CRYSTALLINE EVOLUTIONS IN CHESSBOARD–LIKEMICROSTRUCTURES

ANNALISA MALUSA† AND MATTEO NOVAGA‡

Abstract. We describe the macroscopic behavior of evolutions by crystalline curva-ture 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 confine-ment effects on the geometric motion.

Contents

1. Introduction 1Acknowledgments. 32. Setting of the problem 3Notation 3The crystalline curvature 3Forced crystalline flows 43. Calibrability conditions 54. Forced crystalline flows and their effective motion 8Evolution of coordinate squares 9Evolution of coordinate rectangles 12References 16

1. Introduction

We are concerned with the asymptotic behavior of motions of planar curves accordingto the law

(1) v = κ+ g

(x

ε,y

ε

),

where v is the normal velocity, κ is the crystalline curvature, g : R2 → R is a periodicforcing term, with average g, modeling a chessboard-like structure, and ε > 0 is a smallparameter which takes account of the frequency of oscillation.

Date: October 16, 2017.2010 Mathematics Subject Classification. Primary 53C44, Secondary 35B27.Key words and phrases. Crystalline flow, homogenization, facet-breaking, pinning.† Dipartimento di Matematica “G. Castelnuovo”, Sapienza Università di Roma, Piazzale Aldo Moro

2, 00185 Roma, Italy, email: malusa@mat.uniroma1.it.‡ Dipartimento di Matematica, Università di Pisa, Largo B. Pontecorvo 5, 56217 Pisa, Italy, email:

matteo.novaga@unipi.it.1

2 A. MALUSA AND M. NOVAGA

Crystalline evolutions provide simplified models for describing several phenomena inMaterials Science (see [23, 25, 26] and references therein) and have been significantlystudied in recent years (see for instance [1, 20, 3, 4, 15, 16, 17, 19]).

The forcing term g models a rapidly oscillating heterogeneous medium and, in thehomogenization limit ε → 0, the oscillations of the medium affect the velocity of theevolving front. The geometric motion (1) corresponds to the gradient flow of the energy

Fε(E) =∫∂E

(|νE1 |+ |νE2 |

)dH1 +

∫Eg(xε,y

ε

)dL2, E ⊂ R2,

where we identify the evolving curve with the boundary of a set E. Since the volumeterm converges to gL2(E) as ε→ 0, the Γ-limit of the functionals Fε is given by

F (E) =∫∂E

(|νE1 |+ |νE2 |

)dH1 + gL2(E).

Hence, our analysis can be set in a large class of variational evolution problems dealingwith limits of motions driven by functionals Fε depending on a small parameter (see [7],and the references therein), and we wander whether the limit motion can be directlyrelated to the Γ-limit F (E).

For oscillating functionals, the energy landscape of the energies Fε can be quite differ-ent from that of their Γ-limit and the related motions can be influenced by the presenceof local minima which may give rise to pinning phenomena, or to effective homogenizedvelocities (see [24, 2, 9, 10]). In the case of geometric motions, a general understandingof the effects of microstructure is still missing. Recently, some results have been obtainedfor two-dimensional crystalline energies, for which a simpler description can be given interms of a system of ODEs (see for instance [9, 11, 8, 12, 10]).

Coming back to our specific problem (1), we shall assume for simplicity that g takesonly two values α < 0 < β, its periodicity cell is [0, 1]2, and that g = α+β

2 . We recallthat, in the previous paper [10], we considered a similar homogenization problem wherethe periodic function g depends only on the horizontal variable, so that the medium hasa stratified, opposite to a chessboard-like, structure.

As a consequence of our analysis, it turns out that curves evolving by (1) undergoa microscopic “facet-breaking” phenomenon at a scale ε, with small segments of lengthproportional to ε being created and, in some cases, subsequently reabsorbed. The macro-scopic effect of this behavior is a “pinning effect” for the limit evolution, correspondingto the possible onset of new edges, with slope of 45 degrees and zero velocity (dependingon the initial set and on the values of α, β). On the other hand, the horizontal and thevertical edges always travel with the limit velocity κ+ g.

In particular, due to the possible presence of these new edges, the limit flow doesnot coincide with the gradient flow of the limit functional F , which is simply given byv = κ+ g.

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 evolvingcurve, so that (1) becomes a forced curvature flow. However, as such evolution cannotbe described in terms of a system of ODEs, different techniques would be needed (partialresults in this direction can be found in [14, 13]).

CRYSTALLINE EVOLUTIONS IN CHESSBOARD–LIKE MICROSTRUCTURES 3

The plan of the paper is the following: in Section 2 we introduce the notion of crys-talline curvature and the evolution problem we are interested in. In Section 3 we in-troduce the notion of calibrable edge, that is, an edge which does not break during theevolution, and we recall the calibrability conditions proved in [10]. Finally, in Section 4we characterize the limit evolution as ε→ 0, first for squares and then for rectangles.

Acknowledgments. The authors wish to thank Andrea Braides for useful discussionson the topic of this paper. M.N. was partially supported by the Italian CNR-GNAMPAand by the University of Pisa via grant PRA-2017 “Problemi di ottimizzazione e dievoluzione in ambito variazionale”.

2. Setting of the problem

Notation. The canonical basis of R2 will be denoted by e1 = (1, 0), e2 = (0, 1).The 1–dimensional Hausdorff measure and the 2–dimensional Lebesgue measure in

R2 will be denoted by H1 and L2, respectively.We say that a set E ⊆ R2 is a Lipschitz set if its boundary ∂E can be written, locally,

as the graph of a Lipschitz function (with respect to a suitable orthogonal coordinatesystem). The outward normal to ∂E at ξ, that exists H1–almost everywhere on ∂E, willbe denoted by νE = (νE1 , νE2 ).

The Hausdorff distance between two sets E, F ∈ R2 will be denoted by dH(E,F ).

The crystalline curvature. We briefly recall a notion of curvature κE on ∂E whichis consistent with the requirement that a geometric evolution E(t), reducing as fast aspossible the energy

Pϕ(E) :=∫∂E

(|νE1 |+ |νE2 |

)dH1,

has normal velocity κE(t) H1–almost everywhere on ∂E(t).The surface tension ϕ◦(x, y) = |x| + |y| is the polar function of the convex norm

ϕ(x, y) = max{|x|, |y|}, (x, y) ∈ R2, so that Pϕ(E) turns out to be the perimeter as-sociated to the anisotropy ϕ(x, y), that is, the Minkowski content obtained by consid-ering (R2, ϕ) as a normed space. The sets {φ(ξ) ≤ 1} and {φ◦(ξ) ≤ 1} are the squareK = [−1, 1]2, and the square with corners at (±1, 0) and (0,±1), respectively.

Given a nonempty compact set E ⊆ R2, if we denote by dE the oriented ϕ–distancefunction to ∂E, negative inside E, that is,

dE(ξ) := infη∈E

ϕ(ξ − η)− infη 6∈E

ϕ(ξ − η), ξ ∈ R2.

The normal cone at ξ ∈ ∂E is well defined whenever ξ is a differentiability point for dE ,and it is given by Tφ◦

(∇dE(ξ)

), where

Tϕ◦(ξ◦) := {ξ ∈ R2, ξ · ξ◦ = (ϕ◦(ξ))2}, ξ◦ ∈ R2 .

The notion of intrinsic curvature in (R2, ϕ) is based on the existence of regular selectionsof Tφ◦

(∇dE

)on ∂E.

Definition 2.1 (ϕ–regular set, Cahn–Hoffmann field, ϕ–curvature). We say that a setE ⊆ R2 is ϕ–regular if ∂E is a compact Lipschitz curve, and there exists a vector fieldnϕ ∈ Lip(∂E;R2) such that nϕ ∈ Tϕ◦(∇dE) H1–almost everywhere in ∂E.

4 A. MALUSA AND M. NOVAGA

Any selection of the multivalued function Tϕ◦(∇dE) on ∂E is called a Cahn–Hoffmannvector field for ∂E, and κ = divnϕ is the related ϕ–curvature (or crystalline curvature)of ∂E.

Remark 2.2 (Edges and vertices). A direct computation gives that Tφ◦(ξ◦) is a singletonif ϕ◦(ξ◦) = 1, and ξ◦ is not a coordinate vector. Moreover one gets

Tϕ◦(e1) = [[(1, 1), (1,−1)]],Tϕ◦(e2) = [[(−1, 1), (1, 1)]],Tϕ◦(−e1) = [[(−1, 1), (−1,−1)]],Tϕ◦(−e2) = [[(−1,−1), (1,−1)]].

(Here and in the following [[ξ, η]] is the closed segment joining the vector ξ with η). Theboundary of a ϕ–regular set E is given by a finite number of maximal closed arcs withthe property that Tϕ◦(∇dE) is a fixed set TA in the interior of each arc A. This setTA is either a singleton, if the arc A is not a horizontal or vertical segment, or one ofthe closed convex cones described above. The maximal arcs of ∂E which are straighthorizontal or vertical segments will be called edges, and the endpoints of every arc willbe called vertices of ∂E.

The requirement of Lipschitz continuity keeps the value of every Cahn–Hoffmannvector field fixed at vertices. Hence, in order to exhibit a Cahn–Hoffmann vector fieldnϕ on ∂E it is enough to construct a field nA ∈ Lip(A;R2) on each arc A, with thecorrect values at the vertices, and satisfying the constraint nA ∈ TA. In what follows,with a little abuse of notation, we shall call nA the Cahn–Hoffmann vector field on thearc A.

Forced crystalline flows. Let α < 0 < β, and let g : R2 → R be the function definedin [0, 1]2 by

g(x, y) =

α, in

]0, 1

2

[2⋃]12 , 1

[2,

β, in(]1

2 , 1[×]0, 1

2

[)⋃(]0, 1

2

[×]1

2 , 1[)

,

and extended by periodicity in R2. For ε > 0, let gε(x, y) = g(xε ,yε ).

We will denote by Aε (resp. Bε) the union of all closed squares Q of side length ε suchthat gε = α (resp. gε = β) in the interior of Q. The set of discontinuity points of gε willbe denoted by Ξ. A discontinuity line is a straight line contained in Ξ.

We define the multifunction Gε in R2, by setting Gε = [α, β] on Ξ, and Gε(x, y) ={gε(x, y)} in R2 \ Ξ.

We want to introduce our notion of geometric evolution E(t), obeying to the law(2) V = κ+ gε, on ∂E,

where V is the normal velocity, and κ is the crystalline curvature on ∂E(t).In order to make a sense to (2) 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 aϕ–regular set, the crystalline curvature on ∂E may not be uniquely determined, due tothe infinitely many choices for the Cahn–Hoffmann vector field on the edges of ∂E.

CRYSTALLINE EVOLUTIONS IN CHESSBOARD–LIKE MICROSTRUCTURES 5

This ambiguity can be overcome introducing an additional postulate, which is consis-tent with the notion of forced curve shortening flow (see [3], [4], [5], [21], [22]).

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 forevery edge L of ∂E the restriction nL of n on L is the unique minimum of the functional

NL(n) =∫L|gε − divn|2 dH1

in the set

DL ={n ∈ L∞(L,R2), n ∈ TL, divn ∈ L∞(L), n(p) = n0, n(q) = n1

}where p, q are the endpoints of L and n0, n1 are the values at p, q assigned to everyCahn–Hoffmann vector field (see Remark 2.2).

Remark 2.4. If the minimum nL in DL of the functional NL satisfies the strict constraintnL(ξ) ∈ intTL for every ξ ∈ L, then the velocity gε − divnL is constant along the edge,that is the flat arc remains flat under the evolution. This is always the case for unforcedcrystalline flows, since the unique minimum is the interpolation of the assigned valuesat the vertices of L, and the constant value of the ϕ–curvature is given by

κL = χL2`

on L,

where ` is the length of the edge L and χL is a convexity factor: χL = 1,−1, 0, dependingon whether E(t) is locally convex at L, locally concave at L, or neither.

Definition 2.5 (Forced crystalline evolution). Given T > 0, we say that a family E(t),t ∈ [0, T ], is a forced crystalline curvature flow (or forced crystalline evolution) in [0, T )if

(i) E(t) ⊆ R2 is a Lipschitz set for every t ∈ [0, T );(ii) there exists an open set A ⊆ R2 × [0, T ) such that

⋃t∈[0,T ) ∂E(t)× {t} ⊆ A, and

the function d(ξ, t) .= dE(t)(ξ) is locally Lipschitz in A;(iii) there exists a function n ∈ L∞(A,R2), with divn ∈ L∞(A), such that the

restriction of n(t, ·) to ∂E(t) is a variational Cahn–Hoffmann vector field for∂E(t) for every t ∈ [0, T ];

(iv) ∂td− divn ∈ Gε H1–almost everywhere in ∂E(t) and for all t ∈ [0, T ).

3. Calibrability conditions

In this section we deal with the minimum problem in Definition 2.3 for a given ϕ–regular set E, and we characterize the edges of ∂E having constant velocity vL := κL+gε.

The results concern edges L ∈ ∂E not lying on a discontinuity line of the forcing term,in such a way that gε is defined H1–almost everywhere on L. We will use the notationL = [p, q]× {y} or L = {x} × [p, q], with x, y 6∈ ε

2Z, so that ` = q − p.

6 A. MALUSA AND M. NOVAGA

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:

(3) (BV ) =

n(p) = n(q) = n0 ∈ {±1} if χL = 0;n(p) = −1, n(q) = 1, if χL = 1;n(p) = 1, n(q) = −1, if χL = −1.

Moreover, we denote by γε : R→ R the restriction of gε on the straight line containingL, and we distinguish two different type of discontinuity points for γε:Iβ,α = {s ∈ R : γε = α in (s, s+ ε/2)}, Iα,β = {s ∈ R : γε = β in (s, s+ ε/2)}.

With these notation, the requirement that κ + gε is constant on L can be rephrasedin the following 1D problem.

Definition 3.1 (Calibrability conditions). L is a calibrable edge of ∂E if and only ifthere exists a Lipschitz function n : [p, q]→ R such that the following hold.

(i) n satisfies (3).(ii) |n| ≤ 1 in [p, q].

(iii) n′ + γε = χL2`

+ 1`

∫ q

pγε(s) ds a.e. in [p, q].

In this case, we say that vL = n′ + gε is the (normal) velocity of the edge L.

The calibrability property was studied in [10]. We collect here the results needed inthe rest of the paper.

Denoting by `α, `β ∈ [0, ε/2] the non–negative lengths given by the conditions

(4) `− ε⌊`

ε

⌋= `α + `β,

∫Lγε(s) ds = α+ β

2 (`− `α − `β) + α`α + β`β,

the calibrability condition in Definition 3.1(iii) sets the value of n′ outside the jump setof γε:

n′(x) =

12` (4χL + (β − α)(`− `α + `β)) if γε(x) = α,

12` (4χL − (β − α)(`+ `α − `β)) , if γε(x) = β.

and the feasible velocity of the edge L:

(5) vL = χL2`

+ α+ β

2 + β − α2` (`β − `α).

In conclusion, the calibrability conditions (i) and (iii) in Definition 3.1 fix univocallya candidate field n (and the related velocity of the edge), which is continuous and affinewith given slope in each phase of γε. This field n is the Cahn–Hoffman field whichcalibrates L with velocity (5) if and only if it also satisfies the constraint |n(x)| ≤ 1 forevery x ∈ [p, q].

Remark 3.2. In what follows we will assume 0 < ε <8

β − αin such a way that the small

perturbation χL2`

+ β − α2` (`β − `α) has the same sign of the curvature term χL

2`

.

CRYSTALLINE EVOLUTIONS IN CHESSBOARD–LIKE MICROSTRUCTURES 7

Proposition 3.3 (Edges with zero ϕ–curvature). Let L be an edge with zero ϕ–curvature,let `α, `β be the lengths defined in (4), and let n0 ∈ {±1} be the given value of the Cahn–Hoffmann vector field at the endpoints of L. Then the following hold.

(i) If ` = `α + `β < ε, L is calibrable (with velocity vL = α`α + β`β`α + `β

) if and only if

(ia) n0 = 1, and either γε(p) = β, γε(q) = α, or with an endpoint on Iα,β ;(ib) n0 = −1, and either γε(p) = α, γε(q) = β, or with an endpoint on Iβ,α .

(ii) If ` ≥ ε, L is calibrable (with velocity vL = α+ β

2 ) if and only if(iia) n0 = 1, and (p, y), (q, y) ∈ Iα,β;(iib) n0 = −1, and (p, y), (q, y) ∈ Iβ,α.

Proposition 3.4. Every edge L with positive ϕ–curvature, and such that

(6) `+ `α − `β ≤4

β − α

is calibrable with velocity vL given by (5).

Remark 3.5. Notice that, if L satisfies the condition (6), then

vL ≥2`

+ α+ β

2 + +β − α2`

(`− 4

β − α

)= β > 0

Proposition 3.6. Every edge L with positive ϕ–curvature, and such that p ∈ Iβ,α,q ∈ Iα,β, is calibrable with velocity

vL = 2`

+ α+ β

2 − (β − α)ε4` .

Proposition 3.7. Let L be an edge with positive ϕ–curvature, and such that `+`α−`β >4/(β − α). Then the following hold.

(i) If either γε(p) = β, or γε(q) = β, or p ∈ εIα,β, or q ∈ εIβ,α, then L is notcalibrable.

(ii) If γε(p) = γε(q) = α, let σ1, σ2 ∈ (0, ε/2) be such that p+ ε/2 + σ1 ∈ εIβ,α andq−ε/2−σ2 ∈ εIα,β, and let ˜be the length of the interval [p+ε/2+σ1, q−ε/2−σ2].Setting

m = εβ − α

(β − α)(˜+ ε/2) + 4, h = ε

2(β − α)(˜+ ε/2)− 4(β − α)(˜+ ε/2) + 4

,

and

Σ ={mσ2 + h ≤ σ1 ≤

1mσ2 −

h

m

},

we have m ∈ (0, 1), Σ ∩ [0, ε/2]2 6= ∅, and L is calibrable with velocity

vL = 2`

+ α+ β

2 + β − α2`

(ε

2 − σ1 − σ2

)if and only if (σ1, σ2) ∈ Σ.

8 A. MALUSA AND M. NOVAGA

(iii) if γε(p) = α, and q ∈ εIα,β (resp. p ∈ εIβ,α, and γε(q) = α), let σ ∈ (0, ε/2) besuch that p+ σ + ε/2 ∈ εIβ,α, let `∗ be the length of the interval [p+ ε/2 + σ, q]( resp. of [p, q − ε/2− σ]), and let

σ∗ = ε

2(β − α)(`∗ + ε/2)− 4(β − α)(`∗ − ε/2) + 4 .

Then L is calibrable if and only if σ ≥ σ∗.Remark 3.8 (Calibrability threshold). In the special case when σ1 = σ2 = σ > 0, thecalibrability condition stated in Proposition 3.7(ii) reduces to the unilateral constraintσ ≥ σ, where

σ := ε

2(β − α)(˜+ ε/2)− 4(β − α)(˜− ε/2) + 4

Hence L is calibrable if and only if σ ≥ σ. Moreover, if σ = σ, the edge L is calibrated bya Cahn–Hoffmann vector field n such that n(p) = n(p+ε/2+σ) and n(q) = n(q−ε/2−σ).As a consequence, the same field calibrates both the edges [p, p + ε/2 + σ] × {y} and[q− ε/2− σ, q]×{y} (as edges with zero ϕ–curvature, see Proposition 3.3), and the edge[p+ ε/2 + σ, q− ε/2− σ]×{y} (as edges with positive ϕ–curvature, see Proposition 3.6)with the same velocity.

4. Forced crystalline flows and their effective motion

The results of Section 3 suggest that the forced crystalline curvature flow startingfrom a coordinate polyrectangle (that is a set whose boundary is a closed polygonalcurve with edges parallel to the coordinate axes) remains a coordinate polyrectangle,whose structure changes when either existing edges are squeezing out by the growth oftheir neighbors, or new edges are generated by the splitting of no longer calibrable edges.

Moreover, in every time interval between these events, the motion is determined by asystem of ODEs

vi = χLi

2`i

+ 1`i

∫Li

gε, i = 1, . . . ,m,

with right–hand side continuous outside the set of discontinuity lines of gε. Then somefeatures of the forced crystalline evolution follow from the general theory of differentialequations with discontinuous right–hand sides.

More precisely, given an edge L lying on a discontinuity line of gε, and denoting byν(L) the inner normal vector to L, let us consider the velocity v+

L as in (5) associatedto L+ ε

4ν(L), and the one v−L associated to L− ε4ν(L).

Proposition 4.1. The following hold.(i) If v+ < 0 and v− > 0, then L is pinned.(ii) If v+ > 0 and v− > 0 (or v+ < 0 and v− < 0), then the edge crosses the

discontinuity, and uniqueness is not violated.(iii) If v+ > 0 and v− < 0, then uniqueness fails.

Proof. See [18], Corollary 1 and Corollary 2 in Section 2.10. �

Remark 4.2. As a consequence of Proposition 4.1, if α+β < 0, for every ε > 0 there arenontrivial equilibria of the forced crystalline curvature flow For example, a polyrectangleE such that

CRYSTALLINE EVOLUTIONS IN CHESSBOARD–LIKE MICROSTRUCTURES 9

(a) every vertex of E is also a vertex of of a square Q ∈ Aε, Q ⊆ E,(b) every edge of ∂E with zero ϕ–curvature has length ` = ε/2,(c) every edge of ∂E with positive ϕ–curvature has length ` very closed to−4/(β−α),

is pinned. Namely, requirement (a) implies that every edge of ∂E lies on a discontinuityline, (b) guarantees that v+ = α and v− = β for every edge with zero ϕ–curvature, while(c) guarantees that

v+ = 2`

+ α+ β

2 − (β − α)ε4` < 0, v− = 2

`+ α+ β

2 + (β − α)ε4` > 0.

for every edge with positive ϕ–curvature.In particular, the symmetric equilibria Oε (see Figure 1) converge, as ε → 0 to an

octagon O having horizontal and vertical edges with length ` = −4/(α + β), connectedby diagonal edges.

= α

= β

Figure 1. Microscopic and macroscopic nontrivial equilibrium (α+ β < 0).

We are interested in stressing the macroscopic effect of the underling periodic struc-ture, and the most of the features are revealed by the evolution starting from the simplestcrystals: the coordinate squares.

Evolution of coordinate squares. In what follows S(`) will denote a coordinatesquare with side length ` > 0.

Definition 4.3 (Effective motion). A family of sets E(t), t ∈ [0, T ), is an effective motionstarting from a coordinate square S(`0) if for every ε > 0 and for every coordinate squareS(`ε0) such that dH(S(`0), S(`ε0)) < ε, the forced crystalline curvature flow of S(`ε0) in[0, T ) converges, as ε → 0, to E(t) in the Hausdorff topology and locally uniformly intime.

We will show that for every coordinate square S, there exists a unique forced crys-talline curvature flow starting from S, so that the effective motion of squares is well-defined. Moreover, we will characterize the limit evolution.

Given S(`0) and ε > 0, let S(`ε0) be a square such that dH(S(`0), S(`ε0)) < ε. Based onthe results of Section 3, the forced crystalline curvature flow starting from S(`ε0) dependson the length `0.

10 A. MALUSA AND M. NOVAGA

Case 1: `0 ≤ 4/(β − α) (self-similar shrinking). In this case, by Remark 3.5the square starts shrinking, and, by Proposition 3.4 the forced crystalline evolution ofS(`ε0) is given by calibrable squares S(`ε(t)) with side length governed by the ODE

(7) (`ε)′ = − 4`ε− (α+ β)− β − α

`ε(`εβ − `εα).

Since |`εβ − `εα| ≤ ε/2, we obtain that the effective evolution starting from S(`0) is givenby squares S(`(t)) with side length governed by the ODE

(8) `′ = −4`− (α+ β).

Case 2: either `0 > 4/(β−α) (if α+β ≥ 0), or 4/(β−α) < `0 < −4/(α+β)(if α+ β < 0) (shrinking with temporary breaking).

As a first step, we assume, in addiction, that every vertex of S(`ε0) is also a vertex ofof a square Q ∈ Aε, Q ⊆ S(`ε0) (see Figure 2(I)), so that, by Proposition 3.6, S(`ε0) iscalibrable with velocity of the edges

vε0 = 2`ε0

+ α+ β

2 − (β − α)ε4`ε0

> 0.

Hence the evolution is a square with decreasing side length `ε(t), until the time t0 whenthe edges achieve the calibrability threshold defined in Remark 3.8 (Figure 2(II)).

The edges of the square S(`ε(t0)) cannot be calibrable after the time t0. Nevertheless,by Remark 3.8, the the Cahn–Hoffman vector field calibrating every edge at time t0calibrates also separately its central part (with length `ε0 − 2ε) as an edge with positiveϕ–curvature, and the two ending small edges (with length less then ε) as edges with zeroϕ–curvature. Hence, after t0, the edges break, as in Figure 2(III).

(I) (II) (III) (IV) (V)

Figure 2. The breaking and recomposing phenomenon

By Proposition 4.1(i), the small edges with zero ϕ–curvature lying on a discontinuityline of gε, and appearing after the breaking phenomenon, are pinned. Then, the edgeswith positive ϕ–curvature move inward with constant velocity

vεc = 2`ε0 − 2ε + α+ β

2 + (α− β)ε`ε0 − 2ε ,

while the small edges with zero ϕ–curvature move inward with velocity vεh(t) > vεc , untilthe time t1 when they reach the (stable) position of Figure 2(IV).

Again by Proposition 4.1(i), all the edges with zero ϕ–curvature are pinned until thetime t2 when the evolution becomes a square with side length `ε0 − 2ε (Figure 2(V)).

The (unique) evolution then iterates this “breaking and recomposing” motion in sucha way that it can be approximate, in the Hausdorff topology and locally uniformly in

CRYSTALLINE EVOLUTIONS IN CHESSBOARD–LIKE MICROSTRUCTURES 11

time, by a family of squares with side length satisfying (7), so that the limit motion asε→ 0 is a family of squares S(`(t)) governed by the evolution law (8).

Moreover, the limit evolution does not depend on the choice of the approximating data.Namely, for every square S(`ε0) such that dH(S(`0), S(`ε0)) < ε, the forced crystallineevolution generates and absorbs the small edges near its corners in slightly differentways, but it is always approximable by a family of squares with side length satisfying(7). Hence (8) is the evolution law for the effective motion.

Case 3: α+ β < 0, and `0 ≥ −4/(α+ β) (confinement).If every vertex of S(`ε0) is also a vertex of of a square Q ∈ Aε, Q ⊆ S(`ε0) (see

Figure 3(I)), then S(`ε0) is calibrable with velocity vε0 ≤ 0, so that it cannot shrink.On the other hand, by Proposition 3.7(i), every small enlargement of the initial datumis not calibrable, since gε = β at the vertices. Then the evolution breaks the square(see Figure 3(II)), producing small corners having edges with zero ϕ–curvature, lengthε/2, and pinned (by Proposition 4.1). The long edges with positive ϕ–curvature moveoutward until they reaches 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 theedges with positive ϕ–curvature, so that their (piecewise constant) velocity is given by

vε(t) = 2`ε(t) + α+ β

2 − (β − α)ε4`ε(t) .

Passing to the limit as ε → 0, we obtain an evolution starting from S(`0) and given bythe family E(t) = S(`0)◦ ∩ S(`(t)) of octagons with S(`0)◦ polar square of S(`0) (seeFigure 4), and

`′ = 4`

+ (α+ β)

and increasing to a stable octagon (see Remark 4.2).Notice that the forced crystalline evolution starting from a general initial datum S(`ε0),

with dH(S(`0), S(`ε0)) < ε, reaches a configuration of the type depicted in Figure 3(III)in a time span of order ε. Then the previous macroscopic evolution does not dependon the choice of the approximating initial datum, and it is the effective motion of thesquare S(`0).

The arguments used for the description of the effective motion of squares, essentiallybased on the results of Section 3, and on the general properties of solutions of ODEscollected in Proposition 4.1, can be performed to deal with coordinate polyrectangles(and hence, by approximation, to describe the effective evolution of general sets), but

12 A. MALUSA AND M. NOVAGA

Figure 4. The effective evolution in Case 3 of confinement.

a detailed analysis of the forced crystalline flow in these cases requires considerableadditional computation. Just to appreciate the application of the previous arguments ina slightly more general setting, we devote the end of this section to a coincise descriptionof the motion starting from coordinate rectangles.

Evolution of coordinate rectangles. In what follows R(`1, `2) will denote a coordi-nate rectangle with side lengths `1, `2 > 0. A natural extension of Definition 4.3 is thefollowing.

Definition 4.4 (Effective motion). A family of sets E(t), t ∈ [0, T ), is an effectivemotion starting from a coordinate rectangle R(`1,0, `2,0) if for every ε > 0 and for everycoordinate rectangle R(`ε1,0, `ε2,0) such that dH(R(`1,0, `2,0), R(`ε1,0, `ε2,0)) < ε, the forcedcrystalline curvature flow of R(`ε1,0, `ε2,0) in [0, T ) converges, as ε → 0, to E(t) in theHausdorff topology and locally uniformly in time.

As in the case of coordinate squares, it will become clear that the limit as ε → 0of forced crystalline flows starting from a coordinate rectangle R1 coincides with thelimit of the flow starting from any other coordinate rectangle R with dH(R1, R) < ε.Hence, given R(`1,0, `2,0), we can restrict our attention to the approximating initialdatum R(`ε1,0, `ε2,0) whose vertices are also a vertices of a square Q ∈ Aε, Q ⊆ R(`ε1,0, `ε2,0),so that, by Proposition 3.6, R(`ε1,0, `ε2,0) is calibrable with velocities of the edges

vi,0 = 2`εi,0

+ α+ β

2 − (β − α)ε4`εi,0

, i = 1, 2.

Having already characterized the evolution of a square, without loss of generality we canassume that `1,0 > `2,0.

Case 1: vi,0 ≥ 0, i = 1, 2 (shrinking). We will show that in this case the forcedflow is given by “almost rectangles”, that is rectangles with small perturbations of orderε near the vertices (see Figure 5). Namely, when an edge L with length ` > 4

β−α reachesthe threshold configuration for calibrability, then a new small edge parallel to L andwith starting velocity

v =σα+ ε

2β

σ + ε2

CRYSTALLINE EVOLUTIONS IN CHESSBOARD–LIKE MICROSTRUCTURES 13

is generated (see Figure 5 (I) and (II)). Hence this new small edge moves inward fasterthan the long parallel edge with positive ϕ–curvature (and with constant velocity, since,by Proposition 4.1(i), the orthogonal small edge on the interface is pinned). When iteventually reaches the first discontinuity line of gε, it remains pinned, due to Proposition4.1(i), and the fact that

v− =σα+ ε

2β

σ + ε2≥σα+ ε

2β

σ + ε2

= (β − α)(

(α+ β)˜− (β − α)ε2 + 4)> 0,

v+ =σβ + ε

2α

σ + ε2≤σβ + ε

2α

σ + ε2

< 0.

(σ is the calibrability threshold defined in Remark 3.8). Then the pinned small edge isabsorbed by the contiguous parallel edge with positive ϕ–curvature.

(I) (II) (III) (IV)

Figure 5. Case 1: asymmetric breaking and recomposing.

In conclusion, the forced crystalline flow can be approximated, in Hausdorff topologyand locally uniformly in time, by a family of rectangles R(`ε1(t), `ε2(t)) satisfying

(`ε1)′ = −2( 2`ε2

+ α+ β

2 − (β − α)ε4`ε2

),

(`ε2)′ = −2( 2`ε1

+ α+ β

2 − (β − α)ε4`ε1

),

and then the effective evolution is a family of rectangles R(`1(t), `2(t)) shrinking withlaw

(9)

`′1 = − 4

`2− (α+ β),

`′2 = − 4`1− (α+ β).

Case 2: vi,0 < 0, i = 1, 2 (enlarging confined octagons). Just as in the caseof squares, the forced evolution “cuts” the vertices of the initial datum (Figure 3) in atime lapse of order ε, and, after that, the small edges with zero ϕ–curvature are pinned,and the edges with positive curvature move with velocities

vεi (t) = 2`εi (t)

+ α+ β

2 − (β − α)ε4`εi (t)

, i = 1, 2.

If we denote by Q(R(`1,0, `2,0)) the rotated square touching from outside R(`1,0, `2,0)at its vertices (or, equivalently, the polar set of the square having same center andsame perimeter of R(`1,0, `2,0)), a passage to the limit as ε→ 0 shows that the effective

14 A. MALUSA AND M. NOVAGA

Figure 6. The effective evolution, case 2.

evolution is the family E(t) = Q(R(`1,0, `2,0)) ∩ R(`1(t), `2(t)) of octagons ruled by thelaw

(10)

`′1 = 4

`1+ (α+ β),

`′2 = 4`2

+ (α+ β),

and increasing to a stable octagon (see Figure 6) .Case 3: v1,0 < 0 and v2,0 ≥ 0 (mixed case). In this case, the forced evolution

starts breaking the edge L1, and generating small pinned edges with zero ϕ–curvature(see Figure 7). The edges with positive curvature move with constant velocities vε1(outward) and vε2 (inward). The subsequent evolution depends on the sign of vε1 + vε2,and hence on the sign of the quantity

U0 := 1`1,0

+ 1`2,0

+ α+ β

2 .

(I) (II)

vε2

vε1

Figure 7. How the mixed case starts.

If U0 < 0, so that v2,0 < −v1,0, the evolution “cuts the vertices” and new small pinnededges with zero ϕ–curvature with zero velocity and “slope 45 degrees” appear (see Figure8, left). The edges with positive ϕ–curvature move with velocities

vεi (t) = 2`εi (t)

+ α+ β

2 − (β − α)ε4`εi (t)

, i = 1, 2.

Then, as in Case 2, the effective evolution, in the limit ε → 0, is given by the familyof octagons E(t) = Q(R(`1,0, `2,0)) ∩ R(`1(t), `2(t)), where Q(R(`1,0, `2,0)) the rotated

CRYSTALLINE EVOLUTIONS IN CHESSBOARD–LIKE MICROSTRUCTURES 15

(III)U0 ≥ 0

(III)U0 < 0

Figure 8. How the mixed case carries on.

square touching from outside R(`1,0, `2,0) at its vertices, and `i satisfy (10). Notice that{1/`1 + 1/`2 ≤ −(α + β)/2} is an invariant set for the ODEs system (10), and hencethe effective evolution is given by octagons (not monotonically) converging to a stableoctagon (see Figure 9).

If U0 = 0, then in a time–lapse of order ε the evolution becomes a rectangle with thesame features of the initial datum, but with U ε0 < 0. Then the effective evolution is theone depicted above.

Figure 9. Effective evolutions, case 3 and U0 ≤ 0

If U0 > 0, then the effective evolution maintains the rectangular shape for a shorttime (see Figure 10, left). More precisely, as in Case 1, the forced crystalline flow canbe approximated, in the Hausdorff topology and locally uniformly in time, by a familyof rectangles R(`ε1(t), `ε2(t)) satisfying

(`ε1)′ = −2( 2`ε2

+ α+ β

2 − (β − α)ε4`ε2

),

(`ε2)′ = −2( 2`ε1

+ α+ β

2 − (β − α)ε4`ε1

),

and then the effective evolution starts as a family of rectangles R(`1(t), `2(t)), with(`1, `2) satisfying the system of ODEs (9) with initial datum in the set

A := {U(`1, `2) > 0} ∩{`2 ≤

−4α+ β

≤ `1}, U(`1, `2) := 1

`1+ 1`2

+ α+ β

2 .

16 A. MALUSA AND M. NOVAGA

Notice that the functionJ(`1, `2) = 4(log(`2)− log(`1) + (α+ β)(`2 − `1)

is a constant of motion for system (9). The phase portrait is shown in Figure 10. Inparticular, A is not a positively invariant set for the system, and the behavior of thetrajectories depends on the energy level J(`1,0, `2,0) of the initial datum.

Figure 10. Left: short-time effective evolution, case 3 and U0 > 0.Right: phase portrait of (9), with the region A.

The level set {J = 0} is positively invariant in A, so that, if J(`1,0, `2,0) = 0, theeffective evolution is given by rectangles converging, as t → +∞, to the equilibriumsquare S(−4/(α+ β)).

If J(`1,0, `2,0) < 0, then there exists a unique t0 > 0 such that `′1(t0) = −4/(α + β),and the effective evolution for t > t0 is the one shown in Case 1: rectangles shrinking toa point in finite time.

If J(`1,0, `2,0) > 0, then the solution enters in the region {U < 0} in finite time, sothat the effective evolution becomes a family of octagons, converging to a stable octagonin infinite time.

References[1] F. Almgren and J.E. Taylor. Flat flow is motion by crystalline curvature for curves with crystalline

energies. J. Differential Geometry 42 (1995), 1–22.[2] G. Barles, A. Cesaroni, M. Novaga. Homogenization of fronts in highly heterogeneous media. SIAM

J. Math. Anal. 43 (2011), 212–227.[3] G. Bellettini, M. Novaga, M. Paolini. Characterization of facet breaking for nonsmooth mean curva-

ture flow in the convex case. Interfaces Free Bound. 3 (2001), 415–446.[4] G. Bellettini, M. Novaga, M. Paolini. On a crystalline variational problem, part I: first variation and

global L∞ regularity. Arch. Rational Mech. Anal. 157 (2001), 165–191.[5] G. Bellettini, M. Novaga, M. Paolini. On a crystalline variational problem, part II: BV regularity

and structure of minimizers on facets. Arch. Rational Mech. Anal. 157 (2001), 193–217.[6] A. Braides. Γ-convergence for Beginners. Oxford University Press, 2002.[7] A. Braides, Local Minimization, Variational Evolution and Γ–convergence. Lecture Notes in Mathe-

matics, Springer, Berlin, 2014.[8] A. Braides, M. Cicalese, N. K. Yip. Crystalline Motion of Interfaces Between Patterns. J. Stat. Phys.

165 (2016), 274–319.

CRYSTALLINE EVOLUTIONS IN CHESSBOARD–LIKE MICROSTRUCTURES 17

[9] A. Braides, M.S. Gelli, M. Novaga. Motion and pinning of discrete interfaces. Arch. Ration. Mech.Anal. 95 (2010), 469–498.

[10] A. Braides, A. Malusa, M. Novaga. Crystalline evolutions with rapidly oscillating forcing terms.Preprint arXiv:1707.03342 [math.AP], 2017.

[11] A. Braides, G. Scilla. Motion of discrete interfaces in periodic media. Interfaces Free Bound. 15(2013), 451–476.

[12] A. Braides, M. Solci. Motion of discrete interfaces through mushy layers. J. Nonlinear Sci. 26 (2016),1031–1053.

[13] A. Cesaroni, N. Dirr, M. Novaga. Homogenization of a semilinear heat equation. J. Ec. polytech.Math. 4 (2017), 633–660.

[14] A. Cesaroni, M. Novaga, E. Valdinoci. Curve shortening flow in heterogeneous media. Interfacesand Free Bound. 13 (2011), 485–505.

[15] A. Chambolle, M. Morini, M. Ponsiglione. Existence and uniqueness for a crystalline mean curvatureflow. Comm. Pure Appl. Math., to appear.

[16] A. Chambolle, M. Morini, M. Novaga, M. Ponsiglione. Existence and uniqueness for anisotropic andcrystalline mean curvature flows. Preprint arXiv:1702.03094 [math.AP], 2017.

[17] A. Chambolle, M. Novaga. Approximation of the anisotropic mean curvature flow. Math. ModelsMethods Appl. Sci. 17 (2007), 833–844.

[18] A. F. Filippov. Differential Equations with Discontinuous Righthand Sides, vol. 18 of Mathematicsand Its Applications. Dordrecht, The Netherlands, Kluwer Academic Publishers, 1988.

[19] Y. Giga. Surface evolution equations. A level set approach, vol. 99 of Monographs in Mathematics.Birkhauser Verlag, Basel, 2006.

[20] Y. Giga, M.E. Gurtin. A comparison theorem for crystalline evolution in the plane. Quarterly ofApplied Mathematics 54 (1996), 727–737.

[21] Y. Giga, P. Rybka. Facet bending in the driven crystalline curvature flow in the plane. J. Geom.Anal. 18 (2008),109–147.

[22] Y. Giga, P. Rybka. Facet bending driven by the planar crystalline curvature with a generic nonuni-form forcing term. J. Differential Equations 246 (2009), 2264–2303.

[23] M.E. Gurtin. Thermomechanics of evolving phase boundaries in the plane. Oxford MathematicalMonographs. The Clarendon Press, Oxford University Press, New York, 1993.

[24] M. Novaga, E. Valdinoci. Closed curves of prescribed curvature and a pinning effect. Netw. Heterog.Media 6 (2011), no. 1, 77–88.

[25] J.E. Taylor. Crystalline variational problems. Bull. Amer. Math. Soc. 84 (1978), 568–588.[26] J.E. Taylor, J. Cahn, C. Handwerker. Geometric Models of Crystal Growth. Acta Metall. Mater.

40 (1992), 1443-1474.