grant/cv/grf.pdf · DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS Volume 1,Number1, Xxxx 2000 pp. 1{33 GRAIN SIZES IN THE DISCRETE ALLEN-CAHN AND CAHN-HILLIARD EQUATIONS CHRISTOPHER P

DISCRETE AND CONTINUOUSDYNAMICAL SYSTEMSVolume 1, Number 1, Xxxx 2000 pp. 1–33

GRAIN SIZES IN THE DISCRETE ALLEN-CAHN ANDCAHN-HILLIARD EQUATIONS

CHRISTOPHER P. GRANT

Department of MathematicsBrigham Young University

Provo, UT 84602

Abstract. The discrete Allen-Cahn and Cahn-Hilliard equations are (contin-uous time) lattice differential equations that are analogues of two well-studiedparabolic partial differential equations of materials science, and that modelthe evolution of, respectively, a conserved, or nonconserved, quantity, whichhas two preferred homogeneous phases and which avoids spatial inhomoge-nieties. If the interaction length ν is small, equilibrium states are scatteredthroughout phase space, but as ν increases most of these equilibria disappear.As they do so, the system passes through a transitional stage known as dy-namical metastability, during which true equilibrium solutions are difficult todistinguish from solutions that evolve slowly but eventually traverse long dis-tances. This paper contains results on the grain sizes of genuine equilibria thathelp make this distinction possible. In particular, lower bounds on grain sizesand connections between the grain sizes of adjacent grains and next-nearestneighbors are established.

1. Introduction. Let Ω be a finite one-dimensional lattice identified with the setof integers 0, 1, . . . , n, and consider the (continuous) dynamical systems, havingRΩ as their phase space, generated by the lattice differential equations

u = ν2∆u− f(u) (1.1)

andu = −n2∆(ν2∆u− f(u)). (1.2)

Here, ν > 0 is a constant called the interaction length, ∆ is the diffusion operatorrepresented by the (n+ 1)× (n+ 1) matrix

−1 1 0 · · · 0

1 −2. . . . . .

...

0. . . . . . . . . 0

.... . . . . . −2 1

0 · · · 0 1 −1

,

and f : R → R has zeros at −1, 0, and 1 and is positive on (−1, 0) ∪ (1,∞) andnegative on (−∞,−1)∪(0, 1). (For vector inputs, f is to be applied componentwise.)

1991 Mathematics Subject Classification. 58F21, 34C35.Key words and phrases. lattice differential equations, Allen-Cahn equation, Cahn-Hilliard

equation.The author was partially supported by National Science Foundation grant DMS-9501060.

1

2 CHRISTOPHER P. GRANT

Throughout this paper, if A and B are sets, then AB will represent the set offunctions from B to A, and 2B will represent the collection of subsets of B.

We shall refer to (1.1) and (1.2) as, respectively, the discrete Allen-Cahn equationand the discrete Cahn-Hilliard equation because of their obvious similarities to twopartial differential equations of materials science: the Allen-Cahn equation [3]

ut = ε2∆u− f(u) (1.3)

and the Cahn-Hilliard equation [11]

ut = −∆(ε2∆u− f(u)). (1.4)

(Here ∆ is the standard Laplacian, and ε is a positive constant, usually assumedto be small.) Solutions u(x, t) of (1.3) and (1.4) are often studied for x in somebounded domain D with the boundary condition ∂u/∂η = 0 (where η is the unitoutward normal) and, for (1.4), the additional boundary condition ∂∆u/∂η = 0.While (1.1) and (1.2) could be viewed as crude numerical models of these partialdifferential equations (with ν = nε), they can themselves be viewed as fundamentalphysical models reflecting the underlying particulate nature of matter.

Let W be an antiderivative of f , and apply W componentwise to vectors, justlike f . Let σ : RΩ → RΩ be the shift map defined by

σ(u)i =

ui−1 if i > 0,u0 if i = 0.

The energy function E : RΩ → R defined by

E(u) = W (u) +ν2

2|u− σ(u)|2

serves as a Lyapunov function for both (1.1) and (1.2), in the sense that

dE(u(t))/dt ≤ 0

along solution paths, with equality if and only if u(t) is an equilibrium solution.If, for example, f is bounded away from zero as |u| → ∞, then E(u) becomesunbounded as |u| → ∞ and, together with the fact that E is bounded below, thisimplies that the ω-limit set of every orbit is nonempty and is made up entirely ofequilibria. (See, e.g., [33].) Furthermore, the attractor for (1.1) equals⋃

u∈Ea(ν)

W+(u),

where W+(u) represents the unstable manifold of u and Ea(ν) is the set of equi-librium solutions of (1.1), and a similar result holds for (1.2). (See, e.g., Theorem3.8.5 in [28], as well as the discussion in [10]. For (1.2), it is useful to work on therestricted dynamical system defined on the invariant set

u ∈ RΩ :n∑i=0

ui = m

,

for some constant m, so that the set of equilibria is compact.)Thus, the equilibria are important in describing the long-time dynamics of ar-

bitrary solutions. They are, however, sometimes difficult to identify. There is anextensive literature on the existence, in one space dimension, of extremely-slowlyevolving solutions to (1.3) and (1.4) that are difficult to distinguish from genuineequilibria. (For (1.3), see [9, 12, 13, 14, 15, 23, 24]. For (1.4), see [1, 2, 4, 5, 8, 19,21, 26].) Similar difficulties can arise for the lattice equations. If f(u) = u3 − u

GRAIN SIZES IN DISCRETE EQUATIONS 3

and ν is small, it has been shown that (1.1) has 3n+1 equilibria spread throughoutphase space [20, 29, 31]. Since the set Eh(ν) of equilibria for (1.2) is a superset ofthe set Ea(ν) of equilibria for (1.1), they are also broadly distributed for ν small.In fact, it can be shown that under these conditions Eh(ν) consists of a continuumof states and that

#u ∈ Eh(ν) : ν2∆u− f(u) = β

=

3n+1 if |β| < 2/(3

√3),

2 if |β| = 2/(3√

3),1 if |β| > 2/(3

√3),

if ν is small and positive. As ν →∞ for fixed n, however, most of these equilibriadisappear and slowly-evolving solutions develop as in the continuous equations.Using variational techniques, a bound on the speed of evolution of solutions of(1.1) and (1.2) was obtained in [27], but since that bound was one-sided furtherinformation about Ea(ν) and Eh(ν) is needed in order to differentiate true equilibriafrom pseudo-equilibrium solutions.

In this paper, we examine the grain sizes of the true equilibrium solutions of(1.1) and (1.2). For u ∈ RΩ, we define a grain of u to be a maximal subset G of Ωsatisfying the conditions

1. if i, j ∈ G then uiuj > 0, and2. if i, j ∈ G and i < k < j then k ∈ G;

the grain size will be the cardinality of G. A positive grain will be a grain G suchthat ui > 0 if i ∈ G, and a negative grain is a grain G such that ui < 0 if i ∈ G.We say that G is an interior grain if G ∩ 0, n = ∅; otherwise, we say that G isa boundary grain. Two lattice sites i, j ∈ Ω are adjacent if |i − j| = 1. A grain Gand a lattice site i are adjacent if i 6∈ G but j ∈ G for some lattice site j that isadjacent to i. Two grains G1 and G2 are adjacent if there is a lattice site i ∈ G1

that is adjacent to G2.For fixed u, Ω will be the disjoint union of the collection of grains and the set

of interface points i ∈ Ω : ui = 0. For (1.1) and (1.3), u is typically viewed as anonconserved order parameter, such as that representing the local state of an ice-water system. For (1.2) and (1.4), u is a conserved quantity, normally consideredto be the local concentration of one component of a binary alloy. In either case,locations where u is positive are considered to be in one phase, and locations whereu is negative are considered to be in another. Hence, a grain is considered to be acollection of contiguous lattice sites in the same phase.

While some results on grain size may be obtained for quite general f , moredetailed results about the way that grain size depends on ν are possible whenspecific f are analyzed. Two piecewise linear nonlinearities that have receivedsome recent attention are

f1(ξ) =

(−∞, 1] if ξ = −1,−ξ if −1 < ξ < 1,[−1,∞) if ξ = 1,

and

f2(ξ) =

2(ξ + 1) if ξ < 0,[−2, 2] if ξ = 0,2(ξ − 1) if ξ > 0.

(See Figure 1.)


1−1

1

2

−1

−2f1

1−1

1

2

−1

−2f2

Figure 1. Graphs of f1 and f2.

The nonlinearity f = f1 was studied in [6, 7, 16]. For (1.4) it is known as thedeep quench limit, and (1.3) with this nonlinearity is a double obstacle problem. In[18], equilibrium solutions of (1.1) that take on values confined to the set −1, 0, 1are studied and enumerated for this nonlinearity. Although in some sense simplerthan the cubic nonlinearity, f1 has the advantage of being more physically realisticin the sense that the order parameter (in the Allen-Cahn case) or the single speciesconcentration (in the Cahn-Hilliard case) is forced to stay within reasonable bounds.The choice of f = f2 as a nonlinearity for equations of Allen-Cahn or Cahn-Hilliardtype was suggested by Chow and Mallet-Paret in [17], and it was used in a relatedequation in [32]. The idea behind this nonlinearity is that the stable zeros of f at ±1are more important than the unstable zero at 0 in determining long-time dynamics.In some sense, f2 is the simplest nonlinearity for which the local dynamics near thesestable zeros is asymptotically linear. Because both f1 and f2 are set-valued for someinputs, the differential equations (1.1), (1.2), (1.3), and (1.4) must be interpretedas differential inclusions for them. As will be elaborated on later, examining thesetwo nonlinearities allows us to separate out penalties for small grain size and foraperiodicity.

Our study of the dependence of grain size in equilibria will proceed as follows. In§2, we examine the size of Eh(ν) and of Ea(ν) for small ν for nonsmooth nonlineari-ties. In §3 we provide lower estimates on grain size for (1.1) and on combined sizesof adjacent grains for (1.2) that is valid for a general class of f ’s and for f = f1,in particular. In §4 we focus on Ea(ν) for f = f2. In this context, we describe howthe disappearance of equilibria as ν increases is related to short-range connectionsbetween the sizes of grains.

2. Equilibria when ν 1. While facts about the size of Ea(ν) and Eh(ν) forsmall ν can be obtained without a great deal of difficulty if f is smooth and hasnondegenerate zeros, things are somewhat more involved if f is not smooth, oris even set-valued. Here we present an analysis that is, in particular, sufficientlygeneral to handle f1 and f2. In preparation for this analysis, we present severaldefinitions.

Let A and B be topological spaces, let f be a function from A to 2B, and letaRfb (a ∈ A, b ∈ B) be the relation defined by

(aRf b)⇐⇒ (b ∈ f(a)).


We shall say that f is closed if the graph of Rf is a closed subset of A×B. In otherwords, the closedness of f is equivalent to the statement that whenever there areconvergent sequences ai ⊂ A and bi ⊂ B satisfying bi ∈ f(ai) for every i,

limi→∞

bi ∈ f( limi→∞

ai).

Note that every continuous function from A to B, when interpreted as a set-valuedfunction in the most natural way, is closed.

Let x0 ∈ R be a zero of f : R → 2R (in the sense that 0 ∈ f(x0)). We shallsay that f is two-signed near x0 if for every neighborhood V of x0 there exists aconnected subset U of the graph of Rf such that U ⊂ V×R, U ∩(V×(−∞, 0)) 6= ∅,and U ∩ (V × (0,∞)) 6= ∅.

Given δ > 0 and a point (x, y) ∈ R×R, let

Kδ(x, y) = (x, y) ∈ R×R : |y − y| ≥ δ|x− x|We shall say that f satisfies a cone condition near x0 if there exists a neighborhood(c, d) of x0 and a number δ > 0 such that for any x ∈ (c, d) and any y ∈ f(x)

(x, y) : x ∈ (c, d) and y ∈ f(x) ⊂ Kδ(x, y).

If x0 is a zero of f , f is two-signed near x0, and f satisfies a cone condition nearx0, we shall say that x0 is a nondegenerate zero of f .

Finally, if f satisfies f(x)∩(−∞, 0] = ∅ and f(−x)∩[0,+∞) = ∅ for all (positive)x sufficiently large, then we shall say that f is weakly dissipative.

Theorem 1. Suppose f : R → 2R is closed, has exactly k zeros, all of which arenondegenerate, and is weakly dissipative. Then for ν sufficiently small, #Ea(ν) =kn+1.

Proof. Suppose f satisfies the hypotheses of the theorem. Pick M > 0 such thatf(x) ∩ (−∞, 0] = ∅ if x > M and f(x) ∩ [0,+∞) = ∅ if x < −M . Note that ifu ∈ Ea(ν) then ui ∈ [−M,M ] for every i ∈ Ω. If, for example, ui0 := maxi∈Ω uiwere bigger than M , then the fact that ui0 ≥ ui0±1 if i0 ± 1 ∈ Ω would implythat ν2(ui0−1 − 2ui0 + ui0+1) ≤ 0 if i0 /∈ 0, n, ν2(u1 − u0) ≤ 0 if i0 = 0, andν2(un−1 − un) ≤ 0 if i0 = n; because of the condition satisfied by M , every one ofthese possibilities contradicts ν2∆u ∈ f(u).

Let z1, . . . , zk be the zeros of f (all nondegenerate). Pick δ > 0 such that

(x, y) : x ∈ [zi − δ, zi + δ] and y ∈ f(x) ⊂ Kδ(x, y) (2.1)

for every i from 1 to k, every x ∈ [zi − δ, zi + δ], and every y ∈ f(x), and such that

[zi − δ, zi + δ] ∩ [zj − δ, zj + δ] = ∅ (2.2)

if i 6= j. Let

N =k⋃i=1

(zi − δ, zi + δ),

and let S = NΩ.We claim that if ν is sufficiently small then

Ea(ν) ⊂ S. (2.3)

Suppose (2.3) were false. Because of the boundedness of Ea(ν), this would thenimply the existence of a sequence νj → 0 and a corresponding convergent sequenceof points (u(j)) ∈ Ea(νj) \ S. Let u(0) = limj→∞ u(j). Fix i ∈ Ω, and let xj = u

(j)i

for each j ∈ N. Let yj = ν2j (u(j)

1 − u(j)0 ) if i = 0, and let yj = ν2

j (u(j)n−1 − u

(j)n ) if


i = n; otherwise, let yj = ν2j (u(j)

i−1 − 2u(j)i + u

(j)i+1). Note that since u(j) ∈ Ea(νj),

yj ∈ f(xj) for each j ∈ N. We know that xj → u(0)i and, clearly, yj → 0, so by the

closedness of f , we find that 0 ∈ f(u(0)i ), so u(0)

i is a zero of f . Since this holds foreach i ∈ Ω, we find that u(0) ∈ S. But since S is open, and u(j) is outside S foreach j ∈ N, we have a contradiction. Thus, the claim holds.

Let I be the set of multi-indices 1, 2, . . . , kΩ, and note that #I = kn+1. Foreach p ∈ I, let

Sp =∏i∈Ω

(zp(i) − δ, zp(i) + δ),

and note thatS =

⋃p∈ISp. (2.4)

We claim that for ν sufficiently small, #(Sp ∩Ea(ν)) = 1. In conjunction with (2.3)and (2.4), this claim will prove the theorem.

To verify the claim, fix p ∈ I. Define a map T : Sp → Sp in the following way.Given u ∈ Sp, we will say that another element v of Sp is T (u) if

ν2(u1 − v0) ∈ f(v0), (2.5)

ν2(un−1 − vn) ∈ f(vn), (2.6)and, for i /∈ 0, n,

ν2(ui−1 − 2vi + ui+1) ∈ f(vi). (2.7)We will show that if ν is sufficiently small then these conditions make T a well-defined (single-valued) map from Sp to Sp, it is a contraction map on Sp (for asuitable metric), and the set of its fixed points corresponds exactly to Sp ∩ Ea(ν).The Contraction Mapping Theorem (see, e.g., [30]) then gives the desired cardinal-ity result.

First, we show that if ν2 < δ/2 then it is not possible for two different choices ofv ∈ Sp to satisfy (2.5), (2.6), and (2.7). Suppose v = v(1) and v = v(2) both satisfythese conditions. Fix i ∈ Ω and let

a =

ν2u1 if i = 0,ν2(ui−1 + ui+1) if 0 < i < n,

ν2un−1 if i = n,

(2.8)

and let

b =

−ν2 if i = 0,−2ν2 if 0 < i < n,

−ν2 if i = n.

(2.9)

Note that a + bv(`)i ∈ f(v(`)

i ) for ` = 1, 2. Therefore, by (2.1), (v(1)i , a + bv

(1)i ) ∈

Kδ(v(2)i , a+ bv

(2)i ). But this means that |b||v(1)

i − v(2)i | ≥ δ|v(1)

i − v(2)i | which must

mean that v(1)i = v

(2)i if ν2 < δ/2. Since this is true for every i, v(1) = v(2).

Next, we show that if ν2 is sufficiently small and u ∈ Sp then there is some v ∈ Spfor which (2.5), (2.6), and (2.7) hold. Because f is two-signed near its zeros, foreach i ∈ Ω we can pick a connected subset Ui of the graph of Rf that is containedin [ui − δ, ui + δ]×R and that contains points (x±i , y

±i ) with y−i < 0 and y+

i > 0.Now, suppose that

ν2 <minminy+

i ,−y−i : 1 ≤ i ≤ k2 diam(N )

. (2.10)


Fix i ∈ Ω and let a and b be as in (2.8) and (2.9). Note that because of (2.10)a+bx+

i ≤ 2ν2 diam(N ) < y+i ; similarly, a+bx−i > y−i . The connectedness of Ui then

implies the existence of (vi, yi) ∈ Ui such that a+bvi = yi. Thus, a+bvi ∈ f(vi) andvi ∈ [ui − δ, ui + δ]. Letting i vary over all sites in Ω, we find v = (v1, . . . , vn) ∈ Spsuch that (2.5), (2.6), and (2.7) hold.

Having shown that T : Sp → Sp is well-defined, we next show that it is acontraction on Sp under a suitable metric. For u(1), u(2) ∈ Sp, define

d(u(1), u(2)) = max|u(1)i − u

(2)i | : i ∈ Ω.

Let v(1) = Tu(1) and v(2) = Tu(2). Fix i ∈ Ω \ 0, n. Then

ν2(u(1)i+1 − 2v(1)

i + u(1)i−1) ∈ f(v(1)

i )

andν2(u(2)

i+1 − 2v(2)i + u

(2)i−1) ∈ f(v(2)

i ).By (2.1),

δ|v(2)i − v

(1)i | ≤ ν2|(u(2)

i+1 − 2v(2)i + u

(2)i−1)− (u(1)

i+1 − 2v(1)i + u

(1)i−1)|

≤ 2ν2|v(2)i − v

(1)i |+ ν2(|u(2)

i+1 − u(1)i+1|+ |u

(2)i−1 − u

(1)i−1|)

≤ 2ν2|v(2)i − v

(1)i |+ 2ν2d(u(2), u(1)),

so (since ν2 < δ/2)

|v(2)i − v

(1)i | ≤

2ν2

δ − 2ν2d(u(2), u(1)).

Similarly, if i ∈ 0, n then

|v(2)i − v

(1)i | ≤

ν2

δ − ν2d(u(2), u(1)),

so

d(v(2), v(1)) ≤ 2ν2

δ − 2ν2d(u(2), u(1)).

Consequently, if ν2 < δ/4 then T is a contraction.Clearly, Sp is a complete metric space under the metric d(·, ·), so T has a unique

fixed point in Sp. It is easy to see that any fixed point of T is an element ofEa(ν) ∩ Sp. Because of (2.3) and (2.4) and the fact that for any distinct p, q ∈ I,Sp ∩ Sq = ∅ (which follows from (2.2)), we see that #(Ea(ν) ∩ Sp) = 1, as claimed,and the proof is complete.

Corollary 1. Let f = fi, with i = 1 or 2. Then for ν sufficiently small we have#Ea(ν) = 3n+1, and

#u ∈ Eh(ν) : ν2∆u− β ∈ f(u) =

3n+1 if |β| < i,

2 if |β| = i,

1 if |β| > i.

Proof. That #Ea(ν) = 3n+1 follows immediately from Theorem 1 and the fact thatf1 and f2 are closed and have exactly 3 zeros each, all of them nondegenerate.Similarly, the cardinality results involving Eh(ν) for f = f1 when |β| 6= 1 and forf = f2 when |β| 6= 2 follow immediately from Theorem 1.

In the remaining cases f + β has 2 zeros, one of which is degenerate. There areclearly 2 constant solutions of ∆u ∈ f(u) + β, corresponding to the two zeros off + β; we claim that there are no other solutions of ∆u ∈ f(u) + β. We will verify


this claim for the case f = f2 and β = −2; the three other cases can be handledsimilarly.

Clearly u = 0 and u = 2 are the only constant solutions of ∆u ∈ f2(u) − 2.Suppose u is a nonconstant solution of ∆u ∈ f2(u) − 2. Pick i± ∈ Ω such thatui− = minui : i ∈ Ω and ui+ = maxui : i ∈ Ω. By the minimality ofui− and the maximality of ui+ , (∆u)i− ≥ 0 and (∆u)i+ ≤ 0. This means that(f2(ui−)− 2)∩ [0,∞) 6= ∅ and (f2(ui+)− 2)∩ (−∞, 0] 6= ∅. Consequently, ui+ ≤ 2,and either ui− = 0 or ui− ≥ 2. Since u is nonconstant, the only possibility is to haveui− = 0, so (∆u)i− = 0. But this implies that ui− is equal to the average value of uat neighboring lattice sites, so u = 0 at these neighboring sites as well. Repeatingthis argument inductively, we find that u ≡ 0, which is a contradiction.

3. Minimal grain size. The results of the previous section show that for a fixedlattice, as ν → 0 equilibrium solutions become widely distributed throughout phasespace. In particular, arbitrarily small grains are possible in equilibria. However, if(as when f = f1) the slope of the graph of f is bounded away from vertical nearits unstable zero, a lower estimate on the grain size (in terms of ν) of equilibria ofthe discrete Allen-Cahn equation is obtainable.

Some of the results in this section will involve the function gM : R→ 2R definedby gM (u) = v ∈ R : v sgnu ≥ −M |u|, where

sgnu =

1 if u > 0,0 if u = 0,−1 if u < 0.

In addition, it will be helpful to prove a preliminary result about elements of RZ

that satisfy equations related to (1.1). In particular, we define grains of elements ofRZ in a manner analogous to they way they were defined for elements of RΩ (i.e.,as maximal collections of consecutive sites on which the profile is one-signed), andwe then estimate their size. (In the statement of this result and throughout theremainder of this paper, if H ⊂ Ω, then |H| will denote the number of lattice sitescontained in H.)

Lemma 1. Suppose that M ∈ (0, 2ν2), v ∈ RZ satisfies

ν2(vi+1 − 2vi + vi−1) ∈ gM (vi) (3.1)

for every i ∈ Z, and G is a grain of v. Then

|G| ≥ π

cos−1(1−M/(2ν2))− 1 ≥ 2ν

√2M− 1. (3.2)

Proof. Let M and v satisfy the hypotheses of the lemma. Obviously, (3.2) holds ifthe grain G is unbounded. Assume, then, that G is bounded. Suppose, also, thatG is a positive grain. Let a = inf G and let b = supG. Let θ = π/(b − a+ 2), andnote that kθ ∈ (0, π) for k ∈ [1, b− a+ 1]. By (3.1) and the assumption that G is apositive grain,

ν2(vi+1 − 2vi + vi−1) ≥ −Mvi (3.3)

for every i ∈ G; note also that sin[(i − a + 1)θ] > 0 for every i ∈ G. Multiplyingboth sides of (3.3) by sin[(i− a+ 1)θ] and summing over all i ∈ G yields

ν2b∑i=a

sin[(i− a+ 1)θ](vi+1 − 2vi + vi−1) ≥ −Mb∑i=a

sin[(i− a+ 1)θ]vi.


Rearranging this inequality, and using elementary trigonometric identities and thefact that both va−1 and vb+1 are nonpositive–otherwise, there would be a contra-diction to the maximality of G–we find that

(M − 2ν2(1 − cos θ))b∑i=a

vi sin[(i− a+ 1)θ] ≥ 0,

so M − 2ν2(1− cos θ) ≥ 0. Thus,

|G| = (b− a+ 2)− 1 = π/θ − 1 ≥ π

cos−1(1 −M/(2ν2))− 1. (3.4)

A similar argument yields (3.4) in the case that G is a negative grain.The second inequality in (3.2) is a consequence of the fact that

cos−1(1 − x2)− π

2x ≤ 0 (3.5)

for x ∈ [0, 1]. This fact can be verified by noting that (3.5) holds for x = 0 andx = 1 and that the left-hand side of (3.5) is a convex function of x on [0, 1].

We now use this estimate to get a lower bound on the grain sizes of elements ofEa.

Theorem 2. Suppose that M ∈ (0, 2ν2), f(r) ⊂ gM (r) for every r ∈ R, andu ∈ Ea(ν). If G is a grain of u that is not identical with Ω, then

|G| ≥⌈

π

2 cos−1(1 −M/(2ν2))− 1

2

⌉≥⌈ν

√2M− 1

2

⌉. (3.6)

If, in addition, G is an interior grain, then

|G| ≥⌈

π

cos−1(1−M/(2ν2))− 1⌉≥⌈

2ν

√2M− 1

⌉. (3.7)

Proof. Let M , f , and u be as in the hypotheses of the theorem, and let G be agrain of u. Associate with u an element v ∈ RZ defined by

vk =

uj if k ≡ j (mod (2n+ 2)),uj if k ≡ −1− j (mod (2n+ 2)).

Note that v agrees with u on Ω, is invariant under the reflections vi 7→ v−1−i andvi 7→ v2n+1−i, and satisfies (3.3). Thus, the estimate (3.2) applies to the grains ofv. Furthermore, each grain of u that is not identical with Ω corresponds to eithera half grain of v or a whole grain of v, depending on whether or not the grainintersects 0, n. This observation (along with the fact that the size of a grainmust be an integer) yields (3.6) and (3.7).

As an immediate consequence of Theorem 2, we have:

Corollary 2. If ν > 1/√

2, f = f1, u ∈ Ea(ν), and G is a grain of u that is notidentical with Ω, then

|G| ≥⌈

π

2 cos−1(1− 1/(2ν2))− 1

2

⌉≥⌈ν√

2− 12

⌉.

If, in addition, G is an interior grain, then

|G| ≥⌈

π

cos−1(1− 1/(2ν2))− 1⌉≥⌈2ν√

2− 1⌉.


Theorem 2 and Corollary 2 do not hold, in general, for the discrete Cahn-Hilliardequation. In particular, if Ea(ν) contains an element u with a strict local extremum(e.g., ui > maxui−1, ui+1), and if the graph of f is linear in sections, then it maybe possible to turn u into an element of Eh(ν) with a grain of size 1 by adding thesame constant to each of its components uj. It is possible, however, to obtain alower bound on the diameter of the union of any two grains by placing differentconstraints on f .

Theorem 3. Suppose that M ∈ (0, 2ν2), f(r)−f(s) ⊂ gM (r−s) for every r, s ∈ R,and u ∈ Eh(ν). Suppose that D is a collection of contiguous lattice sites containingat least two grains of u and D 6= Ω. Then

|D| ≥⌈

π

cos−1(1−M/(2ν2))

⌉≥⌈

2ν

√2M

⌉. (3.8)

Proof. Let M , f , u, and D satisfy the hypotheses of the theorem. Since u ∈ Eh(ν),ν2∆u − β ∈ f(u) for some constant β. If we associate an element v ∈ RZ with uby the same procedure described in the proof of Theorem 2, we find that

ν2(vj+1 − 2vj + vj−1)− β ∈ f(vj) (3.9)

for every j ∈ R. Subtracting (3.9) with j = i from (3.9) with j = i+1, and definingw ∈ RZ by setting wi = vi+1 − vi for each i ∈ R, we find that

ν2(wi+1 − 2wi + wi−1) ∈ f(vi+1)− f(vi) ⊂ gM (vi+1 − vi) = gM (wi).

By Lemma 1, (3.2) bounds the size of the grains of w.Now, let a = inf D, and let b = supD. Because, D contains at least two grains of

u, it must contain a maximal collection M of consecutive lattice sites along whichu (and, therefore, v) is strictly monotonic. Let a′ = infM and b′ = supM, andnote that G := i ∈ Z : a′ ≤ i < b′ is a grain of w. Using the bound on the grainsizes of w, and the fact that

|D| = b− a+ 1 ≥ b′ − a′ + 1 = |G|+ 1

yields (3.8).

An immediate consequence is:

Corollary 3. If ν > 1/√

2, f = f1, u ∈ Eh(ν), and D 6= Ω is a collection ofcontiguous lattice sites containing at least 2 grains of u, then

|D| ≥⌈

π

cos−1(1− 1/(2ν2))

⌉≥⌈2ν√

2⌉.

Note that results similar to those in this section cannot be obtained when f = f2.For example, if an integer k ∈ (0, n− 1) is fixed and u ∈ RΩ is defined by

ui =2

3ν2 + 2(δi,k+1 − δi,k),

where δ is the Kronecker delta, then u ∈ Ea(ν) ⊂ Eh(ν), and it has two grains ofsize 1 adjacent to each other.


4. Short-range connections between grains. In the previous section, it wasshown that for f = f1 and for other nonlinearities whose “slopes” are, in a sense,bounded near the unstable zero, a factor leading to disappearance of equilibriaas the interaction length increases is the presence of small grains (or, in the caseof (1.2), pairs of grains contained in a small segment of the lattice). Anotherfactor, relevant for f2 but not for f1, is the presence of marked aperiodicity in grainsizes. (As with the continuous equations [6] [16], as long as the minimal grain sizeestimates from Section 3 are satisfied, f = f1 permits a near arbitrary arrangementof grains in equilibrium solutions.)

To understand the effect of aperiodicity when f = f2, it helps to consider the(multi-valued) spatial dynamical system ϕν generated by the equilibrium problemfor (1.1). More precisely, write the equilibrium equations (for interior sites) point-wise:

ν2(ui+1 − 2ui + ui−1) = f2(ui), (4.1)

and let Fν be the planar map defined by Fν(x, y) = (y, 2y+ ν−2f2(y)− x), so that,in (4.1), (ui+1, ui+2) = Fν(ui, ui+1). Then ϕν will be the discrete dynamical systemgenerated by Fν .

Let π1 : R2 → R be defined by π1(x, y) = x, and let π2 : R2 → R be definedby π2(x, y) = y (here and elsewhere in this paper). Let p0, p1, . . . , pn+1 be a finiteorbit of ϕν such that

π1(pi) = π2(pi), i = 0, n+ 1. (4.2)

Then (π2p0, π2p1, . . . , π2pn) ∈ Ea(ν), and in fact this is a one-to-one correspondence.Hence, understanding the arrangements of grains in elements of Ea(ν) is equivalentto understanding the itineraries of finite orbits of ϕν that begin and end on the line(x, y) ∈ R2 : x = y; these itineraries are relative to the partition E+, E0, E−of R2, where

E+ = (x, y) ∈ R2 : y > 0,E0 = (x, y) ∈ R2 : y = 0,E− = (x, y) ∈ R2 : y < 0.

For the moment, let’s ignore (4.2) and consider the relationship between thenumber of consecutive steps that an orbit of ϕν spends in E+ and the numberof consecutive steps it spends in E−. We will call (m1,m2, . . . ,mk) ∈ Nk anadmissible pattern of ϕν if there exists a finite orbit p0, p1, . . . , pm+1 of ϕν , withm = m1 +m2 + · · ·+mk, satisfying:

• p0 ∈ E−;• pi ∈ E+ if 1 ≤ i ≤ m1;• pi ∈ E+ if 1 +

∑`j=1 mj ≤ i ≤ m`+1 +

∑`j=1mj for some even number `;

• pi ∈ E− if 1 +∑`

j=1 mj ≤ i ≤ m`+1 +∑`j=1 mj for some odd number `;

• π2(pm)π2(pm+1) < 0.

Note that the invariance of (4.1) with respect to the change of index i 7→ n− iimplies that if (m1,m2, . . . ,mk) is an admissible pattern, so is (mk,mk−1, . . . ,m1).Also, the fact that f2 is odd implies that the roles of E+ and E− in the definitionof admissible patterns could be reversed with no consequence. If (m1,m2) is an ad-missible pattern, we will call it an admissible dyad ; if (m1,m2,m3) is an admissiblepattern, we will call it an admissible triad.


Proposition 1. The pattern (m1,m2) is an admissible dyad if and only if either

|m1 −m2| ≤ 2

or

ν <

√2λ0

λ0 − 1,

where λ0 is the unique number λ > 1 solving

λm1+m2+1 − λm1+m2 − 2λmaxm1,m2 + 2λminm1,m2+1 + λ− 1 = 0. (4.3)

Proof. Fix ν > 0, and let x = λ be the larger of the two solutions to

ν2(x2 − 2x+ 1) = 2x, (4.4)

i.e.,λ = 1 + ν−2 +

√2ν−2 + ν−4, (4.5)

and note that x = λ−1 is the other solution of (4.4). A simple calculation shows that(m1,m2) will be an admissible dyad if and only if there are constants A,B,C,D ∈ Rsuch that v+

i := 1−Aλi −Bλ−i and v−i := −1 + Cλi +Dλ−i satisfy1. v+

i > 0 for 1−m1 ≤ i ≤ 0;2. v−i < 0 for 1 ≤ i ≤ m2;3. v+

−m1< 0;

4. v−m2+1 > 0;5. v+

i = v−i for i = 0, 1.Since the functions y 7→ 1−Aλy −Bλ−y and y 7→ −1 +Cλy +Dλ−y each have

at most one strict local extremum, criteria 1 and 2 can be replaced by1. v+

0 > 0 and v+1−m1

> 0;2. v−1 < 0 and v−m2

< 0.

Using criteria 5 to eliminate A andD, and letting B = (λ+1)B/λ and C = (λ+1)C,criteria 1–4 state that

(B, C) ∈6⋂j=1

Pj , (4.6)

where

P1 = (B, C) ∈ R2 : λ2m1+1B − C > λm1+1 + λm1 − 2,P2 = (B, C) ∈ R2 : −λ2m1−1B + C > −λm1 − λm1−1 + 2,P3 = (B, C) ∈ R2 : −λB + C > −λ+ 1,P4 = (B, C) ∈ R2 : B − λC > −λ+ 1,P5 = (B, C) ∈ R2 : B − λ2m2−1C > −λm2 − λm2−1 + 2,

and

P6 = (B, C) ∈ R2 : −B + λ2m2+1C > λm2+1 + λm2 − 2.While there are general approaches to determining the consistency of systems of

linear inequalities (see, e.g., [22]), it appears easier to derive simple results in thiscase by analyzing (4.6) directly. Straightforward, if tedious, calculations comparingthe coordinates of the intersection points of the boundaries Lj of the Pj revealthat, since λ > 1,

⋂4j=1 Pj is a nonempty, open quadrilateral in the B-C plane,

as indicated in Figure 2 (or, in the special case when m1 = 1 and P2 = P3, anonempty, open triangle). Similarly,

⋂6j=3 Pj is a nonempty, open quadrilateral,


as indicated in Figure 2 (or, in the special case when m2 = 1 and P4 = P5, anonempty, open triangle).

L3

L4

L1 L2

B

C

L3

L4

L6

L5

B

C

Figure 2.

⋂4j=1 Pj (left), and

⋂6j=3 Pj (right).

While the figures might not be quantitatively accurate, it can be checked thatthey accurately reflect the ordering of πk(Lj1∩Lj2) and πk(Lj1∩Lj3) for k ∈ 1, 2.(For example, π2(L1 ∩ L3) ≤ π2(L1 ∩ L2).) Furthermore, if we let m(j) denote theslope of Lj, then, as the figures suggest, m(j1) ≥ m(j2) > 0 if j1 ≤ j2. For thesereasons,

⋂6j=1 Pj will be nonempty if and only if π1(L4 ∩ L6) < π1(L2 ∩ L4) and

π1(L1 ∩ L3) < π1(L3 ∩ L5). The first happens precisely when

λm1+m2+1 − λm1+m2 − 2λm1 + 2λm2+1 + λ− 1 > 0, (4.7)

and the second happens precisely when

λm1+m2+1 − λm1+m2 − 2λm2 + 2λm1+1 + λ− 1 > 0. (4.8)

Since the left-hand side of (4.7) is greater than the left-hand side of (4.8) preciselywhen m2 > m1, these two conditions can be combined into one:

λm1+m2+1 − λm1+m2 − 2λmaxm1,m2 + 2λminm1,m2+1 + λ− 1 > 0. (4.9)

It is easy to see that, since λ > 1, (4.9) is satisfied if |m1−m2| ≤ 1. If |m1−m2| =2 then the right-hand side of (4.9) is (λ − 1)(λ(m1+m2)/2 − 1)2, which must bepositive.

Suppose, now, that |m1 − m2| > 2. We claim that, in this case, (4.3) has aunique solution λ > 1, and

x > 1 : (4.9) is satisfied when λ = x = (λ0,∞),

where λ0 is this unique solution. Since (4.5) is equivalent to the equation ν =√2λ/(λ−1), and

√2λ/(λ−1) is a decreasing function of λ when λ > 1, verification

of these two claims will complete the proof.Let p = maxm1,m2 and q = minm1,m2, and consider the polynomial

h(x) = xp+q+1 − xp+q − 2xp + 2xq+1 + x− 1.

By Descartes’ Rule of Signs, h has either 1 or 3 positive zeros. Note that h(0) =−1 < 0, h(1) = 0, h′(1) = 2(2 + q − p) < 0, and limx→∞ h(x) = +∞. Thus, h hasone zero in (0, 1), a zero at 1, one zero in (1,∞), and no other positive zeros. Thus,the claims hold.


A similar, but more complicated, characterization of admissible triads is avail-able:

Proposition 2. The pattern (m−1,m0,m1) is an admissible triad if and only if

λmi+m0+1 − λmi+m0 − 2λmaxmi,m0 + 2λminmi,m0+1 + λ− 1 > 0

for each i ∈ −1, 1 and

(λ− 1)(λp1−j+2m0+2pj+2j − λ2−2j) + 2(−1)j(λm0 − λp1−j )(λ2pj+m0+1+j − λ2−j)

+ λm0+1[(λp1+1 − λp0)(λpj − 1) + (λp1 − λp0)(λpj+j − λ1−j)] > 0

for each j ∈ 0, 1, where λ = 1 + ν−2 +√

2ν−2 + ν−4, p0 = minm−1,m1, andp1 = maxm−1,m1.

Proof. Let λ = 1 + ν−2 +√

2ν−2 + ν−4. Reasoning as in the proof of Proposition1, it is not hard to verify that (m−1,m0,m1) will be an admissible triad if and onlyif the set

⋂8j=1 Pj is nonempty, where

P1 =

(A,B) ∈ R2 : −A−B > 1− 2(λ−m−1 + λm−1+1)1 + λ

,

P2 =

(A,B) ∈ R2 : λA+ λ−1B > −1 +2(λ1−m−1 + λm−1)

1 + λ

,

P3 =

(A,B) ∈ R2 : λm−1A+ λ−m−1B > 1,

P4 =

(A,B) ∈ R2 : −λm−1+1A− λ−m−1−1B > −1,

P5 =

(A,B) ∈ R2 : −λm−1+m0A− λ−m−1−m0B > −1,

P6 =

(A,B) ∈ R2 : λm−1+m0+1A+ λ−m−1−m0−1B > 1,

P7 =

(A,B) ∈ R2 : λm−1+m0+m1A+ λ−m−1−m0−m1B > −1 +2(λm1 + λ1−m1)

1 + λ

,

and P8 =(A,B)∈R2 : −λm−1+m0+m1+1A− λ−m−1−m0−m1−1B > 1− 2(λm1+1 + λ−m1)

1 + λ

.

We will apply an ad hoc analysis to the inequalities defining the Pj in order toget a much simpler set of conditions for their solvability than would result from theapplication of general techniques.

For every λ > 1,⋂4j=1 Pj ,

⋂6j=3 Pj , and

⋂8j=5 Pj, are qualitatively as in Figures

3 and 4, respectively.The slopes m(j) of the lines Lj bounding the half-planes Pj are ordered as in

these figures, and the figures are also accurate in their depiction of the ordering ofπk(Lj1 ∩ Lj2) and πk(Lj1 ∩ Lj3) for k ∈ 1, 2. In particular, the regions are, infact, nonempty open quadrilaterals, except when mi = 1 for some i ∈ −1, 0, 1, inwhich case they may degenerate to triangles.

Clearly,⋂8j=1 Pj is nonempty if and only if S1 :=

⋂6j=1 Pj and S2 :=

⋂8j=3 Pj

are each nonempty and S1 ∩ S2 6= ∅. Note that S1 and S2 are each the interior ofthe convex hull of the union of a connected subset of (L3∩P5∩P6)∪ (L6∩P3∩P4)and a connected subset of (L4 ∩ P5 ∩ P6) ∪ (L5 ∩ P3 ∩ P4). Because of the shapesof S1 and S2 and the relative magnitudes of the slopes of the lines bounding them,the only way S1 and S2 could be disjoint if each were nonempty would be if

maxπ1(L1 ∩ L3), π1(L1 ∩ L6) ≥ minπ1(L8 ∩ L3), π1(L8 ∩ L6), (4.10)


L1

L2

L3

L4

B

A

L3

L4

L5L6A

B

Figure 3.

⋂4j=1 Pj (left), and

⋂6j=3 Pj (right).

L8L7

L6 L5A

B

Figure 4.

⋂8j=5 Pj .

or

maxπ1(L7 ∩ L4), π1(L7 ∩ L5) ≥ minπ1(L2 ∩ L4), π1(L2 ∩ L5), (4.11)

and S1 and S2 are guaranteed to be disjoint if either of these conditions holds.By using the figures to check cases, it can be seen that (4.10) fails precisely when

π1(L1 ∩ L6) < π1(L8 ∩ L6) (4.12)

andπ1(L1 ∩ L3) < π1(L8 ∩ L3) (4.13)

both hold, and (4.11) fails precisely when

π1(L7 ∩ L4) < π1(L2 ∩ L4) (4.14)

andπ1(L7 ∩ L5) < π1(L2 ∩ L5) (4.15)

both hold.


If m−1 ≤ m1, we claim that (4.12) holds if (4.13) holds. To see why this is so,suppose that λ is such that (4.13) holds but (4.12) does not. By again checkingcases, we can see that this would mean that

π1(L1 ∩ L3) < π1(L1 ∩ L6)

andπ1(L8 ∩ L6) < π1(L8 ∩ L3).

Computation shows that the former implies that

λm−1+m0+2 − λm−1+m0+1 − 2λm0+1 + 2λm−1+1 + λ− 1 > 0, (4.16)

and the latter implies that

−λm1+m0+2 + λm1+m0+1 + 2λm0+1 − 2λm1+1 − λ+ 1 > 0. (4.17)

Multiplying (4.16) by λm1 and multiplying (4.17) by λm−1 and adding yields

(λm−1 − λm1)(2λm0+1 − λ+ 1) > 0.

Since λ > 1, this means that m−1 > m1. This verifies the claim. By a similarargument, it can be shown that if m−1 ≤ m1 and (4.15) holds then (4.14) holds.Furthermore, by symmetry it follows that if m−1 ≥ m1 then (4.15) and (4.13) holdwhenever (4.14) and (4.12) hold, respectively. Using these facts, (4.14) and (4.15)can be collapsed to a single equivalent condition involving p0 := minm−1,m1 andp1 := maxm−1,m1, and the same can be said of (4.12) and (4.13).

Finally, note that applying Proposition 1 to the dyads (m−1,m0) and (m0,m1)gives conditions characterizing those values of λ for which S1 and S2 are eachnonempty. Combining this with the preceding information (and performing somestraightforward calculations) completes the proof.

We now return our focus to elements of RΩ and, more specifically, of Ea(ν).The existence of certain arrangements of grains in an element of Ea(ν) implies theexistence of certain admissible patterns. In particular, it may imply that certaindyads and triads are admissible.

Suppose that G1 and G2 are adjacent grains of an element of Ea(ν) and thatneither grain is adjacent to an interface point. Then, depending on whether thegrains are boundary or interior grains, Table 1 indicates dyads and triads that mustbe admissible.

Table 1. Admissible Patterns Implied by Adjacency of G1 and G2.

G2

Boundary Grain Interior Grain

Boundary Grain(2|G2|, 2|G1|, 2|G2|)

(2|G1|, 2|G2|, 2|G1|)(|G2|, 2|G1|, |G2|)

G1

Interior Grain (|G1|, 2|G2|, |G1|) (|G1|, |G2|)

If, in addition, we suppose that G2 is adjacent to G3 and that G1 and G3 aredistinct and G3 is not adjacent to an interface point, then, depending on whetherG1 and G3 are boundary or interior grains, Table 2 indicates additional triads thatmust be admissible. There are other dyads and triads whose admissibility is, in turn,


Table 2. Additional Admissible Patterns Implied by Adjacencyof G1, G2, and G3.

G3

Boundary Grain Interior Grain

Boundary Grain (2|G1|, |G2|, 2|G3|) (2|G1|, |G2|, |G3|)G1

Interior Grain (|G1|, |G2|, 2|G3|) (|G1|, |G2|, |G3|)

implied by the information in these two tables. For example, if (|G1|, |G2|, 2|G3|) isadmissible, then so are (2|G3|, |G2|, |G1|) and (|G1|, |G2|).

Taking the contrapositive of these implications, we see that if certain dyads andtriads are inadmissible, then there are associated arrangements of grains that cannotpossibly appear in elements of Ea(ν). In a sense, the elements of RΩ \ Ea(ν) thatreflect these patterns are in that set because of short-range interactions betweenlattice sites.

In Table 3 and Table 4, we have summarized some data on what we will call

Table 3. Most Robust Arrangements of Grains, n = 13.

Critical ν

Arrangement of Grains Equilibria Admissible Dyads Admissible Triads

• • • • • • • • • • • • • • ∞ ∞ ∞

• • • • • • • ∞ ∞ ∞

• • • • • • • ∞ ∞ ∞

• • • • • • • 20.9051434 ∞ ∞

• • • • • • • 19.3332028 ∞ ∞

• • • • • • • 12.9572259 ∞ ∞

• • • • • • • • 8.5232392 11.1886856 9.1755408

• • • • • • • 8.5232392 ∞ ∞

• • • • • • • 8.1798188 ∞ ∞

• • • • • • • • • 6.4989036 7.2989941 6.6012422

• • • • • • • 5.9643734 ∞ ∞

• • • • • • • 5.9431886 ∞ 11.3814794

• • • • • • • 5.6160304 ∞ ∞

• • • • • • • 5.5963342 ∞ ∞

• • • • • • 5.5290741 ∞ 6.9365737

critical values of ν when n = 13. Theorem 1 guarantees that, given s ∈ −1,+1Ω,


Table 4. Least Robust Arrangements of Grains, n = 13.

Critical ν

Arrangement of Grains Equilibria Admissible Dyads Admissible Triads

• • • • • • • • • • • • • 1.2247460 2.0002750 1.2247483

• • • • • • • • • • • • • 1.2247541 2.0000687 1.2247726

• • • • • • • • • • • • • 1.2248279 2.0000172 1.2249942

• • • • • • • • • • • • 1.2249387 2.0185271 1.2252443

• • • • • • • • • • • • 1.2250867 2.0185271 1.2257465

• • • • • • • • • • • • 1.2250867 2.0044641 1.2255227

• • • • • • • • • • • 1.2251980 2.0185271 1.2257465

• • • • • • • • • • • • • 1.2254947 2.0000043 1.2270093

• • • • • • • • • • • • 1.2255320 2.0044641 1.2270379

• • • • • • • • • • • • 1.2255320 2.0090457 1.2270949

• • • • • • • • • • • • 1.2255817 2.0185271 1.2272658

• • • • • • • • • • • • 1.2255817 2.0185271 1.2272658

• • • • • • • • • • • 1.2256190 2.0185271 1.2272658

• • • • • • • • • • • 1.2256936 2.0185271 1.2272658

• • • • • • • • • • • • 1.2257474 2.0387067 1.2277793

there exists at least one corresponding element u ∈ Ea(ν) such that sgnu = s ifν is sufficiently small. Similarly, every pattern is admissible for ν 1. As νincreases, most of these patterns will eventually become inadmissible. Now, givens ∈ −1,+1Ω, the associated critical value of ν for equilibria is the supremumof the values of ν for which there is a corresponding element of Ea(ν). Let D(s)be the set of those dyads whose admissibility is implied by the existence of anelement u ∈ Ea(ν) for which sgnu = s (see Table 1). The critical value of ν foradmissible dyads is the supremum of the values of ν for which all dyads in D(s)are actually admissible. If we similarly define T (s) to be the set of those triadswhose admissibility is implied by the existence of an element u ∈ Ea(ν) for whichsgnu = s, then the critical value of ν for admissible triads is the supremum of thevalues of ν for which all triads in T (s) are actually admissible.

Table 3 shows these critical values for the most “robust” arrangements of grains;i.e., those arrangements whose critical values for equilibria are the largest. (A“•” represents a lattice site where the equilibrium is positive; a “” represents alattice site where the equilibrium is negative.) Table 4 shows the critical values forarrangements that are the least robust and disappear for the smallest values of ν.Of necessity, the numbers in the first column of critical values are no greater thanthe numbers in the third column, which are in turn no greater than the numbers inthe middle column. (The computation of these numbers used techniques discussed


in [25] and was done in multiprecision arithmetic. All of the digits shown shouldbe correct.)

Table 3 shows a very poor correspondence between the first column of criticalvalues and the other two columns. This suggests that the mechanism leading to theextinction of these arrangements as ν → ∞ has little to do with admissible dyadsor triads. Table 4 tells a different story, however. While the first two columns ofcritical values continue to be poorly correlated, the critical values for admissibletriads match up well with the critical values for equilibria (although the numbers arenot identical). Furthermore, these two columns also show a strong order correlation,with the sixth least robust arrangement being the only one that is out of order inthe last column.

This seems to suggest that the admissibility of the relevant triads is an importantcharacteristic of the least robust arrangements. The focus on grain patterns can beviewed as run-length encoding of the phase of a element of RΩ. In this encoding,next-nearest neighbor effects or interactions appear to be significant.

5. Conclusion. In [31] it was shown that for continuously differentiable nonlin-earities, equilibrium solutions of (1.1) in the case of no coupling can be continuedfor small positive coupling strength. Here we have shown that equilibria can besimilarly continued in the case of set-valued nonlinearities under suitable nonde-generacy and dissipativity conditions. The same approach permits enumeration ofsubclasses of the set of equilbrium solutions of (1.2).

As the coupling strength is increased, most equilibria disappear, with those hav-ing small grains disappearing early in the process. Also, a connection can be estab-lished between the size of grains and the admissible sizes of neighboring grains andof grains that are next nearest neighbors. These connections appear to be mostrelevant when it comes to the least robust equilibria.

6. Acknowledgements. The author wishes to thank the referee for his sugges-tions.

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grant/cv/grf.pdf · DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS Volume 1,Number1, Xxxx 2000 pp. 1{33 GRAIN SIZES IN THE DISCRETE ALLEN-CAHN AND CAHN-HILLIARD EQUATIONS CHRISTOPHER P