Hoboken, New Jersey, 2004.
    [49] R. J. Mander, Aspects of unsteady flow and variable backwater, In: R. W. Herschy, Hydrometry: principles and practices, Wiley: Chichester, 1978.
    [50] I. D. Mayergoyz, Mathematical models of hysteresis, Springer, 1991. https://doi.org/10.1007/978-1-4612-3028-1
    [51] A. Niemunis, T. Wichtmann, T. Triantafyllidis, Long-term deformations in soils due to cyclic loading, In: W. Wu, H. S. Yu, Modern trends in geomechanics, Springer Proceedings in Physics, Springer, 106 (2006), 427–462. https://doi.org/10.1007/978-3-540-35724-7_26
    [52] J. Y. Parlange, Capillary hysteresis and the relationship between drying and wetting curves, Water Resour. Res., 12 (1976), 224–228. https://doi.org/10.1029/WR012i002p00224 doi: 10.1029/WR012i002p00224
    [53] E. Perret, M. Lang, J. Le Coz, A framework for detecting stage-discharge hysteresis due to flow unsteadiness: application to France's national hydrometry network, J. Hydrol., 608 (2022), 127567. https://doi.org/10.1016/j.jhydrol.2022.127567 doi: 10.1016/j.jhydrol.2022.127567
    [54] A. Petersen-Øverleir, Modelling stage–discharge relationships affected by hysteresis using the Jones formula and nonlinear regression, Hydrol. Sci. J., 51 (2006), 365–388. https://doi.org/10.1623/hysj.51.3.365 doi: 10.1623/hysj.51.3.365
    [55] R. Plagge, G. Scheffler, J. Grunewald, M. Funk, On the hysteresis in moisture storage and conductivity measured by the instantaneous profile method, J. Build. Phys., 29 (2006), 247–259. https://doi.org/10.1177/1744259106060706 doi: 10.1177/1744259106060706
    [56] F. Preisach, Über die magnetische Nachwirkung, Z. Physik, 94 (1935), 277–302. https://doi.org/10.1007/BF01349418 doi: 10.1007/BF01349418
    [57] R. Scarfone, S. J. Wheeler, M. Lloret-Cabot, A hysteretic hydraulic constitutive model for unsaturated soils and application to capillary barrier systems, Geomech. Energy Environ., 30 (2022), 100224. https://doi.org/10.1016/j.gete.2020.100224 doi: 10.1016/j.gete.2020.100224
    [58] T. Schanz, P. A. Vermeer, P. G. Bonnier, The hardening soil model: formulation and verification, In: Beyond 2000 in computational geotechnics, Routledge, 2019,281–296.
    [59] H. Sheta, Simulation von Mehrphasenvorgängen in porösen Medien unter Einbeziehung von Hysterese-Effekten, Ph.D. Thesis, Universität Stuttgart, 1999.
    [60] P. Sitarenios, F. Casini, A. Askarinejad, S. Springman, Hydro-mechanical analysis of a surficial landslide triggered by artificial rainfall: the Ruedlingen field experiment, Géotechnique, 71 (2021), 96–109. https://doi.org/10.1680/jgeot.18.P.188 doi: 10.1680/jgeot.18.P.188
    [61] M. Tafili, T. Wichtmann, T. Triantafyllidis, Experimental investigation and constitutive modeling of the behaviour of highly plastic lower rhine clay under monotonic and cyclic loading, Can. Geotechn. J., 58 (2021), 1396–1410. https://doi.org/10.1139/cgj-2020-0012 doi: 10.1139/cgj-2020-0012
    [62] J. Teng, D. Antai, S. Zhang, X. Zhang, D. Sheng, Freezing-thawing hysteretic behavior of soils, Water Resour. Res., 60 (2024), e2024WR037280. https://doi.org/10.1029/2024WR037280 doi: 10.1029/2024WR037280
    [63] M. Thommes, K. Kaneko, A. V. Neimark, J. P. Olivier, F. Rodriguez-Reinoso, J. Rouquerol, et al., Physisorption of gases, with special reference to the evaluation of surface area and pore size distribution (IUPAC Technical Report), Pure Appl. Chem., 87 (2015), 1051–1069. https://doi.org/10.1515/pac-2014-1117 doi: 10.1515/pac-2014-1117
    [64] A. Tsiampousi, L. Zdravković, D. M. Potts, A three-dimensional hysteretic soil-water retention curve, Géotechnique, 63 (2013), 155–164. https://doi.org/10.1680/geot.11.P.074 doi: 10.1680/geot.11.P.074
    [65] N. Vaiana, L. Rosati, Classification and unified phenomenological modeling of complex uniaxial rate-independent hysteretic responses, Mech. Syst. Signal Pr., 182 (2023), 109539. https://doi.org/10.1016/j.ymssp.2022.109539 doi: 10.1016/j.ymssp.2022.109539
    [66] M. T. van Genuchten, A closed-form equation for predicting the hydraulic conductivity of unsaturated soils Soil Sci. Soc. Amer. J., 44 (1980), 892–898. https://doi.org/10.2136/sssaj1980.03615995004400050002x
    [67] K. Wilmanski, B. Albers, Continuum thermodynamics, part II: applications and examples, World Scientific, Singapore, 2015.
    [68] T. Wichtmann, Soil behaviour under cyclic loading-experimental observations, constitutive description and applications, Vol. 181, Habilitation thesis, 2016.
    [69] K. Wilmanski, Lagrangean model of two-phase porous material, J. Non-Equilibrium Thermodyn., 20 (1995), 50–77. https://doi.org/10.1515/jnet.1995.20.1.50 doi: 10.1515/jnet.1995.20.1.50
    [70] Y. Zhou, J. Zhou, X. Shi, G. Zhou. Practical models describing hysteresis behavior of unfrozen water in frozen soil based on similarity analysis, Cold Reg. Sci. Technol., 157 (2019), 215–223. https://doi.org/10.1016/j.coldregions.2018.11.002 doi: 10.1016/j.coldregions.2018.11.002
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