A STATIONARY CORE-SHELL ASSEMBLY IN A TERNARY INHIBITORY SYSTEM

. A ternary inhibitory system motivated by the triblock copolymer theory is studied as a nonlocal geometric variational problem. The free energy of the system is the sum of two terms: the total size of the interfaces separating the three constituents, and a longer ranging interaction energy that inhibits micro-domains from unlimited growth. In a particular parameter range there is an assembly of many core-shells that exists as a stationary set of the free energy functional. The cores form regions occupied by the ﬁrst constituent of the ternary system, the shells form regions occupied by the second constituent, and the background is taken by the third constituent. The constructive proof of the existence theorem reveals much information about the core-shell stationary assembly: asymptotically one can determine the sizes and locations of all the core-shells in the assembly. The proof also implies a kind of stability for the stationary assembly.

1. Introduction. A pattern forming, multi-constituent, inhibitory physical or biological system is characterized by two properties: growth and inhibition. A deviation from homogeneity has a strong positive feedback on its further increase. In the meantime a longer ranging confinement mechanism prevents unlimited spreading. Together they lead to a locally self-enhancing and self-organizing process.
Nakazawa and Ohta introduced a ternary inhibitory system in [15] to study triblock copolymers. A Gamma-limit of the system was derived by Ren and Wei in [20] for the sharp interface case.
This Gamma-limit is a nonlocal geometric variational problem. Let D be a bounded domain in R 2 . A functional is defined on non-overlapping pairs of subsets of D with the fixed size; namely on the admissible set (1.1) The numbers ω i ∈ (0, 1) are given and ω 1 + ω 2 < 1. They are the first set of the parameters in this problem. One denotes by |Ω i | and |D| the Lebesgue measures of Ω i and D respectively.
For each pair (Ω 1 , Ω 2 ) ∈ A, let Ω 3 = D\(Ω 1 ∪ Ω 2 ). The free energy of the system is given by In the first term P D (Ω i ) is the perimeter of Ω i in D. If the boundary of Ω i in D, ∂Ω i ∩ D, is C 1 , then ∂Ω i ∩ D is the interface of Ω i and P D (Ω i ) is simply the length of ∂Ω i ∩ D. Note that ∂Ω i is the boundary of Ω i in R 2 . Part of ∂Ω i may overlap with the domain boundary ∂D, but the interface of Ω i does not include this part. The interface is ∂Ω i ∩ D, which is the part of the boundary of Ω i that is inside D.
For a general measurable set Ω i , perimeter is defined in (2.3).
There are possibly three types of interfaces: ∂Ω 1 ∩∂Ω 2 ∩D, the interface separating Ω 1 and Ω 2 , ∂Ω 2 ∩∂Ω 3 ∩D, the interface separating Ω 2 and Ω 3 , and ∂Ω 3 ∩∂Ω 1 ∩D, the interface separating Ω 3 and Ω 1 . The first term in (1.2) is the combined length of all three type interfaces in D. The half is put there to eliminate double counting since each type of interface is counted twice in the sum.
The numbers γ ij in the second term of (1.2) form a two by two symmetric and positive definite matrix. The operator (−∆) −1/2 is defined as follows. Let 2) is its positive square root. The latter acts on χ Ωi − ω i where χ Ωi is the characteristic function of Ω i (χ Ωi (x) is 1 if x ∈ Ω i and 0 otherwise). A stationary point of J is an element in (1.1) where the first variation of J vanishes. In this paper we are interested in one type of stationary points: stationary core-shell assemblies. A core-shell assembly is a collection of many core-shells; each core-shell consists of a perturbed disc, called a core, surrounded by a perturbed ring, called a shell; see the second plot in Figure 1. The union of the cores is Ω 1 and the union of the shells is Ω 2 . The many core-shells in the assembly do not intersect and they do not touch the domain boundary ∂D. In a core-shell assembly there are only two types of interfaces ∂Ω 1 ∩ ∂Ω 2 and ∂Ω 2 ∩ ∂Ω 3 , namely the interface separating Ω 1 from Ω 2 and the interface separating Ω 2 from Ω 3 ; there is no interface between Ω 1 and Ω 3 .

XIAOFENG REN AND CHONG WANG
We single out a special class of symmetric matrices: (1.10) Note that in (1.10), l starts from 2, not 1. Also Γ > 0 (resp. M l (Γ) > 0) means that the matrix Γ (resp. M l (Γ)) is positive definite. One can show that the interior of S, denoted int(S), is non-empty; see Lemma 4.1.
The main result in this paper is the following existence theorem.
Theorem 1.1. Let D be a bounded, sufficiently smooth domain in R 2 , m ∈ (0, 1), n ∈ N, and ι ∈ (0, 1]. For each compact subset K ⊂ int(S), there exist positive numbers δ, σ depending on D, m, n, K, and ι only, such that if then J admits a stationary assembly of n perturbed core-shells, satisfying the constraints (1.7).
A few remarks are in order. The smoothness assumption on D is only needed to ensure that (1.3) is solvable, (−∆) −1 is well defined, and a Green's function exists (see (1.13)).
The condition 3, ǫ 3 γ ∈ K ⊂ int(S), has several implications. First, γ must be positive definite. Second, seen from the definition of S. This requirement is related to our assumption that cores are formed by the first constituent and shells formed by the second constituent. If the two constituents were reversed, we would have γ 11 > γ 12 . Third, the condition also implies, since K is compact, that there existsσ > 0 such that This upper bound for λ(γ) is by an order (albeit only log 1 ǫ ) greater than the lower bound for λ(γ) in condition 2.
Some of the techniques used in this paper were developed by Xie in [32], where she proved the existence of a single, perturbed core-shell as a stationary point of J , a special case of Theorem 1.1 with n = 1. There are significant differences between the n = 1 case and n ≥ 2 case. The condition 2 is not needed if n = 1. If n ≥ 2, the condition 2 prevents a coarsening phenomenon, where some core-shells grow bigger and others shrink and disappear. Much of our work is to ensure that under the condition 2, n core-shells will keep their shapes and remain stable in an assembly.
Although the focus of this paper is on the existence of a stationary assembly, the construction process in the proof of Theorem 1.1 shows that the stationary assembly has a certain kind of stability.
The proof of Theorem 1.1 contains definitions for the centers of the perturbed core-shells in the stationary assembly and the radii of their interfaces; see the discussion after (4.7). The perturbed core-shells turn out to be of the similar size, so all the inner radii are approximately equal and all the outer radii are also approximately equal. The centers of the core-shells are determined asymptotically by the Green's function of (1.3). Recall that the Green's function of the −∆ operator on D with the Neumann boundary condition is a function G(x, y) that solves as a function of x for each y ∈ D. One can write G as a sum of two terms: (1.14) The first term 1 2π log 1 |x−y| is the fundamental solution of the Laplace operator; the second term R is the regular part of the Green's function, a smooth function of (x, y) ∈ D × D.
For n distinct points ξ k , k = 1, 2, ..., n, in D let of the ξ k 's approaches ∂D, or if the distance of two points ξ k and ξ l approaches 0. This ensures that F is minimized by n distinct points in D.
Theorem 1.2. Let ξ * ,k be the center of the k-th perturbed core-shell of the stationary assembly in Theorem 1.1, and r * ,k 1 and r * ,k 2 be the radii of the inner and outer interfaces, respectively.
Although experimentally an almost unlimited number of architectures can be synthetically accessed in ternary systems like triblock copolymers [3, Figure 5 and the magazine's cover], mathematical study of J is still in an early stage due to its complexity. The core-shell stationary assembly found in Theorem 1.1 is only the second stationary assembly discovered on a general two dimensional domain. The first is an assembly of perturbed double bubbles (the first plot in Figure 1) found by Ren and Wei in [29]. The special case of a single, perturbed double bubble stationary point was found earlier in [26,27]. One dimensional stationary points were found in [20,5].
The functional J has a simpler counterpart in a binary inhibitory system. Let ω ∈ (0, 1) and γ > 0. This time γ is a number. For Ω ⊂ D with the fixed area: |Ω| = ω|D|, the binary free energy of Ω is A stationary set of this functional satisfies the equation on ∂Ω ∩ D. Equation (1.17) or functional (1.16) may be derived from the Ohta-Kawasaki theory [17] for diblock copolymers; see [16,19]. The equation can also be derived from the Gierer-Meinhardt system [25]. This binary problem has been studied intensively in recent years. All solutions to (1.17) in one dimension are known to be local minimizers of J B [19]. There is even a dynamic counterpart of (1.17), and Fife and Hilhorst proved that any time dependent solution converges to one of the local minimizers [8]. Many solutions in two and three dimensions have been found that match the morphological phases in diblock copolymers [18,22,21,23,24,11,12,25,28,31]. Global minimizers of J B are studied in [2,30,14,4,13,10] for various parameter ranges. Applications of the second variation of J B and its connections to minimality and Gamma-convergence are found in [6,1]. A relevant result in [21] states that when ω and γ are in a proper range, (1.17) admits a solution that is an assembly of perturbed discs. The discs have approximately the same size, and the centers of the discs nearly minimize the same function F in Theorem 1.2.
2. Exact core-shell assembly. We reiterate that our data comprises the domain D, the number m ∈ (0, 1), the number of core-shells n in assemblies, a compact subset K of the set S given in (1.10), and finally ι ∈ (0, 1] which bounds the ratio of the eigenvalues of the matrix γ. From D, m, n, K, and ι, one proceeds to find δ and σ. These two numbers specify the range for ǫ and γ, where Theorem 1.1 will hold. Moreover Theorem 1.2 will be proved if ǫ → 0 within the range.
The two terms in J are denoted by J s and J l for short and long range interactions respectively: For Ω 1 , Ω 2 or Ω 3 of a general member (Ω 1 , Ω 2 ) in A defined by (1.1), each perimeter in J s is given by where div g is the divergence of the C 1 vector field g on D with compact support and |g(x)| stands for the Euclidean norm of the vector g(x) ∈ R 2 ; see [7] or [9] for more information on perimeter. Using the Green's function G of (1.13) one can rewrite J l (Ω 1 , Ω 2 ) as Let ξ 1 , ξ 2 , ..., ξ n be n points in D and r 1 1 , r 1 2 , r 2 1 , r 2 2 , ..., r n 1 , r n 2 be 2n numbers such that 0 < r k 1 < r k 2 , k = 1, 2, ..., n. Denote by B k an exact core-shell. More precisely is the core and B k 2 \B k 1 is the shell. Also introduce w k i , i = 1, 2 and k = 1, 2, ..., n, so that The w k i 's belong to the set W which is the closure of For now assume that later h will be restricted to a smaller range. Initially the w k i 's are fixed. Later they will vary in W . Of course w k i can vary only if n > 1.
Here Ξ δ is a subset of Ξ defined as In (2.10) "d" stands for the Euclidean distance in R 2 . The centers ξ k of the coreshells B k = (B k 1 , B k 2 \B k 1 ) will always be in the closure of Ξ δ : The number δ remains fixed throughout this paper. At this point we state our initial requirement on δ of Theorem 1.1, which is the bound for ǫ. The number δ must be small enough so that holds for the radius r k 2 of any B k 2 as long as w = (w k i ) ∈ W . With this choice of δ and with ǫ < δ, let z k ∈ B k 2 . Then for any x ∈ ∂D, (2.14) Hence the core-shells are all inside D and they do not intersect. Moreover with It is an assembly of exact core-shells, which is used as an approximate solution of our problem. For now the centers ξ k and the radii r k i are taken arbitrarily from Ξ δ and W respectively. They will be determined near the end of the paper. Our first result gives J (B 1 , B 2 \B 1 ), the energy of the exact core-shell assembly.
The long range part of J (B 1 , B 2 \B 1 ) is where Here v 1 and v 2 are respectively the solutions of As shown in [21], .
3. Perturbed core-shells. We set up a framework to study perturbed core-shells in this section. Let (φ k 1 , φ k 2 ), k = 1, 2, ..., n, be n pairs of 2π periodic functions, collectively denoted as (3.1) Using φ, we define 2n sets Here S 1 is the unit circle identified with [0, 2π]. Since our domain is in R 2 we often use the complex notation for simplicity. In (3.2) and (3.3) e iθ is just (cos θ, sin θ). The reader will see things like e iθ · x which is the inner product of two vectors e iθ and x in R 2 .
Both φ k 1 (θ) and φ k 2 (θ) must be small compared to (r k and Ω k 2 are respectively then Ω 1 = ∪ n k=1 Ω k 1 and Ω 2 = ∪ n k=1 Ω k 2 satisfy the constraints Let us define some Hilbert spaces. First The inner product on Z is When φ k 1 and φ k 2 are small compared to (r k 1 ) 2 and (r k 2 ) 2 , they define a perturbed core-shell (Ω k 1 , Ω k 2 ). In order to define the energy J on an assembly of such perturbed core-shells, one needs some smoothness on φ k i . Let be a subspace of Z. Here H 1 (S 1 ) is a usual Sobolev space on S 1 . The norm of Y is given by If ξ and w are held fixed, J is viewed as a functional of φ with the domain Recall that for all ǫ < δ, ξ ∈ Ξ δ and w ∈ W , the exact core-shell assembly (B 1 , B 2 \B 1 ) determined by ǫ, ξ, and w has the property that z = (z 1 , z 2 , ..., z n ) ∈ Ξ δ/2 if z k ∈ B k 2 for k = 1, 2, ..., n. Choose β in (3.12) sufficiently small so that for all ǫ < δ, all (ξ, w) ∈ Ξ δ × W , and all φ ∈ Dom(J ), the perturbed core-shell assembly (Ω 1 , Ω 2 ) specified by ǫ, (ξ, w), and φ has the property that z = (z 1 , z 2 , ..., Hence the perturbed core-shells Ω k in (Ω 1 , Ω 2 ) do not intersect, and they all stay inside D, away from ∂D.
One writes J (φ) for J (Ω 1 , Ω 2 ). Then J (φ) = J s (φ) + J l (φ), where J s (φ) and J l (φ) are given in terms of φ as (3.14) Next consider the first variation of the functional J with respect to φ. For convenience introduce functions L k i : for φ ∈ Dom(J ) and ψ ∈ Y. The first variation of J is the directional derivative where Here D 1 L k i (·, ·) is the derivative of L k i with respect to its first argument and D 2 L k i (·, ·) the derivative with respect to its second argument. We would like to have operators S s and S l so that This is always possible for J l . However for J s , one must restrict φ to a smaller, more smooth space. Define with the norm Clearly X ⊂ Y ⊂ Z. One can define S = S s + S l on where β is the same as the number in (3.12). Therefore Dom(S) ⊂ Dom(J ). The nonlinear operators S s and S l map from Dom(S) to Z as follows. The component S k s,i (φ) is given as is the curvature operator, and Λ s,i (φ i ) is a number, depending on Note that Λ s,i (φ i ) is the same for all k = 1, 2, ..., n. The components of S l (φ) are We need to write S k l,i (φ) more explicitly. Let When l = k, let Similarly, Then we obtain Here λ 1 (φ) and λ 2 (φ) are two numbers, independent of k, so chosen that When ξ and w are fixed, the exact core-shell assembly is represented by φ = 0. The next lemma estimates S k i (0), up to the constants λ i (0).
4. Linear analysis. The Fréchet derivative of S at φ ∈ Dom(S) is denoted by S ′ (φ). It can also be interpreted as the second variation of J because for every φ ∈ Dom(S), u ∈ X , and v ∈ Y. Note that the left side of (4.1) is meaningful provided φ ∈ Dom(J ), u ∈ Y, and v ∈ Y, while the right side of (4.1) is defined if φ ∈ Dom(S), u ∈ X , and v ∈ Z.
In this section we study S ′ (0), i.e. the linearized operator at the exact core-shell assembly. Because of (3.29), we consider the Fréchet derivative of each of the terms 2π 0 u l 2 (η)G(ξ k + r k 2 e iθ , ξ l + r l 2 e iη )dη.
Let us separate S ′ (0) to a dominant part E and a minor part F : S ′ (0) = E + F . We define E k , the k-th component of E, to be E k = (E k 1 , E k 2 ) and The real valued linear operator e i is independent of k. It is so chosen that E maps from X to Z. The rest of S ′ (0) is denoted by F . Note that E is determined by H and A, and F is determined by B and C.
in X , where the nontrivial term appears in the k-th position. Same is done with Then and the same holds if cos lθ is replaced by sin lθ. Note that Hence 0 is an eigenvalue for E on X and cos θ α 1 1 , ..., cos θ α n 1 , sin θ α 1 1 , ..., sin θ α n 1 (4.4) are the associated eigenvectors.
Although an exact core-shell (B k 1 , B k 2 \B k 1 ) has the well defined center ξ k and the radii r k 1 and r k 2 . After perturbation φ k to (Ω k 1 , Ω k 2 ), one cannot expect the perturbed core-shell always to have the same center and radii. Nevertheless, there is a special class of perturbations that preserve centers and radii.
Let Π be the orthogonal projection operator from Z to a subspace Z ♭ , where

(4.5)
Here by φ k ⊥ cos θ α k 1 we mean When φ ∈ X ♭ (or Y ♭ , or Z ♭ ), the perturbed core-shells in the assembly φ have well defined centers and radii. The k-th perturbed core-shell described by φ k is considered to be centered at ξ k of radii r k 1 for the inner interface and r k 2 for the outer interface. If φ ∈ Z\Z ♭ , then ξ k and r k i cannot be interpreted as centers and radii of the core-shells in the assembly represented by φ.
We are more interested in ΠS ′ (0) and ΠE restricted on X ♭ . By the self-adjointness of E, E maps X ♭ into Z ♭ , so ΠE = E on X ♭ .
If v = ( m n , 1−m n ), then M l (v, Γ) becomes M l (Γ) of (1.9). Regarding M l (Γ) we have the following lemma. Proof. Identify S 2 with R 3 ; namely every Γ ∈ S 2 corresponds to the vector (Γ 11 , Γ 12 , uniformly with respect to Γ ∈ B( 0, t), as l → ∞. The right side of (4.12) is a positive definite matrix. On the other hand, since where 0 is the zero matrix, for each l ≥ 2 one can find B( 0, t l ), t l > 0, such that M l (Γ) > 0 if Γ ∈ B( 0, t l ). The lemma follows from these two properties.
Recall that K is a fixed compact subset of the interior of S. Coming back to M l (v, Γ), we have  Proof. When l = 1, whose associated eigenvectors are respectively .
(4.15) Part 1 follows from the fact that K is compact and Q 1 , Q 2 are bounded. We also used the fact that Γ 22 > Γ 12 when Γ is in S. When l ≥ 2, let h be any number satisfying 0 < h < min{ m n , 1−m n }. If l → ∞, then d4 ] for whenever l < L and Γ ∈ K. Then take d3 ] for all l ≥ L. Finally set d 2 = min{d 3 , d4 2 }. The eigenvalues of E on X ♭ are eigenvalues of the matricesM k l of (4.8) which we denote byλ k l,i , k = 1, 2, ..., n, l = 1, 2, ..., i = 1, 2. Here one must excludeλ 1 1,1 , λ 2 1,1 ,...,λ n 1,1 , because their corresponding eigenvectors are perpendicular to X ♭ . The eigenvaluesλ k l,i are related to λ l, From now on h in the definition (2.6) of W is chosen in accordance with Lemma 4.3. We use c 2 with a lower case c to indicate that the constant is used for the lower bound of the second variation of J at 0. The second part F in S ′ (0) is a minor part.
Lemma 4.4. There exists C 2 > 0 depending on D, m and n only such that for all u ∈ X ♭ , where f 1 (u) and f 2 (u) are real valued and independent of k. They are included so that F (u) is in Z. Because and 2π 0 u k j (η)dη = 0, where 1 ≤ i, j ≤ 2, 1 ≤ k, l ≤ n, we obtain that for s, t = 1, 2 Since the area of the exact core-shell The condition n k=1 F k i (u)(θ) = 0, implies that |f i (u)| ≤ C|γ|ǫ u Z , where i = 1, 2. The lemma then follows. Lemma 4.5. There exists c 2 > 0 such that when δ is small enough, for all u ∈ X ♭ 1.
To prove the third part it suffices to show that ΠS ′ (0) is from X ♭ onto Z ♭ . Note that ΠS ′ (0) is an unbounded self-adjoint operator on Z ♭ with the domain X ♭ ⊂ Z ♭ .
Finally one needs an estimate on the second Fréchet derivative of S, i.e. the third variation of J . Lemma 4.6. There exists C 3 > 0 such that for all φ ∈ Dom(S), the following estimates hold for u = (u 1 , We choose to denote the constants here by C 3 to remind the reader that C 3 is used in the estimate of the third variation of J . The proof of Lemma 4.6, which is skipped, is straight forward estimation, similar to the proofs of [22,Lemma 3.2], and [21, Lemma 6.1].
By Lemma 3.1, S(0) is a sum of a θ independent part, an r k i e iθ part, and a quantity of order O(|γ|ǫ 4 ). After one applies the projection operator Π, the θ independent part and r k i e iθ part vanish due to the fact that For φ ∈ W, by condition 1 of Theorem 1.1, (4.18), (5.5), (5.7), and (5.9) we deduce Let δ be small enough such that δ < bc2 2C1σ and Lemma 4.5 holds. Then T (φ) X ≤ bǫ 2 .
Therefore T maps W into itself.
The proof of this lemma is similar to that of [21,Lemmas 8.3 and 8.4], so we omit the details.