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
$$ \begin{align} E_0(\Omega_1,\Omega_2) = \sum_k e_0(|\Omega_{1,k}|,|\Omega_{2,k}| ), \qquad \Omega_i = \bigcup_k \Omega_{i,k},\quad i=1,2, \end{align} $$
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
$$ \begin{align} E_{\mathrm{ternary},0}^{3d} (v_1,v_2)& :=\begin{cases} {\displaystyle\sum_{k=0}^{\infty}} e_0(m_{1,k},m_{2,k} ), & \text{if } v_i={\displaystyle\sum_{k=0}^{\infty}} m_{i,k} \delta_{x_{i,k}},\ {\displaystyle\sum_{k=0}^{\infty}} m_{i,k}=M_i,\\ +\infty, & \text{otherwise}, \end{cases} \\ M_i&=\text{total mass of type }i \text{ constituent},\qquad i=1,2, \notag \end{align} $$
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
$$ \begin{align} E(\Omega_1,\Omega_2) &:= \sum_{0\le i< j\le 2}{\mathcal{H}}^2(\partial \Omega_i\cap \partial \Omega_j) + \sum_{i,j=1}^{2} \gamma_{ij} \int_{\Omega_i\times \Omega_j} |x-y|^{-1}{\text d} x{\text d} y, \end{align} $$
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:
Theorem 1.1 Given total masses
$M_1$
,
$M_2$
, and interaction coefficients
$\gamma _{11},\gamma _{12},\gamma _{22}$
, there exists a computable constant
$K=K(M_1,M_2,\gamma _{11},\gamma _{22})$
such that any optimal configuration has at most K clusters.
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.
1.1 Notation
Since the position of the clusters is rarely relevant, in this paper, we denote by
$B_m$
, a ball of mass m.
2 Uniform upper bound on the number of clusters
The proof of Theorem 1.1 will be split over several lemmas. Throughout the entire section,
$M_i$
,
$i=1,2$
, will denote the total masses of type i constituent, and
$\gamma _{ij}$
,
$i,j=1,2$
will denote the interaction coefficients. These parameters completely determine the minimization problem for
$ \mathcal {E} $
in 3D. All the
$M_i$
and
$\gamma _{ij}$
will assumed to be given, and do not change throughout the section. Our proof will proceed as follows:
-
(1) First, in Lemma 2.1, we bound from above the number of clusters made purely of one constituent type. Such upper bound will depend only on
$M_i$
,
$\gamma _{ii}$
,
$i=1,2$
. -
(2) Then, in Lemma 2.2, we show that the total mass of the largest clusters cannot be too small. Such lower bound on the mass will depend only on
$M_i$
,
$\gamma _{ii}$
,
$i=1,2$
. -
(3) Finally, in Lemmas 2.3 and 2.4, we show that the total mass of each cluster is bounded from below by a constant depending only on
$M_i$
,
$\gamma _{ii}$
,
$i=1,2$
. Since there is only so much total mass (i.e.,
$M_1+M_2$
), this allows us to infer Theorem 1.1.
As we have no information on the shape of optimal configurations, we will often compare their energy against that of a suitable standard double bubble. Further information about the geometry of standard double bubbles are available in the Appendix.
Lemma 2.1 Consider an optimal configuration, made of clusters
$(\Omega _{1,k},\Omega _{2,k})$
,
$k\ge 1$
. Then
$$\begin{align*}\# \{ k: |\Omega_{1,k}||\Omega_{2,k}|=0 \} \end{align*}$$
is bounded from above by a constant depending only on
$M_i$
,
$\gamma _{ii}$
,
$i=1,2$
.
Proof The proof follows arguments similar to those used to show the sub-addictivity formula (2.1) in [Reference Lu and Otto16]. It is well known (see, e.g., [Reference Bonacini and Cristoferi2, Reference Frank and Lieb9, Reference Knüpfer and Muratov13–Reference Knüpfer, Muratov and Novaga15, Reference Muratov and Knüpfer20], and the references therein) that there exist
$m_{i,B}=m_{i,B}(\gamma _{ii})>0$
,
$i=1,2$
, such that, for all
$m\le m_{i,B}(\gamma _{ii})$
, the minimizer of
$$\begin{align*}\inf_{ |X| = m} \bigg\{ {\mathcal{H}}^2(\partial X ) + \gamma_{ii}\int_{X\times X} |x-y|^{-1}{\text d} x{\text d} y \bigg\}\end{align*}$$
is given by
$B_m$
. Since
${\mathcal {H}}^2(\partial B_m )$
(resp.
$\int _{X\times X} |x-y|^{-1}{\text d} x{\text d} y $
) scales like
$m^{2/3}$
(resp.
$m^{5/3}$
), the perimeter term is dominating for all sufficiently small masses. Thus, there exist geometric constants
$m_{i,S}=m_{i,S}(\gamma _{ii})\le m_{i,B}(\gamma _{ii})$
such that
$$ \begin{align*} {\mathcal{H}}^2(\partial B_{m_1} ) &+ \gamma_{ii}\int_{B_{m_1}\times B_{m_1}} |x-y|^{-1}{\text d} x{\text d} y\\ &\qquad+{\mathcal{H}}^2(\partial B_{m_2} ) + \gamma_{ii}\int_{B_{m_2}\times B_{m_2}} |x-y|^{-1}{\text d} x{\text d} y\\ &> {\mathcal{H}}^2(\partial B_{m_1+m_2} ) + \gamma_{ii}\int_{B_{m_1+m_2}\times B_{m_1+m_2}} |x-y|^{-1}{\text d} x{\text d} y, \end{align*} $$
for all
$m_1,m_2\le m_{i,S}(\gamma _{ii})$
, i.e., combining the two balls is energetically favorable whenever
$m_1,m_2\le m_{i,S}(\gamma _{ii})$
. Thus, we cannot have two balls of the type i constituent, both with masses less than
$m_{i,S}(\gamma _{ii})$
. Since the total mass is
$M_1+M_2<+\infty $
, the proof is complete.
Lemma 2.2 Consider an optimal configuration, made of clusters
$(\Omega _{1,k},\Omega _{2,k})$
,
$k\ge 1$
. Then
$$\begin{align*}m_i^+:=\sup_k m_{i,k} ,\quad m_{i,k}:=|\Omega_{i,k}| , \qquad i=1,2,\end{align*}$$
is bounded from below by
$$\begin{align*}\min\bigg\{ \frac{M_i}{2}, \bigg( \frac{\sqrt[3]{36\pi} M_i }{4\sum_{i=1}^2 [\sqrt[3]{36\pi} M_i^{2/3} + \gamma_{ii}\int_{ B_{M_i}\times B_{M_i} } |x-y|^{-1}{\text d} x{\text d} y]}\bigg)^3\bigg\},\qquad i=1,2 .\end{align*}$$
Note that, curiously, this lower bound is independent of
$\gamma _{12}$
. As it will be clear from the proof, this is due to the fact that an upper bound for the energy of an optimal configuration is given by the energy of two balls of masses
$M_1$
and
$M_2$
, respectively. Such bound is clearly independent of
$\gamma _{12}$
.
Proof The idea is that, for very small masses, the perimeter term is sub-addictive and dominating. Assume
$m_i^+\le M_i/2$
, as otherwise
$M_i/2$
is already a lower bound. Note that
$$\begin{align*}E(\Omega_{1,k},\Omega_{2,k}) \ge \mathcal{S}(m_{1,k},m_{2,k}) \qquad\forall k\ge 1, \end{align*}$$
where
$$ \begin{align} \mathcal{S}(m_{1},m_{2}) = \text{perimeter of the standard double bubble with masses } m_1 \text{ and } m_2, \end{align} $$
and, by [Reference Hutchings11, Theorem 4.2] (applied with
$v_1=m_1$
,
$x=v_2=m_2$
,
$n=3$
), we get the following isoperimetric inequality for clusters
$$\begin{align*}\mathcal{S}(m_1,m_2)\ge {\frac{\sqrt[3]{36\pi}}{2}}\sum_{i=1}^2 m_i^{2/3}. \end{align*}$$
Thus, the total energy of our optimal configuration satisfies
$$ \begin{align*} \sum_{k\ge 1}E(\Omega_{1,k},\Omega_{2,k}) \ge {\frac{\sqrt[3]{36\pi}}{2}}\sum_{i=1}^2 \sum_{k\ge 1} m_{i,k}^{2/3}. \end{align*} $$
By the concavity of the function
$t\mapsto t^{2/3}$
, the sum
$\sum _{k\ge 1} m_{i,k}^{2/3}$
is minimum when
$m_{i,k}\in \{0,m_i^+\}$
for all k. Since
$\sum _{k\ge 1} m_{i,k}=M_i$
, there are at least
$\lfloor \frac {M_i}{m_i^+} \rfloor $
many clusters containing type i constituents, thus
$$ \begin{align*} \sum_{k\ge 1}E(\Omega_{1,k},\Omega_{2,k}) &\ge {\frac{\sqrt[3]{36\pi}}{2}}\sum_{i=1}^2 \sum_{k\ge 1} m_{i,k}^{2/3} \ge {\frac{\sqrt[3]{36\pi}}{2}} \sum_{i=1}^2 \Big\lfloor \frac{M_i}{m_i^+} \Big\rfloor (m_{i}^+)^{2/3}\\ &\ge {\frac{\sqrt[3]{36\pi}}{2}}\sum_{i=1}^2 \frac{M_i-m_i^+}{(m_{i}^+)^{1/3}} \ge {\frac{\sqrt[3]{36\pi}}{4}}\sum_{i=1}^2 \frac{M_i}{(m_{i}^+)^{1/3}}. \end{align*} $$
Since our configuration was an optimal one, its energy does not exceed that of two balls, which we denote by
$B_{M_1}$
and
$B_{M_2}$
, of masses
$M_1$
and
$M_2$
, respectively. Thus, the above line continues as
$$ \begin{align*} {\frac{\sqrt[3]{36\pi}}{4}} \sum_{i=1}^2 \frac{M_i}{(m_{i}^+)^{1/3}} &\le \sum_{k\ge 1}E(\Omega_{1,k},\Omega_{2,k})\\ &\le \sum_{i=1}^2 \Big[\sqrt[3]{36\pi} M_i^{2/3} + \gamma_{ii}\int_{ B_{M_i}\times B_{M_i} } |x-y|^{-1}{\text d} x{\text d} y\Big] , \end{align*} $$
hence
$$\begin{align*}(m_{i}^+)^{1/3}\ge {\frac{\sqrt[3]{36\pi}}{4}} \frac{ M_i }{\sum_{i=1}^2 [\sqrt[3]{36\pi} M_i^{2/3} + \gamma_{ii}\int_{ B_{M_i}\times B_{M_i} } |x-y|^{-1}{\text d} x{\text d} y]}, \end{align*}$$
and the proof is complete.
Lemma 2.3 Consider an optimal configuration, made of clusters
$(\Omega _{1,k},\Omega _{2,k})$
,
$k\ge 1$
. Assume
$\sup _k |\Omega _{1,k}|$
and
$\sup _k |\Omega _{2,k}|$
are achieved on different clusters, i.e., without loss of generality,
$$\begin{align*}|\Omega_{1,1}|=m_1^+=\sup_k |\Omega_{1,k}|,\qquad |\Omega_{2,2}|=n_2^+=\sup_k |\Omega_{2,k}|.\end{align*}$$
Then
$$\begin{align*}\inf_k \sum_{i=1}^2 |\Omega_{i,k}| \end{align*}$$
is bounded from below by a constant depending only on
$M_i$
,
$\gamma _{ii}$
,
$i=1,2$
.
Proof Consider a cluster
$(\Omega _{1,k},\Omega _{2,k})$
, with
$k\ge 3$
, and let
$$\begin{align*}m_2:= |\Omega_{2,1}|,\qquad n_1:=|\Omega_{1,2}|,\qquad \varepsilon_i:=|\Omega_{i,k}|>0,\ i=1,2.\end{align*}$$
Note that
$m_1^+\ge n_1$
,
$n_2^+\ge m_2$
. The construction will be slightly different depending on the values of
$\frac {m_1^+}{m_2}$
,
$\frac {m_2}{n_2^+}$
, and
$\frac {\varepsilon _1}{\varepsilon _2}$
.
Case 1:
$\frac {m_1^+}{m_2}\ge \frac {\varepsilon _1}{\varepsilon _2}$
. Consider the competitor constructed in the following way (see Figure 1).
-
• Move mass
$\varepsilon _1$
(resp.
$rm_2$
, with
$r:=\frac {\varepsilon _1}{m_1^+}\le 1$
) of type I (resp. type II) constituent from the cluster
$(\Omega _{1,k},\Omega _{2,k})$
to
$(\Omega _{1,1},\Omega _{2,1})$
. This is possible, since we are discussing the case
$\frac {m_1^+}{m_2}\ge \frac {\varepsilon _1}{\varepsilon _2}$
, i.e.,
$rm_2=\varepsilon _1\frac {m_2}{m_1^+}\le \varepsilon _2$
. -
• Replace
$(\Omega _{1,k},\Omega _{2,k})$
and
$(\Omega _{1,1},\Omega _{2,1})$
with
$B_{\varepsilon _2 -rm_2}$
(of type II constituent) and
$ (\tilde \Omega _{1,1},\tilde \Omega _{2,1}):= (1+r)^{1/3}(\Omega _{1,1},\Omega _{2,1})$
, while every other cluster remains unaltered.
Figure 1 Schematic representation of the construction of the competitor: original clusters (top), and modified clusters (bottom). Though the objects in question are three-dimensional, for better clarity, we represented the construction in two dimensions. Only the affected clusters are represented here. The clusters are drawn deliberately deformed, to emphasize the fact that we do not know the clusters’ precise shapes.
Now, we estimate the change in energy. Since our initial configuration was optimal,
$$ \begin{align} 0&\le E((1+r)^{1/3}(\Omega_{1,1},\Omega_{2,1}))+E(\emptyset, B_{\varepsilon_2 -rm_2})\notag\\ &\qquad -E(\Omega_{1,1},\Omega_{2,1})-E(\Omega_{1,k},\Omega_{2,k}). \end{align} $$
By a straightforward scaling argument,
$$ \begin{align*} E&((1+r)^{1/3}(\Omega_{1,1},\Omega_{2,1})) \\& = (1+r)^{2/3} \sum_{0\le i< j\le 2}{\mathcal{H}}^2(\partial \Omega_{i,1}\cap \partial \Omega_{j,1}),\qquad\Omega_{0,1}:=(\Omega_{1,1}\cup \Omega_{2,1})^c,\\&\qquad +(1+r)^{5/3} \sum_{i,j=1}^2 \gamma_{ij}\int_{\Omega_{i,1}\times \Omega_{j,1}}|x-y|^{-1}{\text d} x{\text d} y\\&\le (1+r) \sum_{0\le i< j\le 2}{\mathcal{H}}^2(\partial \Omega_{i,1}\cap \partial \Omega_{j,1})+(1+3r) \sum_{i,j=1}^2 \gamma_{ij}\int_{\Omega_{i,1}\times \Omega_{j,1}}|x-y|^{-1}{\text d} x{\text d} y\\&\le (1+3r) \bigg[\underbrace{ \sum_{0\le i< j\le 2}{\mathcal{H}}^2(\partial \Omega_{i,1}\cap \partial \Omega_{j,1})+ \sum_{i,j=1}^2 \gamma_{ij}\int_{\Omega_{i,1}\times \Omega_{j,1}}|x-y|^{-1}{\text d} x{\text d} y}_{ = E(\Omega_{1,1},\Omega_{2,1}) }\bigg], \end{align*} $$
where we used the estimates
$$ \begin{align*} (1+r)^{2/3} \le 1+r\le 1+3r,\qquad (1+r)^{5/3} \le (1+r)^2 \overset{(r\le 1)}{\le}1+3r. \end{align*} $$
Thus, in view of Lemma 2.2,
$$ \begin{align} E((1+r)^{1/3}(\Omega_{1,1},\Omega_{2,1}))& -E(\Omega_{1,1},\Omega_{2,1})\notag\\ & \le 3r E(\Omega_{1,1},\Omega_{2,1})\le \varepsilon_1 H_1(M_1,M_2,\gamma_{11},\gamma_{22}), \\ H_1(M_1,M_2,\gamma_{11},\gamma_{22})&:= \sum_{i=1}^2 \frac{3}{m_1^+}\bigg[\sqrt[3]{36\pi} M_i^{2/3} + \gamma_{ii}\int_{ B_{M_i}\times B_{M_i} } |x-y|^{-1}{\text d} x{\text d} y\bigg] .\notag \end{align} $$
Now, we estimate
$E(\emptyset ,B_{\varepsilon _2 -rm_2}) -E(\Omega _{1,k},\Omega _{2,k})$
:
$$ \begin{align*} E(\emptyset,B_{\varepsilon_2 -rm_2}) &-E(\Omega_{1,k},\Omega_{2,k}) \le \mathcal{S}(0,\varepsilon_2 -rm_2) -\mathcal{S}(\varepsilon_1,\varepsilon_2 )\\ &=\mathcal{S}(0,\varepsilon_2 -rm_2)-\mathcal{S}(\varepsilon_1,\varepsilon_2-rm_2 )+\mathcal{S}(\varepsilon_1,\varepsilon_2-rm_2 ) -\mathcal{S}(\varepsilon_1,\varepsilon_2 )\\ & \le -c_1 \varepsilon_1^{2/3} ,\qquad c_1:=\frac{\sqrt[3]{36\pi}}{2}, \end{align*} $$
where the last line is due to [Reference Hutchings11, Theorem 3.2], which gives
$$\begin{align*}\mathcal{S}(\varepsilon_1,\varepsilon_2-rm_2 ) -\mathcal{S}(\varepsilon_1,\varepsilon_2 ) \le 0,\end{align*}$$
and [Reference Hutchings11, Theorem 4.2] (applied with
$v_1=\varepsilon _1$
,
$x=v_2=\varepsilon _2 -rm_2$
,
$n=3$
), which gives
$$ \begin{align*} \mathcal{S}(\varepsilon_1,\varepsilon_2 -rm_2) & \ge \frac{\sqrt[3]{36\pi}}{2} [ \varepsilon_1^{2/3} + (\varepsilon_2 -rm_2)^{2/3}+(\varepsilon_1+\varepsilon_2 -rm_2)^{2/3}] \\ &\ge \frac{\sqrt[3]{36\pi}}{2} [ \varepsilon_1^{2/3} +2 (\varepsilon_2 -rm_2)^{2/3}] =\frac{\sqrt[3]{36\pi}}{2}\varepsilon_1^{2/3}+ \underbrace{ \sqrt[3]{36\pi}(\varepsilon_2 -rm_2)^{2/3}}_{=\mathcal{S}(0,\varepsilon_2 -rm_2)}. \end{align*} $$
Combining with (2.2) and (2.3) gives the necessary condition
$$ \begin{align} 0&\le E((1+r)^{1/3}(\Omega_{1,1},\Omega_{2,1}))+E(\emptyset, B_{\varepsilon_2 -rm_2}) -E(\Omega_{1,1},\Omega_{2,1})-E(\Omega_{1,k},\Omega_{2,k})\notag\\ &\le \varepsilon_1 H_1(M_1,M_2,\gamma_{11},\gamma_{22}) -c_1 \varepsilon_1^{2/3} , \end{align} $$
hence
$$\begin{align*}\varepsilon_1^{1/3}\ge H_1(M_1,M_2,\gamma_{11},\gamma_{22}) c_1^{-1},\end{align*}$$
thus, completing the proof for this case.
Case 2:
$\frac {n_2^+}{n_1}\ge \frac {\varepsilon _2}{\varepsilon _1}$
. The competitor constructed in a way similar to the previous case.
-
• Move mass
$\varepsilon _2$
(resp.
$rn_1$
, with
$r:=\frac {\varepsilon _2}{n_2^+}\le 1$
) of type II (resp. type I) constituent from the cluster
$(\Omega _{1,k},\Omega _{2,k})$
to
$(\Omega _{1,2},\Omega _{2,2})$
. This is possible since we are discussing the case
$\frac {n_2^+}{n_1}\ge \frac {\varepsilon _2}{\varepsilon _1}$
, i.e.,
$rn_1=\varepsilon _2\frac {n_1}{n_2^+}\le \varepsilon _1$
. -
• Replace
$(\Omega _{1,k},\Omega _{2,k})$
and
$(\Omega _{1,2},\Omega _{2,2})$
with
$B_{\varepsilon _1 -rn_1}$
(of type I constituent) and
$(1+r)^{1/3}(\Omega _{1,2},\Omega _{2,2})$
, while every other cluster remains unaltered.
Then the proof proceeds like in the previous case. With the same arguments from Case 1, we obtain
$$ \begin{align*} E({\kern-1pt}(1{\kern-1pt}+{\kern-1pt}r)^{1/3}(\Omega_{1,2},\Omega_{2,2}))& -E(\Omega_{1,2},\Omega_{2,2}) {\kern-1pt}\le{\kern-1pt} 3r E(\Omega_{1,2},\Omega_{2,2}){\kern-1pt}\le{\kern-1pt} \varepsilon_2 H_2(M_1,M_2,\gamma_{11},\gamma_{22}),\notag\\ H_2(M_1,M_2,\gamma_{11},\gamma_{22})&:= \sum_{i=1}^2 \frac{3}{n_2^+}\bigg[\sqrt[3]{36\pi} M_i^{2/3} + \gamma_{ii}\int_{ B_{M_i}\times B_{M_i} } |x-y|^{-1}{\text d} x{\text d} y\bigg], \end{align*} $$
which is the analog of (2.3), and
$$ \begin{align*} 0&\le E((1+r)^{1/3}(\Omega_{1,2},\Omega_{2,2}))+E(\emptyset, B_{\varepsilon_1 -rn_1}) -E(\Omega_{1,2},\Omega_{2,2})-E(\Omega_{1,k},\Omega_{2,k})\notag\\ &\le \varepsilon_2 H_2(M_1,M_2,\gamma_{11},\gamma_{22}) -c_2 \varepsilon_2^{2/3}, \end{align*} $$
for some computable, purely geometric constant
$c_2>0$
, which is the analog of (2.4). Thus
$$\begin{align*}\varepsilon_2^{1/3} \ge H_2(M_1,M_2,\gamma_{11},\gamma_{22}) c_2^{-1}, \end{align*}$$
concluding the proof for this case.
Finally, note that the above two cases are exhaustive: if Case 1 does not hold, i.e.,
$\frac {\varepsilon _2}{\varepsilon _1}< \frac {m_2}{m_1^+}$
, using
$m_1^+\ge n_1$
,
$n_2^+\ge m_2$
, we get
$$\begin{align*}\frac{\varepsilon_2}{\varepsilon_1}< \frac{m_2}{m_1^+}\le \frac{n_2^+}{n_1}, \end{align*}$$
i.e., Case 2 holds. The proof is thus complete.
Lemma 2.4 Consider an optimal configuration, made of clusters
$(\Omega _{1,k},\Omega _{2,k})$
,
$k\ge 1$
. Assume
$\sup _k |\Omega _{1,k}|$
and
$\sup _k |\Omega _{2,k}|$
are achieved on the same clusters, i.e., without loss of generality,
$$\begin{align*}|\Omega_{i,1}|=m_i^+=\sup_k |\Omega_{i,k}|,\quad i=1,2.\end{align*}$$
Then
$$\begin{align*}\inf_k \sum_{i=1}^2 |\Omega_{i,k}| \end{align*}$$
is again bounded from below by a constant depending only on
$M_i$
,
$\gamma _{ii}$
,
$i=1,2$
.
Proof We rely on Lemma 2.3: Consider another cluster
$(\Omega _{1,k},\Omega _{2,k})$
,
$k\ge 2$
. Let
$|\Omega _{1,k}|=\varepsilon _1>0$
,
$|\Omega _{2,k}|=\varepsilon _2>0$
, and note that one of the following cases must hold.
-
(1) If
$\frac {m_1^+}{m_2^+} \ge \frac {\varepsilon _1}{\varepsilon _2}$
, then we can use the construction from Case 1 of Lemma 2.3. -
(2) If
$\frac {m_1^+}{m_2^+} \le \frac {\varepsilon _1}{\varepsilon _2}$
, i.e.,
$\frac {m_2^+}{m_1^+} \ge \frac {\varepsilon _2}{\varepsilon _1}$
, then we can use the construction from Case 2 of Lemma 2.3.
The proof is thus complete.
A Appendix: geometry of the standard double bubble
In [Reference Hutchings, Morgan, Ritoré and Ros12], it was shown that the three-dimensional standard double bubbles has the least surface area among all sets enclosing two regions of given volumes.
Geometrically, the standard double bubble is a surface of revolution, with all the three surfaces being part of spheres, meeting at 120 degrees (see Figures A.1 and A.2).
Figure A.1 The standard double bubble in
${\mathbb {R}}^3$
: if the two bubbles that meet have equal volumes, the shared surface between them is a flat disk. But in the case of unequal volumes, the smaller bubble, given its larger internal pressure, will bow slightly into the larger bubble. In either scenario, the two bubbles always meet at angles of 120 degrees. Credit: John M. Sullivan, Technical University of Berlin and University of Illinois at Urbana–Champaign.
Figure A.2 Cross section of a standard double bubble.
Below, we collect several results, used in the proof of Theorem 1.1, on the function
$\mathcal {S}$
introduced in (2.1).
Lemma A.1 [Reference Hutchings11, Theorem 3.2] The function
$\mathcal {S}$
is strictly concave: given
$m_i,n_i\ge 0$
,
$i=1,2$
, it holds
$$ \begin{align*} \mathcal{S}( (1-t) m_1+tn_1 , (1-t) m_2+tn_2) &> (1-t) \mathcal{S}( m_1 , m_2) +t \mathcal{S}( n_1 , n_2) \end{align*} $$
for all
$t>0$
.
Lemma A.2 [Reference Hutchings11, Corollary 3.3] The function
$\mathcal {S}( m_1 , m_2)$
is increasing in both variables.
Lemma A.3 [Reference Hutchings11, Theorem 4.2] Suppose that in a minimal enclosure of volumes
$m_1$
and
$m_2$
in
${\mathbb {R}}^3$
, with the latter having a connected component with volume
$x> 0$
. Then
$$ \begin{align*} \frac{2\mathcal{S}(m_1,m_2)}{c_1} &\ge m_2 x^{-1/3}+m_1^{2/3} +(m_1+m_2)^{2/3} ,\\ c_1&:=\sqrt[3]{36\pi} = \text{surface area of the unit ball in } {\mathbb{R}}^3. \end{align*} $$
Acknowledgment
We are grateful to Chong Wang for useful discussions and suggestions.
