Crystalline Evolutions in Chessboard-like Microstructures

We describe the macroscopic behavior of evolutions by crystalline curvature of planar sets in a chessboard--like medium, modeled by a periodic forcing term. We show that the underlying microstructure may produce both pinning and confinement effects on the geometric motion.


Introduction
We are concerned with the asymptotic behavior of motions of planar curves according to the law where v is the normal velocity, κ is the crystalline curvature, g : R 2 → R is a periodic forcing term, with averageḡ, and ε > 0 is a small parameter which takes account of the frequency of oscillation.
The forcing term g models a rapidly oscillating heterogeneous medium and, in the homogenization limit ε → 0, the oscillations of the medium affect the velocity of the evolving front. The geometric motion (1) corresponds to the gradient flow of the energy where we identify the evolving curve with the boundary of a set E. Since the volume term converges toḡ L 2 (E) as ε → 0, the Γ-limit of the functionals F ε (see [6]) is given by . Hence, our analysis can be set in a large class of variational evolution problems dealing with limits of motions driven by functionals F ε depending on a small parameter.
For oscillating functionals like F ε , the energy landscape of the energies can be quite different from that of their Γ-limit, and the related motions can be influenced by the presence of local minima which may give rise to pinning phenomena, or to effective homogenized velocities (see [25,2,9,7]). In the case of geometric motions, a general understanding of the effects of microstructure is still missing. Recently, some results have been obtained for two-dimensional crystalline energies, for which a simpler description can be given in terms of a system of ODEs (see for instance [9,11,8,12,9,10]).
Coming back to our specific problem (1), we assume for simplicity that g takes only two values α < 0 < β, its periodicity cell is [0, 1] 2 , and thatḡ = α+β 2 . We also assume that g has a chessboard structure, as specified in (2) below.
After a careful analysis, it turns out that curves evolving by (1) undergo a microscopic "facet-breaking" phenomenon at a scale ε, with small segments of length proportional to ε being created and, in some cases, subsequently reabsorbed. The macroscopic effect of this behavior is a "pinning effect" for the limit evolution, corresponding to the possible onset of new edges, with slope of 45 degrees and zero velocity (depending on the initial set and on the values α, β). On the other hand, the horizontal and the vertical edges always travel with the asymptotic velocity κ +ḡ. We thus obtain a characterization of the limit evolution, which is the main result of this paper, and is stated precisely in Theorems 4.7 and 4.8.
We point out that, due to the possible presence of these new edges with zero velocity, the limit flow does not coincide with the gradient flow of the limit functional F , which is simply given by v = κ +ḡ.
We recall that, in a previous paper [10], we considered a similar homogenization problem where the periodic function g depends only on the horizontal variable, so that the medium has a stratified, opposite to a chessboard-like, structure.
It would be very interesting to extend our analysis to the isotropic variant of (1), where the crystalline curvature κ is replaced by the usual curvature of the evolving curve, so that (1) becomes a forced curvature flow. However, as such evolution cannot be described in terms of a system of ODEs, different techniques would be needed (partial results in this direction can be found in [15,14]).
The plan of the paper is the following: in Section 2 we introduce the notion of crystalline curvature and the evolution problem we are interested in. In Section 3 we introduce the notion of calibrable edge, that is, an edge which does not break during the evolution, and we state the calibrability conditions. Finally, in Section 4 we characterize explicitly the limit evolution as ε → 0, for initial data which are squares (Theorem 4.7) and more generally rectangles (Theorem 4.8).
Definition 2.1 (ϕ-regular set, Cahn-Hoffmann field, ϕ-curvature). We say that a set E ⊆ R 2 is ϕ-regular if ∂E is a compact Lipschitz curve, and there exists a vector field n ϕ ∈ Lip(∂E; R 2 ) such that n ϕ ∈ T ϕ • (∇d E ) H 1 -almost everywhere in ∂E. Any selection of the multivalued function T ϕ • (∇d E ) on ∂E is called a Cahn-Hoffmann vector field for ∂E, and κ = div n ϕ is the related ϕ-curvature (or crystalline curvature) of ∂E.

Remark 2.2 (Edges and vertices). A direct computation gives that
(Here and in the following [[ξ, η]] is the closed segment joining the vector ξ with η). The boundary of a ϕ-regular set E is given by a finite number of maximal closed arcs with the property that T ϕ • (∇d E ) is a fixed set T A in the interior of each arc A. This set T A is either a singleton, if the arc A is not a horizontal or vertical segment, or one of the closed convex cones described above. The maximal arcs of ∂E which are straight horizontal or vertical segments will be called edges, and the endpoints of every arc will be called vertices of ∂E.
The requirement of Lipschitz continuity keeps the value of every Cahn-Hoffmann vector field fixed at vertices. Hence, in order to exhibit a Cahn-Hoffmann vector field n ϕ on ∂E it is enough to construct a field n A ∈ Lip(A; R 2 ) on each arc A, with the correct values at the vertices, and satisfying the constraint n A ∈ T A . In what follows, with a little abuse of notation, we shall call n A the Cahn-Hoffmann vector field on the arc A.
We will denote by A ε (resp. B ε ) the union of all closed squares Q of side length ε such that g ε = α (resp. g ε = β) in the interior of Q. The set of discontinuity points of g ε will be denoted by Ξ. A discontinuity line is a straight line contained in Ξ. We define the multifunction G in R 2 , by setting G = [α, β] on Ξ, and We want to introduce our notion of geometric evolution E(t), obeying to the law where V is the normal velocity, and κ is the crystalline curvature on ∂E(t).
In order to make a sense to (3) it would be enough to require that the evolution is a family of ϕ-regular sets. Nevertheless, as underlined in Remark 2.2, even if E is a ϕ-regular set, the crystalline curvature on ∂E may not be uniquely determined, due to the infinitely many choices for the Cahn-Hoffmann vector field on the edges of ∂E.

Definition 2.3 (Variational Cahn-Hoffmann field). A variational Cahn-Hoffmann vec-
tor field for a ϕ-regular set E is a Cahn-Hoffmann vector field n on ∂E such that for every edge L of ∂E not lying on a discontinuity line of g ε , the restriction n L of n on L is the unique minimum of the functional where p, q are the endpoints of L and n 0 , n 1 are the values at p, q assigned to every Cahn-Hoffmann vector field (see Remark 2.2).
Remark 2.4. If the minimum n L in D L of the functional N L satisfies the strict constraint n L (ξ) ∈ int T L for every ξ ∈ L, then the velocity g ε − div n L is constant along the edge, that is the flat arc remains flat under the evolution. This is always the case for unforced crystalline flows, since the unique minimum is the interpolation of the assigned values at the vertices of L, and the constant value of the ϕ-curvature is given by where is the length of the edge L and χ L is a convexity factor: χ L = 1, −1, 0, depending on whether E(t) is locally convex at L, locally concave at L, or neither. = d E(t) (ξ) is locally Lipschitz in A; (iii) there exists a function n ∈ L ∞ (A, R 2 ), with div n ∈ L ∞ (A), such that the restriction of n(t, ·) to ∂E(t) is a variational Cahn-Hoffmann vector field for ∂E(t) for almost every t ∈ [0, T ]; (iv) ∂ t d − div n ∈ G ε H 1 -almost everywhere in ∂E(t) and for almost every t ∈ [0, T ).

Calibrability conditions
In this section we deal with the minimum problem in Definition 2.3 for a given ϕregular set E, and we characterize the edges of ∂E having constant velocity v L := κ L +g ε .
The results concern edges L ∈ ∂E not lying on a discontinuity line of the forcing term, in such a way that g ε is defined H 1 -almost everywhere on L. We will use the notation Setting by n : R → [−1, 1] the unique varying component of the variational Cahn-Hoffmann vector field on L (recall Remark 2.2), the assigned values of n are the following: Moreover, we denote by γ ε : R → R the restriction of g ε on the straight line containing L, and we distinguish two different type of discontinuity points for γ ε : With these notation, the requirement that κ + g ε is constant on L can be rephrased in the following 1D problem. (i) n satisfies (4).
In this case, we say that v L = n + γ ε is the (normal) velocity of the edge L.
The calibrability property was studied, in its full generality, in [10]. We collect here the results needed in the rest of the paper, sketching the proofs for sake of completeness.
Denoting by α , β ∈ [0, ε/2] the non-negative lengths given by the conditions the calibrability condition in Definition 3.1(iii) sets the value of n outside the jump set of γ ε : so that n needs to be where v L is the feasible velocity of the edge L In conclusion, the calibrability conditions (i) and (iii) in Definition 3.1 fix univocally a candidate field (7) which is continuous and affine with given slope in each phase of γ ε . This field n is the Cahn-Hoffman field which calibrates L with velocity (8) if and only if it also satisfies the constraint |n(x)| ≤ 1 for every x ∈ [p, q].
In what follows we will assume 0 < ε < 8 β − α in such a way that the small has the same sign of the curvature term χ L 2 .

Proposition 3.3.
Let L be an edge with zero ϕ-curvature, and let n 0 ∈ {±1} be the given value of the Cahn-Hoffmann vector field at the endpoints of L. Then the following hold.
Proof. If n 0 = 1 the Cahn-Hoffman vector field n needs to be neither increasing near p nor decreasing near q. This occurs only if L is the union of three consecutive segments then the candidate field (7) always satisfied the constraint in Definition3.1(ii), proving (i).

Proposition 3.4.
Let L be an edge with positive ϕ-curvature. If either or p ∈ I β,α , q ∈ I α,β , then L is calibrable with velocity v L given by (8).
Proof. Under the assumption (9), the candidate Cahn-Hoffmann vector field (7) is an increasing function in [p, q]. Hence the constraint in Definition 3.1(ii) is fulfilled, and L is calibrable.
Proposition 3.6. Let L be an edge with positive ϕ-curvature, and such that . Then the following hold.
Then L is calibrable if and only if σ ≥ σ * . (6) the candidate Cahn-Hoffmann field n is strictly decreasing in the β phase. Hence, under the assumptions in (i), n does not satisfy the constraint |n| ≤ 1 at least near an endpoint, and L is not calibrable. If both the endpoints belong to the α phase, then the requirement (σ 1 , σ 2 ) ∈ Σ is equivalent to the conditions that guarantee the calibrability of the edge. Setting (10) The proof of (iii) follows the same arguments.

Forced crystalline flows and their effective motion
The results of Section 3 prescribe a velocity to every calibrable edge not lying on a discontinuity line of g ε , and suggest that the forced crystalline curvature flow starting from a coordinate polyrectangle (that is a set whose boundary is a closed polygonal curve with edges parallel to the coordinate axes) remains a coordinate polyrectangle, whose structure changes when either existing edges disappear by the growth of their neighbors, or new edges are generated by the splitting of no longer calibrable edges.
In every time interval between these events, the motion is determined by a system of ODEs, and hence the behavior of the evolution on the discontinuities can be described using the general theory of differential equations with discontinuous right-hand side [19].
Concerning the changes of geometry, it is clear what is meant by "disappearing edges", that is edges whose length becomes zero in finite time, but the notion of "appearing edges", that is how a no longer calibrable edge breaks, has to be specified.
We focus our attention to coordinate polyrectangles whose edges have non-negative ϕ-curvature. In the sequel we will use the abuse of notation L = [p, q] when the edge L is of the form L = [p, q] × {y} or L = {y} × [p, q], and n(p), n(q) will denote the prescribed values of the Cahn-Hoffman vector field at the endpoints of L.  For every edge L lying on a discontinuity line of g ε , with (inner) normal ν(L), consider the values If v in L > 0 and v out L < 0, then we set M (L) ∈ {1, M (L+ ε 4 ν(L)), M (L− ε 4 ν(L)). Otherwise, the craking multiplicity M (L) assigned to L is given by Concerning the velocities, assume that p = p b . If either γ ε (p) = β or p ∈ I α,β , we Remark 3.7). The inequality v − > v c then follows from the fact that the function v(σ) is strictly monotone decreasing. The same arguments can be used for v + in the case q b = q.
Recalling Proposition 3.3, we can perform a similar splitting for edges with zero ϕcurvature. If q − p ≥ ε, then, if n(p) = n(q) = 1, we have In both cases the remaining parts L ± are either empty or calibrable with velocities Since v c = α+β 2 , the strict inequalities in (ii) and (iii) hold true.
The case of L with zero ϕ-curvature and q − p < ε is similar and is left to the reader. We would like to stress that the edges of this type appearing as L ± , due to the cracking set-up, are always calibrable.
We associate to E a breaking configuration (ν 1 , . . . , ν m ), (s 1 , . . . , s m ) based on the cracking set-up of its edges in the following way.
By Definition 2.5, a forced crystalline flow E(t), t ∈ [0, T ] given by calibrable coordinate polyrectangles with edges L 1 (t), . . . , L m (t) and with normal direction (ν 1 , . . . , ν m ), is then identified with a solution s(t) = (s 1 (t), . . . , s m (t)) to the system of ODEs and defined outside Σ by Notice that the fictitious edges with zero length, possibly added in the breaking configuration of E, are contained on discontinuity lines of g ε . Then either E is calibrable, or it corresponds to a string s ∈ Σ.
System (12) fits therefore into Filippov's theory of discontinuous dinamical systems (see [19], [13]): the field V is extended on Σ by the multifunction (where we denote by co(A) the convex envelope of a set A) and a solution of (12) is, by definition, a solution of the differential inclusion s ∈ F (V )(s). Since V : R m → R m is measurable and essentially bounded, then there exists at least a solution of such a differential inclusion, starting from any initial datum s. In order to deal with the uniqueness of solutions, we need an explicit computation of the multifunction F (V ) on Σ.
For every s ∈ Σ, and for every component s i of s such that s i ∈ ε 2 Z, and s i−1 = s i+1 , so that i (s) > 0, let V + i (s) and V + i (s) be the values and let I(V − i (s), V + i (s)) be the interval with endpoints V − i (s) and V + i (s).
For every i = 1, . . . , m, V i (s) depends only on s i−1 , s i , and s i+1 , and it is discontinuous only in the s i variable, then F (V )(s) in (13) is the convex set if s i ∈ ε 2 Z, and i = 0, Assume now that every element of the breaking configuration of ∂E with positive length satisfies one of the conditions (1), (2), (3). Concerning the edges with zero length, notice that if L i (t) is an edge starting from (1). Hence we can split the indices {1, . . . , m} = N 1 ∪ N 2 ∪ N 3 in such a way L i (t) satisfies (j) for every i ∈ N j , locally near t = 0.
Let s 0 be the string corresponding to the breaking configuration of ∂E, let E(t) be any evolution obtained by solving the differential inclusion (12) with initial datum s(0) = s 0 , and let t ∈ (0, T ] be such that L i (t) satisfies (j) in (0, t) for every i ∈ N j , j = 1, 2, 3. The family E(t) is then identified with s(t) = (s 1 (t), . . . , s m (t)) solving s i (t) ∈ I i (s(t)) a.e. in (0, t), i = 1, . . . , m.
On the other hand, for every i ∈ N 2 ∪ N 3 , then either s i > 0 or s i < 0 in (0, t), and hence s i (t) = {V i (s(t))} a.e. in (0, t).
In conclusion, since the function V is Lipschitz continuous outside Σ, the solution of the differential inclusion is unique in (0, t), and it is fully determined by the law a.e. in (0, t).
In terms of the breaking configuration of ∂E, we can conclude that every edge L i , with i ∈ N 2 ∪ N 3 crosses immediately the discontinuity line, moving inward (respectively outward) if i ∈ N 2 (respectively i ∈ N 3 ), while every edge L i with i ∈ N 1 is pinned on the discontinuity line.
for every edge L with positive ϕ-curvature.
In particular, the symmetric equilibria O ε (see Figure 1) converge, as ε → 0 to an octagon O having horizontal and vertical edges with length = −4/(α + β), connected by diagonal edges. In conclusion, the forced crystalline evolutions defined in Definition 2.5 and starting from a polyrectangle are obtained by the following procedure: we set-up the initial datum and we obtain the evolution E(t) by solving the system of ODEs (12), for t ∈ [0, T ], where T > 0 is the first time when an edge either disappear or is no more calibrable. The subsequent evolution is obtained by inizializing E(T ) according to Definition 4.1 as an initial datum for the new system of ODEs of the form (12).
We are interested in stressing the macroscopic effect of the underling periodic structure on the geometric evolutions, depicting clearly the forced crystalline flows and passing to the limit as ε → 0, and the most of the features are revealed by the evolution starting from the simplest crystals: the coordinate squares.
In what follows S( ) will denote a coordinate square with side length > 0.
Theorem 4.7 (Effective motion of coordinate squares). Let S( 0 ) be a given coordinate square. For every ε > 0, let S( ε 0 ) be a coordinate square such that d H (S( 0 ), S( ε 0 )) < ε. Then there exists a forced crystalline curvature flow E ε (t), t ∈ [0, T ), starting from S( ε 0 ). Moreover, there exists a family of sets E(t), t ∈ [0, T ), such that E ε (t) converges to E(t) in the Hausdorff topology and locally uniformly in time, as ε → 0. The limit evolution E(t) is independent of the choice of the approximating initial data S( ε 0 ), but its geometry depends on 0 in the following way.
is the solution to (14). In particular, the moving edges of E(t) have length governed by the ODE and then E(t) is increasing in [0, +∞), and converging to a stationary octagon as t → +∞.
By Proposition 3.4, the breaking configuration of ∂S( ε 0 ) has no breaking points, and, by Remark 3.5 is on a discontinuity line of g ε , or v L > 0, otherwise. Then, by Remark 4.4, and Proposition 4.5, there exists a unique forced crystalline flow E ε (t) starting from S( 0 ), and it is given by calibrable squares S( ε (t)) with side length governed by the ODE Since | ε β − ε α | ≤ ε/2, a passage to the limit in (16) as ε → 0 shows that E ε (t) converges in the Hausdorff topology and locally uniformly in time to the family of squares S( (t)) with side length governed by the ODE (14).
As a first step, we assume, in addition, that every vertex of S( ε 0 ) is also a vertex of of a square Q ∈ A ε , Q ⊆ S( ε 0 ) (see Figure 2(I)), so that the edges lie on discontinuity lines of g ε and they have (the same) velocities v in 0,ε , v out 0,ε ≥ 0. Hence, by Definition 4.1 and Proposition 3.4, we have and, by Proposition 4.5, there exists a unique forced crystalline flow starting from S( ε 0 ), given by squares S( ε (t)) with side length governed by the ODE (16), and defined in (0, t 0 ) where t 0 = sup{t > 0 : S( (s)) is calibrable for every s ∈ (0, t)} (see Figure 2(II)). By symmetry, the breaking set-up of every edge of E(t 0 ) = S( (t 0 )) is the same, and it is given by Proposition 4.2(i). More precisely, every edge L ε i (t 0 ) has cracking multiplicity M (L ε i (t 0 )) = 5, and set-up whereσ is the calibrability threshold defined in (10). Moreover, by Remark 3.7, using the notation of Proposition 4.2, we have L ε Then, by Proposition 4.5, the evolution admits a unique extension E ε (t), given by polyrectangles with 20 edges, and defined for t ∈ (t 0 , t 1 ), where t 1 is the first time when an edge of E ε (t) touches a discontinuity line of g ε . By simmetry, the evolution in (t 0 , t 1 ) is fully depicted by its behavior near a vertex of S( ε (t 0 )) (see Figure 2(III)): pinned edges with zero ϕ-curvature are generated in the normal direction of the edges of S( ε (t 0 )) at the breaking points, while the edges parallel to the edges of S( ε (t 0 )) move inward. More precisely, the edges with positive ϕ-curvature move inward with constant velocity while the small edges with zero ϕ-curvature move inward with velocity v ε ± (t) > v ε c , and reach a discontinuity line at time t 1 (see Figure 2(IV)). Every edge S( ε (t 1 )) lying on a discontinuity line of g ε has zero ϕ-curvature, and v in = α, v out = β, and, by Proposition 4.5, there exists a unique extension of the evolution after t 1 , given by polyrectangles with pinned edges with zero ϕ-curvature, and moving edges with positive ϕ-curvature. If we denote by t 2 the first time when those edges reach the discontinuity lines, in such a way that E ε (t 2 ) becomes again a square S( ε 0 − 2ε), then we have , where k is a constant independent of ε, and hence the evolution recomposes the square in a time lapse of order ε (see Figure 2(V)). Since E ε (t 2 ) is a square with every vertex which is also a vertex of of a square Q ∈ A ε , Q ⊆ S( ε 0 ), the (unique) evolution then either iterates this "breaking and recomposing" motion, if ε (t 2 ) = ε 0 − 2ε > 4/(β − α), or it is a family of shrinking squares, if ε (t 2 ) ≤ 4/(β − α). In any case, E ε (t) can be approximate, in the Hausdorff topology and locally uniformly in time, by a family of squares with side length satisfying (16), so that the limit motion as ε → 0 is a family of squares S( (t)) governed by the evolution law (14).
Moreover, for every square S( ε 0 ) such that d H (S( 0 ), S( ε 0 )) < ε, the forced crystalline evolution generates and absorbs the small edges near its corners in slightly different ways, but it is always approximable by a family of squares with side length satisfying (16). In particular, the limit evolution does not depend on the choice of the approximating data.
If every vertex of S( ε 0 ) is also a vertex of of a square Q ∈ A ε , Q ⊆ S( ε 0 ) (see Figure  3(I)), then the edges of the square lie on discontinuity lines of g ε , and they have the same velocities v in 0,ε , v out 0,ε < 0. Hence, by Definition 4.1 and Proposition 3.4, we have and, by Proposition 4.2, every edge of S( ε 0 ) is splitted as Then, by Proposition 4.5, there exists a unique forced crystalline flow starting from S( ε 0 ) (see Figure 3(II)), producing small pinned corners having edges with zero ϕ-curvature and length ε/2, while he long edges with positive ϕ-curvature move outward with constant velocity v c given by (17) until they reach, the next discontinuity line (see Figure 3(III)).
(I) (II) (III) Figure 3. The cutting phenomenon Then the process iterates, "cutting" the square and reducing the length ε (t) of the edges with positive ϕ-curvature, so that their (piecewise constant) velocity is given by  Finally, notice that the forced crystalline evolution starting from a general initial datum S( ε 0 ), with d H (S( 0 ), S( ε 0 )) < ε, reaches a configuration of the type depicted in Figure 3(III) in a time span of order ε. Then the macroscopic limit E(t) does not depend on the choice of the approximating initial datum, and it is the effective motion of the square S( 0 ).
The arguments used in the proof of Theorem 4.7 can be performed to deal with every polyrectangle (and hence, by approximation, to describe the effective evolution of general sets), but a detailed analysis of the forced crystalline flow in these cases requires considerable additional computation. Just to appreciate the application of the previous arguments in a slightly more general setting, we devote the end of this section to a coincise description of the motion starting from coordinate rectangles.
is a family of rectangles E(t) = R( 1 (t), 2 (t)) with i (t) governed by the system of ODEs is the rotated square touching from outside R( 1,0 , 2,0 ) at its vertices, and (˜ 1 ,˜ 2 ) is the solution to (18). In particular, the lengths of the moving edges of E(t) are governed by the system of ODEs is a family of rectangles E(t) = R( 1 (t), 2 (t)) with i (t) governed by the system of ODEs (18). If T = +∞, then E(t) converges to an equilibrium (either a point or the square S(−4/(α + β))) for t → +∞. If T < +∞, E(t), t ∈ [0, T ), is the family of octagons following the rules of case (ii).
Proof. When either v i,0 ≥ 0, or v i,0 < 0, i = 1, 2, the proof is very similar to the one of Theorem 4.7, hence we address our attention to initial data with v 1,0 ≤ 0 and v 2,0 > 0 (mixed case). Assume that every vertex of the approximating initial datum R( ε 1,0 , ε 2,0 ) is also a vertex of a square Q ∈ A ε , Q ⊆ R( ε 1,0 , ε 2,0 ). By symmetry, it is enough depict the evolution of two contiguous edges L ε 1 (horizontal) and L ε 2 (vertical) starting as in Figure  5. We have that v in Therefore the forced evolution starts breaking the edge L ε 1 , and generating small pinned edges with zero ϕ-curvature (see Figure 5(II)). The edges with positive curvature move with constant velocities v ε 1 (outward) and v ε 2 (inward).
If U 0 < 0, so that v ε 2 < −v ε 1 , the small pinned edges with zero ϕ-curvature and "slope 45 degrees" are not absorbed (see Figure 6, left), and the evolution "cuts the vertices". On the other hand, the edges with positive ϕ-curvature move with velocities v ε i (t) = 2 ε i (t) Then, the effective evolution E(t), in the limit ε → 0, is given by the family of octagons whose moving edges have lengths ( 1 (t), ( 2 (t)) solution to (19). Since the level set {U ≤ 0} is invariant under the flow of the ODEs system (19), E(t) are octagons (not monotonically) converging to a stationary octagon as t → +∞ (see Figure 7). If U 0 = 0, then in a time-lapse of order ε the evolution becomes a rectangle with the same features of the initial datum, but with U ε 0 < 0. Then the effective evolution is the one depicted above. If U 0 > 0, then the effective evolution maintains the rectangular shape for a short time (see Figure 8, left). Namely, the forced crystalline flow becomes a rectangle after a time lapse of order ε (see Figure 6), and then the geometric motion E ε (t) is given by "almost rectangles", that is rectangles with small perturbations of order ε near the vertices, until the lengths of the edges with positive ϕ-curvature have velocities v ε 2 (t) > −v ε 1 (t). Hence there exists T > 0 such that E ε (t) it can be approximated, in the Hausdorff topology and locally uniformly in [0, T ], by a family of rectangles R( ε 1 (t), ε 2 (t)) satisfying Passing to the limit as ε → 0, we obtain a family of rectangles E(t) = R( 1 (t), 2 (t)), t ∈ [0, T ] where ( 1 , 2 ) is the solution of the system of ODEs (18) with initial datum in the set Notice that the function J( 1 , 2 ) = 4(log( 2 ) − log( 1 )) + (α + β)( 2 − 1 ) is a constant of motion for system (18). The phase portrait is shown in Figure 8. In particular, A is not a positively invariant set for the system, and the behavior of the trajectories depends on the energy level J( 1,0 , 2,0 ) of the initial datum. The level set {J = 0} is positively invariant in A, so that, if J( 1,0 , 2,0 ) = 0, the effective evolution is given by rectangles converging, as t → +∞, to the equilibrium square S(−4/(α + β)).
If J( 1,0 , 2,0 ) > 0, then the solution enters in the region {U ≤ 0} in finite time, so that the effective evolution becomes a family of octagons, converging to a stationary octagon in infinite time.