1 Introduction

Energy functionals entailing a direct competition between an attractive short-range force and a repulsive Coulombic long-range force have been studied intensively in recent years, to understand physical problems such as Gamow’s liquid-drop problem, and self-assembly of block copolymers. In Gamow’s liquid-drop model [Reference Gamow10], the volume of the nucleus $\Omega \subset \mathbb {R}^3$ is fixed, i.e., $|\Omega | = m$ with the parameter m being referred to as “mass.” The binding energy is given by

$$ \begin{align*} \mathcal{E}_{\text{liquid}}(\Omega) := \text{Per}(\Omega) + \frac{1}{8\pi} \int_{\Omega \times \Omega} \frac{{\text d} x {\text d} y}{|x-y|} , \end{align*} $$

where the first term is the perimeter (or surface area) of $\Omega $ , which arises due to the lower nucleon density near the nucleus boundary; the second term is a Coulomb-type one, introduced to account for the presence of positively charged protons [Reference Choksi, Muratov and Topaloglu3].

In Ohta and Kawasaki’s diblock copolymer model [Reference Ohta and Kawasaki22], the free energy is given by

$$ \begin{align*} \mathcal{E}_{\text{diblock}} (\Omega) := \text{Per}(\Omega) + \gamma \int_{\Omega \times \Omega} G(x,y) \; {\text d} x {\text d} y, \end{align*} $$

where the first term, i.e., the perimeter, favors a large ball; the second term prefers splitting, and models long-range interactions between monomers due to the connectivity of different subchains in copolymer molecules. Here,

$$ \begin{align*} G(x,y) = \frac{1}{4\pi |x-y| } + R(x,y) \end{align*} $$

is the zero-average Green’s function of the Laplace operator in $\mathbb {R}^3$ , $R(x,y)$ is the regular part of $G(x,y)$ , and $\gamma $ is the long-range interaction coefficient, determined by the percentage of each type monomer, the total number of monomers in a chain molecule, the repulsion between different monomers, and the average distance between two adjacent monomers [Reference Choksi and Ren6]. During each experiment, the total mass of each type monomer is fixed. So the energy is minimized under the mass constraint $|\Omega | = m$ .

In this paper, we study a model in ternary systems, introduced by Nakazawa and Ohta to study triblock copolymers [Reference Nakazawa and Ohta21], with vanishing mass fraction. A triblock copolymer is a chain molecule consisting of three types of subchains: a subchain of type A monomers is connected to a subchain of type B monomers, and then connected to a subchain of type C monomers. Block copolymers can be used as a material in artificial organ technology and controlled drug delivery.

The free energy of triblock copolymers, for the sharp interface model in $\mathbb {R}^2$ , was derived by Ren and Wei in [Reference Ren and Wei24, Reference Ren and Wei25] as the $\Gamma $ -limit of Nakazawa and Ohta’s diffuse interface model:

$$ \begin{align*} \mathcal{E}_{\text{triblock}} (\Omega_1, \Omega_2) := \frac{1}{2} \sum_{i=0}^2 \text{Per}(\Omega_i) + \sum_{i,j=1}^2 \gamma_{ij} \int_{\Omega_i \times \Omega_j} G(x,y) \; {\text d} x {\text d} y. \end{align*} $$

Here, $\Omega _0 = (\Omega _1\cup \Omega _2)^c$ , the perimeter term is defined by

$$\begin{align*}\frac{1}{2} \sum_{i=0}^2 \text{Per}(\Omega_i) = \sum_{0\le i< j\le 2}{\mathcal{H}}^2(\partial \Omega_i\cap \partial \Omega_j), \end{align*}$$

with $\partial $ denoting the (reduced) boundary, and the long-range interaction coefficients $\gamma _{ij} $ form a $2\times 2$ symmetric matrix. The choice of $\gamma _{ij}$ is nontrivial, as it must ensure that the perimeter and long-range interaction terms must be of comparable order. Therefore, finding the correct order of $\gamma _{ij}$ can be somewhat delicate, and can depend on the total masses. Using a “droplet” scaling argument, as done by Choksi and Peletier in [Reference Choksi and Peletier4, Section 3], [Reference Choksi and Peletier5, Section 3], and by Alama, Bronsard, the first author, and Wang in [Reference Alama, Bronsard, Lu and Wang1], it can be shown that the leading order of the free energy takes the form

(1.1) $$ \begin{align} E_0(\Omega_1,\Omega_2) = \sum_k e_0(|\Omega_{1,k}|,|\Omega_{2,k}| ), \qquad \Omega_i = \bigcup_k \Omega_{i,k},\quad i=1,2, \end{align} $$

with

$$ \begin{align*} e_0 &:[0,+\infty) \times [0,+\infty)\longrightarrow {\mathbb{R}},\\ e_0(m_1,m_2)&:=\inf \bigg\{ \sum_{0\le i< j\le 2}{\mathcal{H}}^2(\partial \Omega_i\cap \partial \Omega_j)\\ &\qquad + \sum_{i,j=1}^{2}\frac{ \Gamma_{ij} }{4\pi} \int_{\Omega_i\times \Omega_j} \frac{{\text d} x {\text d} y }{|x-y| } : |\Omega_i |=m_i,\ i=1,2 \bigg\}, \end{align*} $$

where $\Gamma _{ij}$ is a suitable scaling of $\gamma _{ij}$ . That is, $E_0$ seeks the optimal partition $\Omega _i = \bigcup _k \Omega _{i,k}$ , with each couple $(\Omega _{1,k},\Omega _{2,k})$ minimizing $e_0$ . Note that, generally, $\Omega _0$ represents the “background,” while $\Omega _i$ , $i=1,2$ represent the two types of “materials.” The fact that $e_0$ does not penalize interactions between $\Omega _0$ and $\Omega _i$ , $i=1,2$ , thus, corresponds to the (rather natural) assumption that neither type of material interacts with the background.

The fact that $\Omega _{i,k}$ and $\Omega _{j,l}$ do not interact when $k\neq l$ , is due to the fact that these represent masses located in different clusters. In the small volume-fraction limit, as argued in [Reference Alama, Bronsard, Lu and Wang1, Reference Choksi and Peletier4], the interaction terms between different mass clusters become negligible compared to the interaction within the same cluster.

Choksi and Peletier showed in [Reference Choksi and Peletier4, Theorem 4.2] that, when the domain is the unit torus $\mathbb {T}^3$ , in the small mass volume fraction regime, the first-order $\Gamma $ -limit of the free energies (see [Reference Choksi and Peletier4, Equation (1.8)])

$$ \begin{align*} E_\eta^{3d} (v):=\begin{cases} \eta \int_{\mathbb{T}^3} |\nabla v|{\text d} x +\eta \big\| v-\frac{1}{|\mathbb{T}^3|} \int_{\mathbb{T}^3}v{\text d} x \big\|_{{\mathcal{H}}^{-1}(\mathbb{T}^3),}^2 & \text{if } v\in BV(\mathbb{T}^3;\{0,\eta^{-3}\}),\\ +\infty,&\text{otherwise}, \end{cases} \end{align*} $$

is of the form

$$\begin{align*}\text{perimeter} + \text{long-range interaction},\end{align*}$$

i.e., (see [Reference Choksi and Peletier4, Equation (4.1)], and more in general [Reference Choksi and Peletier4, Section 4]),

$$ \begin{align*} E_0^{3d} (v) :=\begin{cases} {\displaystyle\sum_{k=0}^{\infty}} e_0(m_k), & \text{if } v={\displaystyle\sum_{k=0}^{\infty}} m_k \delta_{x_k}, \ {\displaystyle\sum_{k=0}^{\infty}} m_k=M=\text{total mass},\notag \\ +\infty, & \text{otherwise},\notag \end{cases} \end{align*} $$

with

$$\begin{align*}e_0^{3d}(m)= \inf\bigg\{ \int_{{\mathbb{R}}^3} |\nabla z| {\text d} x+ \|z\|_{H^{-1}({\mathbb{R}}^3)}^2:z\in BV({\mathbb{R}}^3;\{0,1\}), \ \|z\|_{L^1({\mathbb{R}}^3)}=M\bigg\}. \end{align*}$$

The $H^{-1}$ norm can be made explicit:

$$\begin{align*}\|z\|_{H^{-1}({\mathbb{R}}^3)}^2 = \int_{ {\mathbb{R}}^3\times{\mathbb{R}}^3 } G(|x-y|) z(x)z(y){\text d} x{\text d} y, \end{align*}$$

where G denotes the Green’s function of the Laplacian in ${\mathbb {R}}^3$ . That is, the minima seeks the optimal partition, in which each component minimizes the energy $e_0^{3d}$ . An analogous result, but for ternary systems in the two-dimensional torus, was obtained by Alama, Bronsard, the first author, and Wang in [Reference Alama, Bronsard, Lu and Wang1, Theorem 3.2].

With the same arguments from [Reference Alama, Bronsard, Lu and Wang1, Reference Choksi and Peletier4], it is possible to show that, again, with the domain being the unit torus $\mathbb {T}^3$ , in the small mass volume fraction regime, the first-order $\Gamma $ -limit of the free energies (which are the analog of [Reference Alama, Bronsard, Lu and Wang1, Equation (1.8)] for ternary systems in 3D)

$$ \begin{align*} E_{\mathrm{ternary},\eta}^{3d} (v_{1,\eta},v_{2,\eta}) &:=\begin{cases} f_{\eta}(v_{1,\eta},v_{2,\eta}), & \text{if } v_{1,\eta},v_{2,\eta}\in BV(\mathbb{T}^3;\{0,\frac{1}{\eta^{3}}\}),\\ +\infty,&\text{otherwise}, \end{cases}\\ f_{\eta}(v_{1,\eta},v_{2,\eta})&:= \frac{\eta}{2}\sum_{i=0}^2\int_{\mathbb{T}^3} |\nabla v_{i,\eta}|{\text d} x \\ &\qquad+\eta^4\sum_{i,j=1}^2 \gamma_{ij}\int_{\mathbb{T}^3\times\mathbb{T}^3}G_{{\mathbb{T}^3}}(|x-y|) v_{i,\eta}(x) v_{i,\eta}(y){\text d} x{\text d} y, \\ G_{{\mathbb{T}^3}} & :=\text{Green's function of the Laplacian in } \mathbb{T}^3 \text{ with zero average}, \end{align*} $$

can be again written in the form

(1.2) $$ \begin{align} E_{\mathrm{ternary},0}^{3d} (v_1,v_2)& :=\begin{cases} {\displaystyle\sum_{k=0}^{\infty}} e_0(m_{1,k},m_{2,k} ), & \text{if } v_i={\displaystyle\sum_{k=0}^{\infty}} m_{i,k} \delta_{x_{i,k}},\ {\displaystyle\sum_{k=0}^{\infty}} m_{i,k}=M_i,\\ +\infty, & \text{otherwise}, \end{cases} \\ M_i&=\text{total mass of type }i \text{ constituent},\qquad i=1,2, \notag \end{align} $$

and $\Gamma _{ij}\eta ^{-3}=\gamma _{ij}\ge 0 $ are coefficients penalizing the Coulomb interaction. Observe that the problem of minimizing $ E_{\mathrm{ternary},0}^{3d} $ is again fully determined once we fix the total masses $M_i$ and the interaction coefficients $\Gamma _{ij}$ . Each couple of sets $(\Omega _1,\Omega _2)$ , with the appropriate masses, and minimizing $e_0$ , is referred to as a “cluster.”

Next, we introduce the main energy of this paper: given connected sets $\Omega _i$ , with $\mathbf {1}_{\Omega _i} \in BV({\mathbb {R}}^3;\{0,1\})$ , $i=1,2$ , and $|\Omega _1\cap \Omega _2|=0$ , define the energy

(1.3) $$ \begin{align} E(\Omega_1,\Omega_2) &:= \sum_{0\le i< j\le 2}{\mathcal{H}}^2(\partial \Omega_i\cap \partial \Omega_j) + \sum_{i,j=1}^{2} \gamma_{ij} \int_{\Omega_i\times \Omega_j} |x-y|^{-1}{\text d} x{\text d} y, \end{align} $$

where $ \Omega _0 = (\Omega _1\cup \Omega _2)^c$ . Here, $\gamma _{ij}$ denote the interaction strengths, and are positive, of order $O(1)$ . Note that E is the analog of $e_0$ from (1.1), (1.2), and [Reference Alama, Bronsard, Lu and Wang1], and of $e_0^{3d}$ from [Reference Choksi and Peletier4], for ternary systems with domain ${\mathbb {R}}^3$ . Then, given disjoint unions

$$\begin{align*}\Big(\bigsqcup_k \Omega_{1,k},\bigsqcup_k \Omega_{2,k}\Big), \end{align*}$$

with $\Omega _{i,k}$ being the connected components, the total energy of this configuration is defined by

$$\begin{align*}\mathcal{E}\Big(\bigsqcup_k \Omega_{1,k},\bigsqcup_k \Omega_{2,k}\Big):= \sum_k E(\Omega_{1,k},\Omega_{2,k}). \end{align*}$$

Observe that $\mathcal {E}$ is the analog of [Reference Alama, Bronsard, Lu and Wang1, Equation (3.5)] and [Reference Choksi and Peletier4, Equation (4.1)], for 3D ternary systems. It is also worthy noting that $\mathcal {E}$ is similar to $\mathcal {E}_{\text {liquid}}$ , $\mathcal {E}_{\text {diblock}}$ , and $\mathcal {E}_{\text {triblock}}$ , as they are all of the form

$$\begin{align*}\text{perimeter} + \text{long-range interaction},\end{align*}$$

with the main difference being that $\mathcal {E}$ suppresses the interaction between different connected components.

In the following, when we say “optimal configuration,” unless otherwise specified, we mean a configuration $(\bigsqcup _k \Omega _{1,k},\bigsqcup _k \Omega _{2,k})$ minimizing $\mathcal {E}$ .

Existence of minimizers has been proven in [Reference Choksi and Peletier4], whose Sections 4 and 5 were dedicated to the 3D case, whereas Section 6 deal with the 2D case. In 2D, due to the fact that the Green’s function is a logarithmic term, the interaction was simply the product of the masses, hence it was equivalent to minimize the perimeter, subject to two mass constraints. It is well known that the double bubble is the unique such minimizer (see, e.g., [Reference Foisy, Garcia, Brock, Hodges and Zimba8, Reference Morgan and Wichiramala19] for the 2D case, and [Reference Hutchings, Morgan, Ritoré and Ros12] for the 3D case, and also [Reference Dorff, Lawlor, Sampson and Wilson7, Reference Milman and Neeman17, Reference Morgan and Ritoré18, Reference Reichardt, Heilmann, Lai and Spielman23]). In the ternary 3D case, however, such simplification is not available, and the shape of the minimizers is unclear, even for small masses. This is a significant hurdle, and studying the shape of minimizers is hindered by the lack of a quantitative isoperimetric inequality with two mass constraints in 3D

Therefore, a priori, it is even unclear whether optimal configurations have finitely many clusters, as we cannot exclude the presence of infinitely many components with very small masses. Our main result is to show that this is not the case:

Existence of optimal configurations can be shown using the same arguments as in Sections 4 and 5 of [Reference Choksi and Peletier4]. The proof of Theorem 1.1 is split into several steps: first, in Lemma 2.1, we show that the number of clusters made of one constituent type is bounded from above. Then, in Lemma 2.2, we show that there is at least one cluster that is relatively massive, i.e., we bound from below the mass of the largest cluster. Finally, in the crucial Lemma 2.3, and the subsequent Lemma 2.4, we bound from below the mass of any cluster. Since there the total amount of masses is given a priori, this allows us to infer Theorem 1.1.