数理解析研究所講究録
第 1924 巻 2014 年 78-96

78

Pattern formation in multi-component inhibitory systems
Xiaofeng Ren *\dagger
Department of Mathematics
The George Washington University
Washington, DC 20052, USA
September 15, 2014

Abstract
Pattern formation can be understood as an outcome of competing short range and long range interactions. This principle is demonstrated with a minimal, geometric model that assigns the free energy
to each partition of a domain into two disjoint sets. The two sets represent the two components of abinary system. The free energy consists of two terms: a short range term that is the perimeter between
the two sets, and a long range term that is given by a nonlocal quantity. The nonlocal term behaves as:
an inhibitor that limits growth and spreading. Because of this property, the system is termed a binary
inhibitory system. The free energy functional is naturally defined on the class of Cacciopolli sets with afixed volume, where one can also define the notion of stationary points. When the preblem is posed in
one dimension, all stationary points can be found. In higher dimensions, many stationary points havebeen discovered.
A pattern observed in the physical world is a stationary point with additional properties. First, it
must be stable in some sense; second it must be somewhat periodic. Typically a physical pattern is
an assembly of many copies of a geometric object. An example is a stationary assembly of discs. The
construction of this assembly involves several intricate ideas, which can be applied to other stationary

assemblies.
Recent studies also deal with a ternary inhibitory system of three components. The free energy ofthis system is again a sum of a short interaction energy of perimeter and a long range nonlocal energy.
Unique in the ternary system is the phenomenon of triple junction where the three components come tomeet at the same place. In an appropriate parameter range, there is a stable stationary assembly of two
dimensional double bubbles.

A binary inhibitory system

1

Patterns appear in physical and biological systems as outcomes of self-organization principles. One finds
examples in morphological phases of block copolymers, animal coats, and skin pigmentation. Common inthese pattern-forming systems is that a deviation from homogeneity has a strong positive feedback on its
further increase. On its own, it would lead to an unlimited increase and spreading. Pattern formation
requires in addition a longer ranging confinement of the locally self-enhancing process.
Therefore at the minimum pattern formation requires two mechanisms: growth and inhibition. One of
the most elegant inhibitory systems is a sharp interface limit model derived from the Ohta-Kawasaki [21]density functional theory. It was first introduced to the mathematical community by Nishiura and Ohnishi[20]. The system is binary because it consists of two components. It is a geometric variational problem on
a bounded, open, and suﬃciently smooth set $D$ of
This set is partitioned into two disjoint subsets$\mathbb{R}^{n}$$\Omega$’Phone: 1202994-6791; Fax: 1202994-6760; -mail: ren@gwu.edu
Supported in part by NSF grant DMS-1311856 and Simons Foundation Collaboration Grant for Mathematicians #245729.
$E$$\dagger$79, each of which is occupied by one of the two components of the system. The free energy functionalandtakes the form
(1.1).
$D\backslash \Omega$$\mathcal{J}_{B}(\Omega)=\frac{1}{n-1}\mathcal{P}_{D}(\Omega)+\frac{\gamma}{2}\int_{D}|(-\Delta)^{-1/2}(\chi_{\Omega}-\omega)|^{2}dx$

for subsets$\Omega$of $D$ of prescribed measure. Namely$\mathcal{A}=${$\Omega$is in

is Lebesgue measurable and$\Omega\subset D:\Omega$$|\Omega|=\omega|D|$

(1.2)

}

where $\omega\in(0,1)$ is a parameter of the system. Hereand $|D|$ stand for the Lebesgue measures of and$D$ respectively. The constantin (1.1) and the half in are not essential. We put them in (1.1) to havea better looking Euler-Lagrange equation later.
in $D$ . If
has
Growth is generated by the first term in (1.1). Hereis the perimeter ofboundary, thenis simply the area of the part of the boundary of that is inside $D$ , i.e. the area of. For a general Lebesgue measurable set , the perimeter is defined by$\Omega$$|\Omega|$$\frac{1}{n-1}$$f2$$C^{1}$$\Omega$$\Omega$$\mathcal{P}_{D}(\Omega)$$\Omega$$\mathcal{P}_{D}(\Omega)$$\partial\Omega\cap D$$\Omega$(1.3)$\mathcal{P}_{D}(\Omega)=\sup\{\int_{\Omega}divg(x)dx:9\in C_{0}^{1}(D, \mathbb{R}^{n}) , |g(x)|\leq 1\forall x\in D\}$

where $divg$ is the divergence of thevector field on $D$ with compact support and $|g(x)|$ stands for the
Euclidean norm of the vector
The reader who is familiar with BV functions will recognize that,and $\chi_{\Omega}(x)=0$ ifmeans that , the characteristic function of $\Omega(\chi_{\Omega}(x)=1$ if$D$is a BV function in . If
is called a Caccioppoli set. See for instance [7, 9] for more on
BV functions and Caccioppoli sets. To make this term small, likes to form a large region bounded by asurface of small area. and shares the boundary with $D$ as much as possible.
The number in (1.1) is a positive constant, the second parameter after . The second term in (1.1) is
responsible for inhibition. The operator $(-\Delta)^{-1/2}$ is defined by the Poisson’s equation. Given $f\in L^{2}(D)$such that $\int_{D}f(x)dx=0$ , let
be the solution of the following Poisson’s equation with the Neumannboundary condition:$C^{1}$$g$$g(x)\in \mathbb{R}^{n}$$\mathcal{P}_{D}(\Omega)<0$$x\in\Omega$$\chi_{\Omega}$$\mathcal{P}_{D}(\Omega)<\infty,$$x\in D\backslash \Omega)$$\Omega$$\Omega$$\omega$$\gamma$$u$$-\Delta u=f$

in $D,$$\partial_{n}u=0$on$\partial D,$(1.4)$\int_{D}u(x)dx=0$ .

In (1.4). Because the integral of is , the partial
stands for the outward normal derivative at
diﬀerential equation with the boundary condition is solvable. The solution is unique up to an additive
constant. The condition $\int_{D}u(x)dx$ fixes this constant and gives us a unique solution. The map $farrow u$ fromthe space of $\{f\in L^{2}(D) : \int_{D}f(x)dx=0\}$ to itself given above is the operator
Since this operatorin (1.1). Likeis bounded, self-adjoint, and positive definite, it has a positive square root, which ismust
is a nonlocal operator. For the second term in (1.1) to be small, the function
have frequent fluctuation.
Therefore the two terms in (1.1) have diﬀerent preferences. On the other hand, the two terms are eﬀectiveat diﬀerent length scales. The perimeter term is more central at a small length scale so it forces the boundaryof to be everywhere close to a surface of constant mean curvature. The second term is more important at
a longer distance so that it often makes to have a near periodic shape.
, whereOne can also study
with the periodic boundary condition. In other words take $D$ to be
is a discrete subgroup ofhasisomorphic toThis simplifies the problem somewhat because, and one often needs to find ways
no boundary. On the other hand there is translation invariance on
to remove any degeneracy related to this invariance.
$\partial D$$\partial_{n}$

$f$

$0$

$(-\triangle)^{-1}$$(-\triangle)^{-1/2}$$(-\Delta)^{-1},$$(-\triangle)^{-1/2}$$\chi_{\Omega}$$\Omega$$\Omega$$\mathbb{R}^{n}/\Lambda$$\mathcal{J}_{B}$$\Lambda$$\mathbb{R}^{n}$$\mathbb{Z}^{n}$$\mathbb{R}^{n}/\Lambda$$\mathbb{R}^{n}/\Lambda$

There is a related variational problem that allows the two components to mix. Instead of using a subset
of $D$ , one uses a function defined on $D$ that represents the concentration of one of the two components. If$u(x)=1$ , then the point $x\in D$ is occupied by the first component; if $u(x)=0$ , then the point is occupiedby the second component; if $0<u(x)<1$ , then the point is taken by a mixture of the two components.
The situation $u(x)\not\in[0$ , 1 is not physical and can be mathematically ruled out when one studies stationary$\Omega$

$u$

$x$

$]$

80
points. The free energy of this more complex system is(1.5)$\mathcal{I}_{B,\epsilon}=\int_{D}[\frac{\epsilon^{2}}{2}|\nabla u|^{2}+\frac{u^{2}(1-u)^{2}}{4}+\frac{\epsilon\gamma}{2}|(-\Delta)^{-1/2}(u-\omega)|^{2}]dx$

where

$u$

is in the admissible set$\mathcal{A}_{I}=\{u\in H^{1}(D) : \frac{1}{|D|}\int_{D}u(x)dx=\omega\}$

.

(1.6)

The additional parameter is positive. It is large if the temperature is high and small if the temperature islow. A stationary point of
is a solution to the following integro-diﬀerential equation.
$\epsilon$$\mathcal{I}_{B,\epsilon}$

in $D,$$-\epsilon^{2}\Delta u+u(u-1/2)(u-1)+\epsilon\gamma(-\Delta)^{-1}(u-\omega)=\lambda$$\partial_{n}u=0$on$\partial D,$$\int_{D}u(x)dx=0$ .

(1.7)

In this model there are no longer clearly defined interfaces separating the two components in a stationary
point. Instead, when is small, one can often identify regions whose width is of the order where thefunction changes quickly between values close to and values close to 1, These regions are viewed asinterfaces, or more precisely, diﬀusive interfaces. The problem
is called a diﬀusive interface model.
De Giorgi’s Gamma-convergence theory connects
to. As
Gamma-converges to. The reader may find the details of this story in [25]. One consequence of this theory is that asa global minimizer ofconverges to a global minimizer of. Conversely, if
has an isolated local
minimizer , then for small
has a local minimizer
such that$\epsilon$$\epsilon$

$0$

$u$

$\mathcal{I}_{B,\epsilon}$$\epsilonarrow 0,$$\mathcal{J}_{B}$$\mathcal{I}_{B,\epsilon}$$\epsilon^{-1}\mathcal{I}_{B,\epsilon}$$\mathcal{J}_{B}$$\epsilonarrow 0,$$\mathcal{J}_{B}$$\mathcal{I}_{B,\epsilon}$$\mathcal{J}_{B}$$\Omega$$\epsilon,$$\mathcal{I}_{B,\epsilon}$$u_{\epsilon}$$\int_{D}|u_{\epsilon}-\chi_{\Omega}|dxarrow 0$2

as$\epsilonarrow 0$

.

(1.8)

Global minimizer and first variation

I start this section by showing that

Theorem 2.1 There exists$\Omega_{*}\in \mathcal{A}$$\mathcal{J}_{B}$always has a global minimizer.

such that$\mathcal{J}_{B}(\Omega_{*})= inf\mathcal{J}_{B}(\Omega)$.

$\Omega\in A$

Proof. The functional$\mathcal{J}_{B}$is bounded from below by , so there exists a sequence of$0$$\lim_{jarrow\infty}\mathcal{J}_{B}(\Omega_{j})=\inf_{\Omega\in A}\mathcal{J}_{B}(\Omega)$$\Omega_{j}$.

in$\mathcal{A}$such that(2.1)

Recall that the BV norm is defined as follows. For any function $f\in L^{1}(D)$ , let$[f]_{BV}= \sup\{\int_{D}f(x)divg(x)dx:g\in C_{0}^{1}(D,\mathbb{R}^{n}) , |g(x)|\leq 1\forall x\in D\}$

Clearly$[f]_{BV}=\mathcal{P}_{D}(\Omega)$if$f=\chi_{\Omega}$. If $[f]_{BV}<\infty$ , we say that

$f$

.

(2.2)

is a BV function (a function of boundedvariation) and define the BV norm of $f$ to be$\Vert f\Vert_{BV}=[f]$ $BV$ $+\Vert f\Vert_{L^{1}(D)}.$(2.3)

It follows from (2.1) that the’s have bounded BV norms.
By the compactness theorem of BV functions [7, Theorem 4, Page 176] there exists $u\in BV(D)$ suchthat
in $L^{1}(D)$ . One can also assume thatat every $x\in D$ . Then $u(x)=0$ or 1 for$\chi_{\Omega_{j}}$$\chi_{\Omega_{j}}arrow u$$\chi_{\Omega_{j}}(x)arrow u(x)$81.

every , and one can identify withfor some measurable set. The convergenceinalso implies that
, i.e.
. Next apply the lower semi-continuity of BV functions [7,Theorem 1, Page 172] to conclude that
and
$x$

$u$

$L^{1}(D)$$\Omega_{*}\subset D$$\chi_{\Omega_{*}}$$|\Omega_{*}|=\omega|D|$$\chi_{\Omega_{j}}arrow\chi_{\Omega}$$\Omega_{*}\in \mathcal{A}$$\chi_{\Omega_{r}}\in BV(D)$$\mathcal{P}_{D}(\Omega_{*})\leq\lim_{jarrow}\inf_{\infty}\mathcal{P}_{D}(\Omega_{j})$Since$\chi_{\Omega_{j}}$is uniformly bounded by 1,

.

(2.4)

in $L^{1}(D)$ implies thatin $L^{2}(D)$ and$\chi_{\Omega_{j}}arrow\chi_{\Omega_{*}}$$(-\Delta)^{-1/2}(\chi_{\Omega_{j}}-\omega)arrow(-\Delta)^{-1/2}(\chi_{\Omega_{*}}-\omega)$$\chi_{\Omega_{j}}arrow\chi_{\Omega}$$\lim_{jarrow\infty}\int_{D}|(-\triangle)^{-1/2}(\chi_{\Omega_{j}}-\omega)|^{2}dx=\int_{D}|(-\triangle)^{-1/2}(\chi_{\Omega_{*}}-\omega)|^{2}dx$. in$L^{2}(D)$. Then.

(2.5)

Combining (2.4) and (2.5) we deduce that$\mathcal{J}_{B}(\Omega_{*})\leq\lim_{jarrow}\inf_{\infty}\mathcal{J}_{B}(\Omega_{j})$

.

(2.6)

This shows thatis a minimizer ofin
It is diﬃculty to determine the exact shape of the global minimizer
We will see that in one dimension,i.e. $D=(0,1)$ , a global minimizer is a periodic union of intervals; see Theorem 3.2. In higher dimensions,this question. is far from being settled, although recent years have seen a number of results on this problem[2, 41, 17, 4, 16, 10] for various ranges of and , and some special shapes of $D.$
The functional
has a complicated landscape. The global minimizer is not the only object of interest.
This article is more about stationary points than global minimizers. Letand :be a smooth function.
is termed a deformation of $D$ if$\Omega_{*}$$\mathcal{A}.$$\mathcal{J}_{B}$$\square$$\Omega^{*}$$\omega$$\gamma$$\mathcal{J}_{B}$$\Phi$$\epsilon_{0}>0$$\mathbb{R}^{n}\cross(-\epsilon_{0}, \epsilon_{0})arrow \mathbb{R}^{n}$$\Phi$

1. $\Phi(x, 0)=x$ for every$\Omega$2.

$\Phi$

3.

$\Phi$

Letif$\epsilon)$:$\mathbb{R}^{n}arrow \mathbb{R}^{n}$$x\in \mathbb{R}^{n},$is one-to-one and onto for every

leaves $D$ invariant, i.e. for every$\Omega\in \mathcal{A}$, such that$\mathcal{J}_{B}(\Omega)<\infty$$\epsilon,$$\Phi(x, \epsilon)\in D$, and let$\Phi$$\epsilon\in(-\epsilon_{0}, \epsilon_{0})$if and only if $x\in D.$

be a deformation. We say that$|\Phi(\Omega, \epsilon)|=|\Omega|, \forall\epsilon\in(-\epsilon_{0}, \epsilon)$

The first variation of$\mathcal{J}_{B}$,under a volume preserving deformation$\Phi$$\Phi$

preserves the volume of

.

(2.7)

is
(2.8)

$\frac{d\mathcal{J}_{B}(\Phi(\Omega,\epsilon))}{d\epsilon}|_{\epsilon=0}$

Since is a Caccioppoli set, i.e.,, the Riesz representation theorem asserts that the distributional
derivative
can be regarded as a vector valued measure on $D$ . The total variation of this measure is areal valued, finite measure, denoted by
There are various natural ways to define the boundary of a Cacciopolli set measure theoretically. For
instance one can use the support of the measure
as the boundary of . This support, denoted by$supp(|D\chi_{\Omega} is$ defined $by its$ complement, $i.e.,$ supp (is the subset of $D$ whose complement in $D$ is$\Omega$$\mathcal{P}_{D}(\Omega)<\infty$$D\chi_{\Omega}$$|D\chi_{\Omega}|.$$\Omega$$|D\chi_{\Omega}|$$|D\chi_{\Omega}|)${ $x\in D:\exists\rho>0$ such that

$B_{\rho}(x)\subset D$

In (2.9),is the open ball centered at of radius
Another concept for a measure theoretic boundary of
A point $x\in D$ is inif$B_{\rho}(x)$$x$$\partial^{*}\Omega$

1. $|D\chi_{\Omega}|(B_{\rho}(x)\cap D)>0$ for any $\rho>0,$

and $|D\chi_{\Omega}|(B_{\rho}(x))=0$ }.

(2.9)$\rho.$$\Omega$

is called the reduced boundary, denoted by$\partial^{*}\Omega.$

82
2. the limit

exists where$\nu(x)=\lim_{\rhoarrow 0}\nu_{\rho}(x)$$\nu_{\rho}(x)=\frac{D\chi_{\Omega}(B_{\rho}(x)\cap D)}{|D\chi_{\Omega}|(B_{\rho}(x)\cap D)},$

3. $|v(x)|=1.$
The reduced boundaryis always a subset of $supp(|D\chi_{\Omega}$ but the converse is not always true. A simple
counter-example is a rectangle in a two dimensional $D$ . The four corner points are in $supp(|D\chi_{\Omega}|)$ but notboundary, then
. Of course, if the set has
and $supp(|D\chi_{\Omega}|)$ all agree with the topologicalinboundary. I refer to [7, 9] for more on reduced boundaries.
$\partial^{*}\Omega$$\partial^{*}\Omega$$C^{1}$$\Omega$$\partial^{*}\Omega$$\partial\Omega\cap D$

Although$supp(|D\chi_{\Omega}|)\backslash \partial^{*}\Omega$

may not be empty, this set is always small. Actually the measurenothing but the $n-1$ dimensional Hausdorﬀ measure restricted to$\partial^{*}\Omega$$|D\chi_{\Omega}|$is:(2.10)$|D\chi_{\Omega}|(E)=H^{n-1}(E\cap\partial^{*}\Omega)$

for any $E\subset D$ . The unit vector $v(x)$ in the definition of the reduced boundary is the normal vector ofat , inward with respect to . With this normal vector, one obtains a tangent space toat every$\partial^{*}\Omega$$\partial^{*}\Omega$$\Omega$$x$$x\in\partial^{*}\Omega.$

We introduce the term inhibitor variable, denoted$-\Delta I_{\Omega}=\chi_{\Omega}-\omega$

In other words,
Let$I_{\Omega}=(-\triangle)^{-1}(\chi_{\Omega}-\omega)$

i

$n$

$D,$

$I_{\Omega}$, for each$\partial_{n}I_{\Omega}=0$

on$\Omega\in \mathcal{A}$to be the solution of$\int_{D}u(x)dx=0.$$\partial D,$.
(2.11)$X(x)= \frac{\partial\Phi(x,\epsilon)}{\partial\epsilon}|_{\epsilon=0}, \forall x\in \mathbb{R}^{n}$be the infinitesimal vector field of a deformation$\Phi$. Sinceleaves $D$ invariant,$\Phi$$X(x)\in T_{\partial D}(x) , \forall x\in\partial D$

where$T_{\partial D}(x)$is the tangent space of$\partial D$

at

$x$

,

(2.12). Calculations show that
(2.13)$\frac{d}{d\epsilon}|_{\epsilon=0}\mathcal{P}_{D}(\Phi(\Omega, \epsilon))=\int_{\partial’\Omega}div_{\partial^{r}\Omega}XdH^{n-1}$

Hereis the divergence on
precisely, with $X=(X_{1}, X_{2}, \ldots, X_{n})$ ,$\partial^{*}\Omega$$div_{\partial^{\bullet}\Omega}$, made possible by the properties of the reduced boundary. More
$div_{\partial\Omega}X=\nabla_{j}^{\partial\Omega}X_{j}$where$\nabla_{j}^{\partial\Omega}$is the j-th component of the gradient$\nabla^{\partial\Omega}f=\sum_{k=1}^{n-1}D_{\tau_{k}}f\tau_{k}$Hereis the usual directional derivative inan orthonormal basis of the tangent space ofinAlso$D_{\tau_{k}}$$\mathbb{R}^{n}$$\partial^{*}\Omega$$\nabla^{\partial\Omega}$

and

on

(2.14),$\partial^{*}\Omega$, namely,$\nabla_{j}^{\partial\Omega}f=\nabla^{\partial\Omega}f\cdot e_{j}$.

(2.15)

along the direction , the $\tau_{k}’ s(k=1,2, \ldots, n-1)$ form’s $(j=1,2, n)$ are the standard unit vectors$\tau_{k}$, and the$e_{j}$$\mathbb{R}^{n}$$\frac{d}{d\epsilon}|_{\epsilon=0}\frac{1}{2}\int_{D}|(-\triangle)^{-1/2}(\chi_{\Phi(\Omega,\epsilon)}-\omega_{\epsilon})|^{2}dx=-\int_{\partial\Omega}I_{\Omega}\nu\cdot XdH^{n-1}$

where$\omega_{\Xi}=\frac{|\Phi(\Omega,\epsilon)|}{|D|}$(2.16). Moreover$\frac{d|\Phi(\Omega,\epsilon)|}{d\epsilon}|_{\epsilon=0}=-\int_{\partial\Omega}\nu\cdot XdH^{n-1}$(2.17)

83
See, for example [40], for the derivations of these formulas. Hence for a volume preserving deformation, the infinitesimal vector field $X$ satisfies$\Phi$

of$\Omega$$\int_{\partial\Omega}v\cdot XdH^{n-1}=0$

Motivated by (2.13)-(2.17), we say that a Cacciopolli set.

(2.18)$\Omega\in \mathcal{A}$is a stationary point of$\mathcal{J}_{B}$if(2.19)$\int_{\partial^{*}\Omega}(div_{\partial^{t}\Omega}X-I_{\Omega}\nu\cdot X)dH^{n-1}=0$

for every
If hasequation,that satisfies (2.12) and (2.18).
boundary and does not share boundary with $D$ , then (2.19) is equivalent to the Euler-Lagrange$X\in C^{\infty}(\mathbb{R}^{n}, \mathbb{R}^{n})$$\Omega$$C^{2}$$H+\gamma I_{\Omega}=k, on\partial\Omega\cap D$.

(2.20)In (2.20)} $H$ is the mean curvature of the surface, and the constantis a Lagrange multipliercorresponding to the constraint $|\Omega|=\omega|D|$ . If shares boundary with $D$ , then one adds another condition
that$\partial\Omega\cap D$$\lambda$$\Omega$(2.21)$\partial\Omega\cap D\perp\partial D$

where
and
meet.
A stationary point withboundary is a called a regular stationary point. Whether
has non-regularstationary points is a diﬃcult open question. This article only deals with regular stationary points.
$\partial\Omega\cap D$$\partial D$$C^{2}$$\mathcal{J}_{B}$Stationary points

3

In one dimension, $D=(0,1)$ , all stationary points of$z_{K}\in(0,1)$ be given by$z_{1},$$\mathcal{J}_{B}$are known. For each positive integer $K$ let$z_{2},$$z_{1}= \frac{1-\omega}{K}, z_{3}=z_{1}+\frac{2}{K}, Z_{5}=z_{3}+\frac{2}{K},$and$z_{2}= \frac{1+\omega}{K}, z_{4}=z_{2}+\frac{2}{K}, z_{6}=z_{4}+\frac{2}{K},$Also define$z_{1}’,$$z_{2}’,$$z_{K}’\in(0,1)$(3.1)$\cdots$by$z_{1}’= \frac{\omega}{K}, z_{3}’=z_{1}’+\frac{2}{K}, z_{5}’\cdot=z_{3}’+\frac{2}{K},$$\cdots$and$z_{2}’= \frac{2-\omega}{K}, z_{4}’=z_{2}’+\frac{2}{K}, z_{6}’=z_{4}’+\frac{2}{K}$

,

(3.2)

For each $K$ we have two sets$\Omega_{K}=(z_{1}, z_{2})\cup(z_{3}, z_{4})\cup(z_{5}, z_{6})\cup\ldots$

and

(3.3)

$\Omega_{K}’=(0, z_{1}’)\cup(z_{2}’, z_{3}’)\cup(z_{4}’, z_{5}’)\cup\ldots$

It was proved by Ren and Wei [25] that the stationary points of$\mathcal{J}_{B}$on $(0,1)$ are precisely these$\Omega_{K}$’s and$\Omega_{K}’s.$

Theorem 3.1 ([25]) If $D=(0,1)$ , thenis a stationary point of
if and only if$K=1$ , 2,sets
andMoreover every stationary point is a local minimizer.
$\Omega\in \mathcal{A}$$\Omega_{K}$$\Omega_{K}’,$$\cdots$$\mathcal{J}_{B}$$\Omega$

is one of the84
on $(0,1)$ are also known. The energy of
The global minimizers ofdenote by $J(K)$ . Calculations show that$\mathcal{J}_{B}$$J(K)=K+ \frac{\gamma\omega^{2}(1-\omega)^{2}}{6K^{2}}$$\Omega_{K}$and$\Omega_{K}’$are equal, which we

(3.4)

.

If we minimize among positive integers $K$ , there are two possibilities: (1) is minimized by one integer
is minimized by two consecutive integers
and $K_{*}+1$ . The first case is generic.
$J$

$J$

$K_{*};(2)J$$K_{*}$

Theorem 3.2 ([25]) If is minimized by one integer,, thenhas two global minimizers:; if is minimized by two integers,
and $K_{*}+1$ , thenhas four global minimizers:
, and
$J$

$\Omega_{K}’.$$K_{*}$$J$$\Omega_{K.+1}$$\mathcal{J}_{B}$$K_{*}$$\Omega_{K}$. and$\Omega_{K}.,$$\mathcal{J}_{B}$$\Omega_{K}’.$’$\Omega_{K.+1}’.$The proofs of these theorems depend on the special structure of Caccioppoli sets in one dimension.
Caccioppoli set
is, up to a set of Lebesgue measure , just a finite union of intervals. It can be$x_{K}\in(0,1)$ .
identified by the endpoints of the intervals. Say there are $K$ endpoints denoted by
They are the interfaces separating from. Then the measureisjust the sum of delta measures$K$ , namely$k=1$ , 2,centered at
$A$

$0$

$\Omega\subset(0,1)$$x_{1},$ $x_{2},$$\Omega$$(0,1)\backslash \Omega$$|D\chi_{\Omega}|$$x_{k},$$|D \chi_{\Omega}|=\sum_{k=1}^{K}\delta_{x_{k}}$

.

(3.5)

Within the class of Caccioppoli sets in with $K$ endpoints the energy
is a function of. For
a stationary point, the equation (2.20) becomes a system of equations for these. One can solve theseequations and find two solutions, which are exactly (3.1) and (3.2).
To discuss whether a stationary point is a local minimizer, one needs a topology on . The natural
topology is given by the $L^{1}(D)$ norm; namely, forthe distance between the two sets is$\mathcal{A}$$\mathcal{J}_{B}$$x_{1},$ $x_{2},$$\cdot$$x_{K}$$x_{k}’ s$$\mathcal{A}$$\Omega,$$\Omega’\in \mathcal{A}$$d( \Omega, \Omega’)=\int_{D}|\chi_{\Omega}-\chi_{\Omega’}|dx$

.

(3.6)

Under this topology, in one dimension, the subclass of Cacciopolli sets in
has an unusual structure, sothat all stationary points ofare local minimizers. There are no other type stationary points.
$\mathcal{A}$$\mathcal{J}_{B}$$n\geq 2$ , is a much more diﬃcult problem.
The functionalon a higher dimensional set
One
can certainly extend the one dimensional stationary points
trivially to a stationary point on aand$D\in \mathbb{R}^{n},$$\mathcal{J}_{B}$$\Omega_{K}$$\Omega_{K}’$rectangle(3.7)$D=(0,1)\cross(0, w)$

where $w>0$ is the width. They would be good candidates for the stripe pattern observed in the lamellar
phase of diblock copolymers, provided one can show that they were stable in $D$ . It turns out that with given
and $K$ , the stability of
depends on the width of the rectangle $w.$
andThere are several works addressing this issue, where stability is discussed from diﬀerent perspectives.
First, Ren and Wei [27] considered the diﬀusive interface problem (1.5). It was shown in [25] by De Giorgi’s$Gamma_{r}$ convergence theory, that near each(or
) of Theorem 3.1, there exists a local minimizers
, or
, of
. When such a local minimizer on $(0,1)$ is trivially extended to a stationary point on$(0,1)\cross(0, w)$ , one can study its stability from the eigenvalue problem$\omega,$$\Omega_{K}$$\gamma$$w$$\Omega_{K}’$$\Omega_{K}$$U_{\epsilon,K}$$U_{\epsilon,K}’$$\Omega_{K}’$$\mathcal{I}_{B,\epsilon}$$- \epsilon^{2}\Delta\phi+(3U_{\epsilon_{)}K}^{2}-3U_{\epsilon,K}-\frac{1}{2})\phi+\epsilon\gamma(-\Delta)^{-1}\phi=\lambda\phi+Const.$$\partial_{n}\phi=0$

on$\partial D,$

,

(3.8)

$\int_{D}\phi(x)dx=0.$

In [27] the authors found asymptotic formulas for all the eigenvalues in terms of the small parameter . It
turns out that only when is small, all the eigenvalues are positive. If is large, one can find a negative
eigenvalue, and consequently
is unstable.
$\epsilon$

$w$

$w$

$U_{\epsilon,K}$

85
Knowing the dependence of the eigenvalues on , the authors were able to find bifurcation solutions
crosses some critical values [31]. These solutions have wiggled interfaces. They are the first real two$\gamma$as$\gamma$dimensional solutions.
Next, Choksi and Sternberg [6] considered the second variation of. Let
be a Caccioppoli set,
and a deformation be a volume preserving deformation mentioned in the last section. The second variationof along is$\Omega\in \mathcal{A}$$\mathcal{J}_{B}$$\Phi$$\Omega$$\Phi$(3.9)$\frac{d^{2}\mathcal{J}_{B}(\Phi(\Omega,\epsilon))}{d\epsilon^{2}}|_{\epsilon=0}$

They proved that whenis small, the second variation of(or) along any deformation
is nonnegative, and whenis large, there is always some deformation
long which, the second variation isnegative. Actually they considered the periodic boundary condition case, but their result is also valid if thedomain is a rectangle inIn [1], Acerbi et $al$ showed that the positivity of the second variation at a regular stationary point indeed
implies that the stationary point is a local minimizer with respect to the metric given in (3.6). Using thisresult, Morini and Sternberg showed in [16] that when the width $w$ is small,andare local minimizers
on $(0,1)\cross(0, w)$ . Moreover, when is small, a global minimizer on $(0,1)\cross(0, w)$ must be the extension ofa global minimizer on $(0,1)$ .
Another one dimensional result was obtained by Ren and Wei [26] that there are stationary points on adisc inwhich are unions of concentric annuli. There are also solutions of wiggled interfaces bifurcatingout of these radial solutions [30].
$w$$\Omega_{K}$$\Phi$$\Omega_{K}’$$w$$\Phi$$\mathbb{R}^{n}$$\Omega_{K}$$\Omega_{K}’$$w$$\mathbb{R}^{2}$

The first higher dimensional result on a general domain, not by the bifurcation theory, was proved by Oshita[22].

Theorem 3.3 ([22]) Let $D$ be a bounded and smooth domain in
There exists
depending on $D$
only such that if, there isso that when
admits a stationary point which is close
to a disc of radius
and is centered near a global minimum of the junction $zarrow R(z, z)$ .
$\mathbb{R}^{2}$$\gamma_{0}>0$$\omega<\omega_{0}$$\gamma<\gamma_{0},$$\omega_{0}>0$$\mathcal{J}_{B}$$\sqrt{\frac{\omega|D|}{\pi}}$

The function $R$ in this theorem is the regular part of Green’s function of with the Neumann boundary
condition. Recall that the Green’s function, denoted $G(x, y)$ , is the solution of the following problem$\triangle$$- \triangle G(\cdot, y)=\delta(\cdot-y)-\frac{1}{|D|}$for each$y\in D\subset \mathbb{R}^{2}$

One can write

$G$

in

$D,\cdot$$y)=0$ on$\partial_{n}G$$\partial D$;$\int_{D}G(x, y)$$dx=0$(3.10)

as a sum of two terms:$G(x, y)= \frac{1}{2\pi}\log\frac{1}{|x-y|}+R(x, y)$.

(3.11)

The first termis the fundamental solution of the Laplace operator in two dimensions; the second
term is the regular part of the Green’s function, a smooth function of $(x, y)\in D\cross D$ . It is known that$\frac{1}{2\pi}\log\frac{1}{|x-y|}$$R$$R(z, z)arrow\infty$, as$zarrow\partial D$,(3.12)

so the function $zarrow R(z, z)$ has a global minimum in $D.$
The idea of Oshita’s work is to view the nonlocal part ofas a small perturbation of the perimeter
term in. Then a disc centered at any general point $\xi\in D$ is a stationary point of the perimeter functional
according to the isoperimetric inequality. With the addition of a small nonlocal term, one can make a small
perturbation to the disc and find a set that almost solves (2.20). By “almost solves”’ we mean that this set isstationary with respect to all but two deformations. These two deformations are related to the translationsof the set. However if one adjusts the center of the disc in $D$ , it turns out that there is a special placein such that if the disc is centered at , then the perturbed disc is a stationary point with respect to all
deformations. Asymptotically converges to a minimum of $zarrow R(z, z)$ as$\mathcal{J}_{B}$$\mathcal{J}_{B}$

$\xi$

$D$

$\xi^{*}$$\xi^{*}$$\xi^{*}$$\gammaarrow 0.$

86
The next breakthrough came in [33] by Ren and Wei. They observed that a disc is not only a stationary
point of the perimeter functional, it is also stationary with respect to the singular part of the nonlocal energy.
can be written, with the help of Green’s function, as
Recall that in (2.20), the function$I_{\Omega}$$I_{\Omega}(x)= \int_{\Omega}G(x, y)dy$(3.13).

But by (3.11) we have
$I_{\Omega}(x)= \int_{\Omega}\frac{1}{2\pi}\log\frac{1}{|x-y|}dy+\int_{\Omega}R(x, y)dy$(3.14).

If we take to be a discof radius , centered at any $\xi\in D$ and insert it into the equation (2.20), thenis obviously , a constant. Moreover, the first term of
the curvature $H$ of the boundary of$\Omega$$B_{\rho}$$\rho$$B_{\rho}(\xi)$$I_{B_{\rho}},$$\frac{1}{\rho}$$\int_{B_{\rho}(\xi)}\frac{1}{2\pi}\log\frac{1}{|x-y|}dy,$

is also constant for all$x\in\partial B_{\rho}(\xi)$. Only the term$\int_{B_{\rho}(\xi)}R(x_{\rangle}y)dy$. Based on this observation, Ren and Wei showed that an approximate disc exists
is not constant on
as a stationary point for a much larger range of the parameter and$\partial B_{\rho}(\xi)$$\omega$

Theorem 3.4 ([33]) Let $\rho>0$ be such thatand satisfy$\pi\rho^{2}=\omega|D|$$\gamma.$

For any $\eta>0$ there exists $\delta>0$ such that if.

$\rho$$\gamma$1). $\gamma\rho^{3}<12-\eta$ ,then$\mathcal{J}_{B}$2$\gamma\rho^{4}<\delta$,

3).

$\rho<\sqrt{\frac{\omega_{0}|D|}{\pi}},$admits a stationary point that is close to a disc of radius , and is centered near a minimum of
In some sense, this stationaw point is stable.
$\rho$$zarrow R(z, z)$ .

Hereis the same as the one in Theorem 3.3. So to have a disc-like stationary point, does not have tobe small. If is small enough, we can again have a disc-like stationary point even if is large.
The claim of stability lies in the proof of the theorem. First, take a disc of radius and place it inside$D$ with the center being. At this point, is arbitrary and is not where the center of the solution is. Next,, so that the boundary of the perturbed disc is parametrized bywe perturb the disc by a function$\omega_{0}$$\gamma$$\omega$$\gamma$$\rho$

$\xi$

$\xi$

$\phi(\theta)$(3.15)$\theta\in \mathbb{S}^{1}arrow\sqrt{\rho^{2}+2\phi(\theta)}e^{i\theta}$

In (3.15),
foris the unit circle, or the interval $[0, 2\pi]$ with identified endpoints. We write
for simplicity. The function
is really the radius variable in the polar coordinates centeredbecause theat . We do not use the radius variable to describe perturbation, but use the variable
constraint $|\Omega|=\omega|D|$ is a simple aﬃne condition$e^{i\theta}$$\mathbb{S}^{1}$$(\cos\theta, \sin\theta)$$\thetaarrow\sqrt{\rho^{2}+2\phi(\theta)}$$\thetaarrow\phi(\theta)$$\xi$(3.16)$\int_{0}^{2\pi}\phi(\theta)d\theta=0$

in terms of . Another condition on$\phi$$\phi$

is that$\int_{0}^{2\pi}\phi(\theta)\cos\theta d\theta=\int_{0}^{2\pi}\phi(\theta)\sin\theta d\theta=0$This ensures that the set perturbed by$\phi$

is(centered’) at$\xi$(3.17).

. We denote this set by$\Omega_{\phi}.$

87
Since our set
is centered at an arbitrary point , it is not realistic to find a so thatsolves (2.20).
Instead we look for a that solves the $(projected”$ equation, namely
is in the linear span of. In other words we look for , and three numbers, andsuch that$\Omega_{\phi}$$\phi$$\xi$$\phi$$\Omega_{\phi}$$H(\partial\Omega_{\phi})+\gamma I_{\Omega_{\phi}}$$\{1, \cos\theta, \sin\theta\}$$\phi$$\lambda,$$A_{2}$$A_{1}$$H(\partial\Omega_{\phi})+\gamma I_{\Omega_{\phi}}=\lambda+A_{1}\cos\theta+A_{2}\sin\theta$.

(3.18)

To solve this equation, we use the exact disc $B(\xi)$ centered at as an approximate solution. It corresponds
to $\phi=$ O. We analyze the linearized operator of the equation (2.20) at $\phi=$ O. This linear operator turns
out to be invertible and positive definite. The equation (3.18) may be written in a fixed point form near theapproximate solution and is solved by the contraction mapping principle.
Denote this solution of (3.18) by
and the corresponding set by. Recall that our starting pointis a fixed center , so thisalso depends on . We emphasize this dependence by writing)and. The final step is to show that there is a particular
in $D$ such that when
, the
) and
corresponding
solves (3.18) with $A_{1}=A_{2}=0.$
To do this, one minimizes the energy ofwith respect to . It turns out that sincewasfound near the exact disc $B(\xi)$ , the energy ofis close to the energy of $B(\xi)$ . The energy of $B(\xi)$ canbe computed explicitly:$\xi$$\phi^{*}$$\xi$$\Omega_{\phi^{*}}$$\phi^{*}$$\phi^{*}=\phi^{*}$$\xi$$\Omega_{\phi^{\bullet}}=\Omega_{\phi^{*}(\cdot,\xi)}$$\phi^{*}$$\xi^{*}$$\xi$$\xi=\xi^{*}$$\xi^{*}$$\Omega_{\phi^{\nu}(\cdot,\xi^{*})}$$\xi$$\Omega_{\phi^{*}(\cdot,\xi)}$$\Omega_{\phi^{*}(\cdot,\xi)}$$\Omega_{\phi^{*}(\cdot,\xi)}$$\mathcal{J}_{B}(B(\xi))=2\pi\rho+\frac{\pi^{2}\gamma\rho^{4}}{2}[\frac{1}{2\pi}\log\frac{1}{\rho}+\frac{1}{8\pi}+R(\xi, \xi)+\frac{\rho^{2}}{4|D|}]$

Clearlythat
As a minimum,$\mathcal{J}_{B}(B(\xi))$$\mathcal{J}_{B}(\Omega_{\phi^{*}(\cdot,\xi)})$(3.19).

is minimized, with respect to , at a minimum of the function $\xiarrow R(\xi, \xi)$ . It also implies
is minimized, with respect to , at a point
which is close the minimum of $\xiarrow R(\xi, \xi)$ .
satisfies
$\xi$

$\xi$

$\xi^{*}$$\xi^{*}$$\frac{\partial \mathcal{J}_{B}(\Omega_{\phi^{*}(\cdot,\xi)})}{\partial\xi_{i}}|_{\xi=\xi^{*}}=0, i=1, 2$(3.20).

A careful reading of (3.20) shows that it implies that $A_{1}=A_{2}=0$ in (3.18) at
This shows the existence of a stationary point. The stability of this point is ascertained from two facts.
First, we constructed a fixed point) with the help of the ljnearized operator at the exact disc $B(\xi)$ .
One can show that this linear operator is similar to the one at. Hence the latter linear operator
is also positive definite. This shows that among the perturbations satisfying (3.16) and (3.17),) islocally energy minimizing. We emphasize thatis only locally energy minimizing. among these special
perturbations. Second, we minimized the energy of) with respect to . As we vary
) can beviewed as a family of perturbations. These perturbations do not satisfy the condition (3.17), so they arebeyond the perturbations in the first step. Since) is obtained from a minimum , our stationary
point locally minimizes in both steps. We therefore conclude that in this sense,) is stable.
$\xi=\xi^{*}$$\phi^{*}$$\xi$$\Omega_{\phi^{*}(\cdot,\xi)}$$\phi^{*}$$\xi$$\phi^{*}(\cdot, \xi)$$\phi^{*}$$\phi^{*}$$\xi$$\xi,$$\xi$$\xi^{*}$$\phi^{*}$$\xi$$\xi^{*}$$\phi^{*}$$\xi^{*}$The sense of stability discussed above should be compared to the more standard notions of stability. One
can define a stationary point to be if it is a local energy minimizer with respect to the metric given in (3.6).
Or one can define a stationary point to be stable if
has positive second variation, (3.9), at the stationary
point along any deformation whose infinitesimal element is non-trivial. Although we expect that these three
notions of stability are more or less equivalent, the only known fact now is that the positivity of the second
variation implies the local minimality of the energy functional by Acerbi et $al[1]$ mentioned earlier.
$d$$\mathcal{J}_{B}$For pattern formation problems, one really likes to find stationary points that are nearly periodic setsof multiple components. A stationary point that models a pattern should be an assembly of many similargeometric shapes. We call such a stationary point a stationary assembly. The single disc solutions in
Theorems 3.3 and 3.4 are not stationary assemblies.
Fortunately Ren and Wei discovered that the techniques they developed in [33] can be elaborated toconstruct stationary assemblies. They showed that for each positive integer $K$ there is an assembly of $K$
small discs as a stationary point ofwhen is small and is in a particular range. To state this result,$\mathcal{J}_{B_{\rangle}}$$\omega$$\gamma$88
define a function

$F$

by

$F( \xi_{1}, \xi_{2}, \xi_{K})=F(\xi_{1}, \xi_{2}, \xi_{K})=\sum_{k=1}^{K}R(\xi_{k}, \xi_{k})+\sum_{k=1}^{K}\sum_{l=1,l\neq k}^{K}G(\xi_{k}, \xi_{l})$(3.21),

where $\xi_{k}\in D$ , andif $j\neq k$ , with the help of Green’s function and its regular part. This function
approaches the boundary of $D$ or if two distinct
tends to infinity if one
and become too close, so $F$
always has a global minimum.
$\xi_{j}\neq\xi_{k}$$\xi^{k}$$\xi^{k}$Theorem 3.5 ([32]) Let $K\geq 2$ be an integer and$\rho$$\xi^{l}$be such that $K\pi\rho^{2}=\omega|D|.$

1. For every $\eta>0$ there exists $\delta>0$ , depending on$\eta,$

and $D$ only, such that if$K$$\frac{1+\eta}{\rho^{3}\log\frac{1}{\rho}}<\gamma<\frac{12-\eta}{\rho^{3}}$,(3.22)

and$\rho<\delta$,(3.23)

then there exists a stationary point, stable in some sense.

2. This stationary point is a union of $K$ sets, each of which is close to a disc of radius
3. Let the centers of these approximate discs be$\xi_{1}^{*},$$\xi_{2}^{*}$,$\cdots$,$\xi_{K}^{*}$. Then$\xi^{*}=$$\rho.$

is close to a$(\xi_{1}^{*},\xi_{2}^{*}, \xi_{K}^{*})$

global minimum of the function $F.$

As in the previous case, one takes $K$ discs and place them inside $D$ with centers at
unlike the last case, the radius of these discs is not fixed at . Instead we introduce
the area of the k-th disc is. The’s satisfy$\rho$$w_{k}$$\xi_{1},$$\xi_{2}$$w_{1},$,$\cdots$,$\xi_{K}$. However, so that$w_{K}$$w_{2},$$w_{k}$(3.24)$\sum_{k=1}^{K}w_{k}=K\pi\rho^{2}$

Perturbed each

$B_{r_{k}}(\xi_{k})$, the disc centered atsatisfies (3.16) and (3.17). One then finds$\xi$

of radius, by a function
such that the set$r_{k}=\sqrt{arrow w\pi}$$\phi^{*}=(\phi i, \phi_{2}^{*}, \phi_{K}^{*})$$\phi_{k}$to. Each. solves$\Omega_{\phi_{k}}$$\phi_{k}$$\Omega^{*}=\bigcup_{k=1}^{K}\Omega_{\phi_{k}}$(3.25)$H(\partial\Omega_{\phi_{k}})+\gamma I_{\Omega_{\phi}}=\lambda_{k}+A_{1,k}\cos\theta+A_{2,k}\sin\theta$on eachAs before
is not a solution of (2.20) yet. There are the termsandin (3.25).
Moreover the constants
there depend on , while the constant in (2.20) does not. We use the ’s andthe’s to fix these problems. Note that) depends on
and $w=(w_{1}, \cdots, w_{K})$One minimizes) with respect to $(\xi, w)$ . This energy is close to the energy of the approximate, which issolution$\partial\Omega_{\phi_{k}}.$$\phi^{*}$$A_{1,k}\cos\theta$$\lambda_{k}$$k$$\phi^{*}=\phi^{*}$$w_{k}$$A_{2,k}\sin\theta$$\lambda$$\xi_{k}$$\xi,$$w$$\xi=(\xi_{1_{\rangle}}\ldots, \xi_{K})$$\mathcal{J}_{B}(\phi^{*} \xi, w)$$\bigcup_{k=1}^{K}B_{f}k(\xi_{k})$$K 2 K$$\mathcal{J}_{B}(\bigcup_{k=1}^{K}B_{r_{k}}(\xi_{k})) = \sum 2\pi r_{k}+\frac{\gamma\pi}{2}[\sum(\frac{r_{k}^{4}}{2\pi}\log\frac{1}{r_{k}}+\frac{r_{k}^{4}}{8\pi}+r_{k}^{4}R(\xi_{k_{\rangle}}\xi_{k}))$$k=1 k=1$$K K K$$+ \sum\sum r_{k}^{2}r_{l}^{2}G(\xi_{k}, \xi_{l})+\sum\sum(\frac{r_{k}^{2}r_{l}^{4}}{8|D|}+\frac{r_{k}^{4}r_{l}^{2}}{8|D|})j.$(3.26)$k=1l\neq k k=1l=1$When is small as in assumption (3.23) of Theorem 3.5 and theleading order quantity of (3.26) is$\rho$$\sum_{k=1}^{K}2\pi r_{k}+\frac{\gamma\pi^{2}}{2}\sum_{k=1}^{K}\frac{r_{k}^{4}}{2\pi}\log\frac{1}{r_{k}}$.

$r_{k}$’s are in a neighborhood of , the$\rho$(3.27)

89

Figure 1: Left: a stationary assembly of
double bubbles.

$\mathcal{J}_{B}$with 40 discs. Right: a stationary assembly of$\mathcal{J}_{T}$with 40

The lower boundensures that (3.27) is locally minimized at $r_{1}=\ldots=r_{K}=\rho$ . One can take all. This means minimizing $F$ of (3.21).
the ’s to be in (3.26) and minimize with respect to the$(\xi,
w)$ atwhere
and
In conclusion,is minimized with respect to
we have the equations
, and
each
is close tois close to a minimum of $F$ . At$\frac{1+\eta}{\rho^{3}\log\frac{1}{\rho}}<\gamma$$r_{k}$$\xi_{k}’ s$$\rho$$(\xi^{*}, w^{*})$$\mathcal{J}_{B}(\Omega_{\phi^{*}(_{\rangle}\xi,w)})$$\pi\rho^{2}$$w_{k}^{*}$$\frac{\partial \mathcal{J}_{B}(\phi^{*}(\cdot,\xi,w))}{\partial w_{k}}|_{(\xi,w)=(\xi^{v},w^{v})}=\lambda,$$\frac{\partial \mathcal{J}_{B}(\phi^{*}(\cdot,’\xi,w))}{\partial\xi_{kj}}|_{(\xi,w)=(\xi^{s},w^{t})}=0,$

The constant$\lambda$$w^{*}=(w_{1}^{*}, \ldots, w_{K}^{*})$$(\xi^{*}, w^{*})$$\xi^{*}$$k=1$ ,$\cdots$, $K,$ $j=1$ , 2.

(3.28). Using these equations we deduce that
corresponds to the constraint (3.24) on the’s and theall’s are actually independent of , and the$w_{k}’ s$in (3.25), when $(\xi, w)=(\xi^{*}, w^{*})$ , thevanish. This proves the theorem.

$k$$\lambda_{k}$$A$$A_{1,k}$$s$

If $D$ is the unit disc, the Green’s function is known explicitly:(3.29)$G(x, y)= \frac{1}{2\pi}\log\frac{1}{|x-y|}+\frac{1}{2\pi}[\frac{|x|^{2}}{2}+\frac{|y|^{2}}{2}+\log\frac{1}{|x\overline{y}-1|}]-\frac{3}{8\pi}$andis the complex product of andwhere denotes the complex conjugate ofConsequently $F$ is also known explicitly. The left plot of Figure 1 shows a stationary point ofthat is an
assembly of 40 discs. The locations of these discs are determined by numerically minimizing $F.$
$y\in D\subset \mathbb{R}^{2}\underline{\simeq}\mathbb{C}$$\overline{y}$$x$$x\overline{y}$$\overline{y}.$$\mathcal{J}_{B}$This result has an analogy in three dimensions.

Theorem 3:6 ([34]) Let$D\subset \mathbb{R}^{3}$

be a bounded and smooth domain, $K$ be an integer, and$\rho$

be such that$\frac{K4\pi\rho^{3}}{3}=\omega|D|.$

1. For every $\eta>0$ there exists$\delta>0$, depending on$\eta,$$K$

and $D$ only, such that if$\frac{1.5+\eta}{\rho^{3}}<\gamma<\frac{15-\eta}{\rho^{3}}$,(3.30)

and$\rho<\delta$,then there exists a stationary point, stable in some sense.

(3.31)

90
2. The stationary point is a union of $K$ sets, each of which is close to a ball of radius
3. The centers of the balls are determined by the analogous$F$$\rho.$

in three dimensions in the same way,

One subtle diﬀerence between the two dimensional case and the three dimensional case is that in thethree dimensions, the lower bound and the upper bound of in (3.30) are of the same order, while thecorresponding bounds in two dimensions are of diﬀerent orders. This diﬀerence can be ultimately attributed
to the diﬀerence in the fundamental solutions of in two and three dimensions: one isand theother is$\gamma$$\Delta$$\frac{1}{2\pi}\log\frac{1}{|x|}$$\frac{1}{4\pi|x|}.$The profile problem and ansatz solutions

4

As we mentioned earlier, Ren and Wei recognized in [33] that a disc is stationary with respect to the perimeter
term and the singular part of the Green’s function. The same idea was also used in [34] for a ball in the three
dimensional analogy. In this section we elaborate on this idea and consider a profile problem that consistsof the perimeter term and the singular part of the Green’s function.
This profile problem is posed on the entire space
Let $m>0$ . For each Lebesgue measurable set
with the fixed Lebesgue measure $m,$$|\Omega|=m$ ,(4.1)$\Omega$$\mathbb{R}^{n}$define the energy$\mathcal{K}_{B}(\Omega)=\frac{1}{n-1}\mathcal{P}(\Omega)+\frac{\gamma}{2}\int_{\Omega}\mathcal{N}(\Omega)(x)dx$

In (4.2),is the perimeter ofof given by$\mathcal{P}(\Omega)$$\Omega$in$\mathbb{R}^{n}$, given by (1.3) with $D$ being.

$\mathbb{R}^{n};\mathcal{N}(\Omega)$(4.2)
is the Newtonian potential$\Omega$(4.3)$\mathcal{N}(x)=\int_{\Omega}\Gamma(x-y)dy, \forall x\in \mathbb{R}^{n}$Here$\Gamma$is the fundamental solution of$\Delta$in$\mathbb{R}^{n}$$\Gamma(x)=\{$, i.e.,$\frac{\frac{1}{2\pi}\log\frac{1}{|x_{1}|}}{n(n-1)\alpha(n)|x|^{\mathfrak{n}-2}}$

if $n=2,$
if $n\geq 3,$

(4.4)

where
is the volume of the unit ball in
A regular stationary point ofsatisfies$\alpha(n)$$\mathbb{R}^{n}$$\Omega$$\mathcal{K}_{B}$(4.5)$H(\partial\Omega)+\gamma \mathcal{N}(\Omega)=\lambda$

on its boundary. Often times we use a solution of (4.5) as a building block to construct a stationary
assembly to (2.20), as in Theorems 3.4, 3.5, and 3.6. We call such a solution of the profile equation (4.5) anansatz solution. We know that the unit disc inand the unit ball inare ansatz solutions.
In order to make a stable stationary assembly of (2.20) from an ansatz, the ansatz itself must be stable.
For equation (4.5) the relevant concept here is the linearly stability. It turns out that the unit disc is linearlystable if$0<\gamma<12$ ,(4.6)$\partial\Omega$$\mathbb{R}^{2}$$\mathbb{R}^{3}$

The number 12 is a bifurcation point. For near 12, (4.5) has an oval shaped bifurcation solution. This ovalsolution may also be used as an ansatz for an oval shaped solution of; see Ren and Wei [35] for detail.
In three dimensions the unit ball is linearly stable if$\gamma$$\mathcal{J}_{B}$$0<\gamma<15$ .

(4.7)

Interestingly, there are other ansatz solutions of (4.5). Kang and Ren proved that in two dimensions
there is a ring like solution.

91
Theorem 4.1 ([12]) There exists
such that if, equation
admits a ring shaped solution
R_{1}<|x|<R_{2}\}$ and $|\Omega|=\pi$ .
(greater than ) such that the ring
There is another value
is linearly stable if
and linearly unstable if.
$\gamma_{0}>0$$(4\cdot 5)$$\gamma>\gamma_{0}$$\Omega=\{x\in \mathbb{R}^{2} :$\gamma_{1}$$\gamma_{0}$$\gamma\in(\gamma_{0}, \gamma_{1})$$\gamma>\gamma_{1}$This ring ansatz was used by Kang and Ren in [13] to build a stationary assembly of rings, and also astationary assembly of both rings and discs. The first assembly is stable, but the second is not.
In three dimensions, the obvious analogy of a ring is a shell, that is the set bounded by two concentric
spheres. However it was proved in [24] by Ren that any shell solution of (4.5) must be linearly unstable. It
is not known at the moment whether (4.5) has any linearly stable solution in, which is not a ball.
There is one solution found by Ren and Wei which may play a role in three dimensions.
$\mathbb{R}^{3}$

Theorem 4.2 ([36]) When is suficiently large, equation
like solution inof volume 1.

$(4\cdot 5)$$\gamma$

has an approximately torus shaped, tube$\mathbb{R}^{3}$

Let $f=f(\gamma)$ be a function given by its inverse$\gamma=\underline{2}$$f^{3} \log\frac{1}{2\pi^{2}f^{3}},$Note$\lim_{\gammaarrow\infty}f(\gamma)=0$. If$p_{\gamma}$and$q_{\gamma}$

are the two radii of the torus solution$\lim_{\gammaarrow\infty}\frac{q_{\gamma}}{f(\gamma)}=1$

and, then$2\pi^{2}p_{\gamma}q_{\gamma}^{2}=1$

and$\lim_{\gammaarrow\infty}2\pi^{2}f^{2}(\gamma)p_{\gamma}=1$

A cross section of this ansatz is approximately a round disc of radiussolution is linearly stable.

5$(p_{\gamma}>q_{\gamma})$$q_{\gamma}$. It is not known whether thisA ternary inhibitory system

The functionalof (1.1) can easily be generailzied to study inhibitory systems with more than two
componetns. We consider a three component system in this section. This ternary system was originallyderived by Ren and Wei in [28] from Nakazawa and Ohta’s density functional formulation for triblock
copolymers [19]. Again let $D$ be a bounded and open set ofSuppose thatare two positive
andnumbers such that $\omega_{1}+\omega_{2}<1$ . For two measurable subsets
of $D$ satisfying
and
, and
, set
. The free energy of the system is$\mathcal{J}_{B}$$\mathbb{R}^{n}$$\omega_{1}$$\Omega_{1}$$|\Omega_{2}|=\omega_{2}|D|$$|\Omega_{1}\cap\Omega_{2}|=0$$\omega_{2}$$|\Omega_{1}|=\omega_{1}|D|,$$\Omega_{2}$$\Omega_{3}=D\backslash (\Omega_{1}\cup\Omega_{2})$$\mathcal{J}_{T}(\Omega_{1}, \Omega_{2})=\frac{1}{2}\sum_{i=1}^{3}\mathcal{P}_{D}(\Omega_{i})+\sum_{i,j=1}^{2}\int_{D}\frac{\gamma_{ij}}{2}((-\triangle)^{-1/2}(\chi_{\Omega_{i}}-\omega_{i}))((-\Delta)^{-1/2}(\chi_{\Omega_{j}}-\omega_{j}))dx$

(5.1)

.

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], the mathematical study ofis still in an early stage due to its complexity. Found by Ren and Wei in [29] is a one dimensional local, consisting of alternating $A,$ $B$ , and $C$ micro-domains. The functional
minimizer ofis posed on theunit interval with the periodic boundary condition there. The matrix
is assumed to be positive
definite in [29].
Another one dimensional stationary point, again a local minimizer of, was found by Choksi and Ren
in [5]. This time one eigenvaule of is positive but the other one is whose eigenvector is. It models
a diblock copolymer/homopolymer blend. Such a blend is a mixture of an $AB$ diblock copolymer with ahomopolymer of monomer species $C$ , where the species $C$ is thermodynamically incompatible with both theand $B$ monomer species. In this blend there is a macroscopic phase separation into homopolymer-rich andcopolymer-rich domains followed by micro-phase separation within the copolymer-rich domains into -richand $B$ -rich subdomains. The stationary point has the ABAB ABACC $C$ pattern.
$\mathcal{J}_{T}$$\mathcal{J}\tau$$\mathcal{J}\tau$$\gamma=[\gamma_{ij}]$$\mathcal{J}\tau$$0$$\gamma$$(\omega_{1}, \omega_{2})$

$A$

$A$

$\cdots$$\ldots$

92
A new challenge in a higher dimensional ternary system is the phenomeno1l of triple junction, where
the three components come to meet. In two dimensions a triple junction occurs on points and in three
dimensions a triple junction occurs on curves.
Two dimensional results came only recently in [37, 38, 39] by Ren and Wei. The stationary points theyfound are related to the fascinating structure of the double bubble; see Figure 2. This structure arises as
the optimal configuration of the two component isoperimetric problem. The usual isoperimetric problem isa one component problem which asserts that the round sphere is the least-perimeter way to enclose a given
volume. For the two component isoperimetric problem, one finds two disjoint sets
andinof theprescribed Lebesgue measures such that the size ofis minimum. The double bubble is the uniquesolution to this isoperimetric problem by the works of Foisy et $al[8]$ , Hutchings et $al[11]$ , and Reichardt[23]. In two dimensions the planar double bubble is enclosed by three circular arcs that meet at two triple$E_{1}$$E_{2}$$\mathbb{R}^{n}$$\partial E_{1}\cup\partial E_{2}$junction points, or triple points. The angles between the arcs at a triple point are a11120 degrees.
To state Ren and Wei’s two dimensional results, introduce a fixed number $m\in(0,1)$ and a small sothat
and $\omega_{2}|D|=\epsilon^{2}(1-m)$ . The area constraints
andtakes the
form
and
.
(5.2)
$\epsilon$$\omega_{1}|D|=\epsilon^{2}m$$|\Omega_{1}|=\omega_{1}|D|$$|\Omega_{1}|=\epsilon^{2}m$$|\Omega_{2}|=\omega_{2}|D|$$|\Omega_{2}|=\epsilon^{2}(1-m)$

Instead of
andbecomes one parameter of our problem. The fixed number $m$ measures the relative
area ofvssince
The other parameter is the matrix . It must be positive definite and satisfy a uniform positivity
condition. Namely, there exists $\iota>0$ so that
where
and
are the two eigenvalues ofsuch that. The matrix must also have a lower bound and an upper bound.
$\omega_{2},$$\omega_{1}$$|\Omega_{1}|$$|\Omega_{2}|$$\epsilon$$\frac{|\Omega_{1}|}{|\Omega_{2}|}=\frac{m}{1-m}.$$\gamma$$\iota\overline{\overline{\lambda}}(\gamma)\leq\overline{\lambda}(\gamma)$$\overline{\overline{\lambda}}(\gamma)$$\overline{\lambda}(\gamma)$$0<\overline{\lambda}(\gamma)\leq\overline{\overline{\lambda}}(\gamma)$$\gamma$$\gamma$

Theorem 5.1 ([39]) Let $D$ be a bounded and connected open set ofwith smooth boundary, $m\in(0,1)$ ,, and $\iota\in(0,1$ ]. There exist positive numbers, and depending on $D,$ $m,$ , and only, such that
if the following three conditions hold$\mathbb{R}^{2}$$n\in \mathbb{N}$

1.
2.
3.

$\delta,\tilde{\sigma}$$n$$\sigma$$\iota$$0<\epsilon<\delta,$$\frac{\tilde{\sigma}}{\epsilon^{3}\log\frac{1}{\epsilon}}\leq\overline{\lambda}(\gamma)\leq\overline{\overline{\lambda}}(\gamma)<\frac{\sigma}{\epsilon^{3}},$$\iota\overline{\overline{\lambda}}(\gamma)\leq\overline{\lambda}(\gamma)$,

then there is a stationary assembly of perturbed double bubbles, satisfying the constraints (5.2). Each
perturbed double bubble is bounded by three smooth curves that meet at two triple junction points.
This solution is stable in some sense. If $n=1$ , the lower boundis not needed.
$n$$\frac{\overline{\mathring{\sigma}}}{\epsilon^{3}1g\frac{1}{e}}\leq\overline{\lambda}(\gamma)$

All the perturbed double bubbles in the stationary assembly have almost the same size and shape. The
locations of the double bubbles are determined by the same function $F$ of (3.21). This theorem was firstproved f$orthen=lands$ ymmetric (case i $n[37]_{)}$ then fand asymmetric $(m\in(0,1))$$orthen.=1$case i $n[38],andf$inally f$orthegenera1$ case n $\geq 2andm\in(0, l))in[39]$The right plot of Figure 1 illustrates this theorem with an assembly of 40 double bubbles in $D$ whichis the unit disc. As in Theorem 3.5 the locations of the double bubbles in the assembly are obtained byminimizing $F$ . Theorem 5.1 does not tell what the directions of the double bubbles are, so the directions of
the double bubbles in the figure are not the true directions in the stationary assembly.
In two dimensions, a regular stationary pointofis a solution to the following equations:$m= \frac{1}{2,(})$$(\Omega_{1}, \Omega_{2})$$\mathcal{J}\tau$onon$\partial\Omega_{1}\backslash \partial\Omega_{2}$(5.3)$\partial\Omega_{2}\backslash \partial\Omega_{1}$(5.4)$\kappa_{1}+\gamma_{11}I_{\Omega_{1}}+\gamma_{12}I_{\Omega_{2}}$$=$$\lambda_{1}$$\kappa_{2}+\gamma_{12}I_{\Omega_{1}}+\gamma_{22}I_{\Omega_{2}}$$=$$\lambda_{2}$$\kappa_{0}+(\gamma_{11}-\gamma_{12})I_{\Omega_{1}}+(\gamma_{12}-\gamma_{22})I_{\Omega_{2}}$$=$$\lambda_{1}-\lambda_{2}$$\nu_{1}+\nu_{2}+\nu_{0}$$=$$\vec{0}$aton$\partial\Omega_{1}\cap\partial\Omega_{2}$$\partial\Omega_{1}\cap\partial\Omega_{2}\cap\partial\Omega_{3}$

.

(5.5)
(5.6)

93
Here we assume that
do not touch the boundary of $D$ . Otherwise we need to add anotherandcondition that the boundary of(or) meets the boundary of $D$ perpendicularly.
In $(5.3)-(5.5)\kappa_{1},$, andare the curvatures of the curves, andrespectively. These are signed curvatures defined with respect to a choice of normal vectors. For instance acircle has positive curvature if the normal vector is inward pointing. On
the normal vector points
inward into . On
the normal vector points inward into . On
, the normal vector
points from, i.e. inward with respect totowards
and outward with respect to
As in the binary caseandare inhibitors of, respectively. The constants
and
are
and
Lagrange multipliers corresponding to the constraints (5.2).
In the last equation, (5.6),, andare the inward pointing, unit tangent vectors of the curves, andat triple points. The requirement that the three unit vectors sum tozero is equivalent to the condition that the three curves meet at 120 degree angles.
The proof of Theorem 5.1 consists of several steps. In the first step, one constructs an assembly of exact
double bubbles and compute its energy. Take $K$ exact double bubbles
whose two bubbles areandfor $k=1$ , 2, $K$ . The area ofis. Take $K$ distinct points
in $D$ and $K$ angles, whereis the unit circle. Scale down eachby a factor , rotate by the angle
and place it in $D$ centered atMore precisely, let
be the aﬃne transformation$\Omega_{1}$$\Omega_{2}$$\Omega_{1}$$\kappa_{2}$$\Omega_{2}$$\partial\Omega_{1}\backslash \partial\Omega_{2},$$\kappa_{0}$$\partial\Omega_{1}\cap\partial\Omega_{2},$$\partial\Omega_{2}\backslash \partial\Omega_{1}$$\partial\Omega_{1}\backslash \partial\Omega_{2}$$\Omega_{1}$$\partial\Omega_{2}\backslash \partial\Omega_{1)}$$\Omega_{2}$$\Omega_{1}$$\Omega_{1}$$I_{\Omega_{1}}$$\partial\Omega_{2}\backslash \partial\Omega_{1}$$\Omega_{1}$$I_{\Omega_{2}}$$v_{2}$$\nu_{1},$$\partial\Omega_{1}\backslash \partial\Omega_{2},$$\partial\Omega_{1}\cap\partial\Omega_{2}$$\Omega_{2}$$\Omega_{2}.$$\Omega_{2}$$\lambda_{1}$$\lambda_{2}$$\nu_{0}$$\partial\Omega_{1}\cap\partial\Omega_{2}$$B^{k}$$B_{2}^{k}$$B_{l}^{k}$,$B^{k}$$\mathbb{S}^{1}$$B_{1}^{k}$$\theta^{k}\in \mathbb{S}^{1}$$\xi^{k}$$w_{i}^{k}$$\theta^{k}$$\epsilon$$\xi^{k}$$T^{k}=T_{\epsilon,\xi^{k},\theta^{k}}$$T^{k}(\hat{x})=T_{\epsilon,\xi^{k},\theta^{k}}(\hat{x})=\epsilon e^{i\theta^{k}}\hat{x}+\xi^{k}$and then map

into $D$ by$B^{k}$$\xi=(\xi^{1}, \ldots, \xi^{K})$$\mathcal{J}_{T}(T(B))$$=$,$\theta=(\theta^{1}, .

(5.7)

The image is a small double bubble denoted, and the collection
is an assembly of exact double bubbles denoted by $T(B)$ . This $T(B)$ depends, and $w=\{w_{i}^{k}\}$ . One finds the energy of $T(B)$ as follows.

$T^{k}$$(T^{1}(B^{1}), T^{2}(B^{2}), \cdots, T^{K}(B^{K}))$on,$T^{k}(B^{k})$\theta^{K})$$\epsilon\sum_{k=1}^{K}\sum_{i=0}^{2}2a_{il}^{k_{r\prime}k}+(\log\frac{1}{\epsilon})\epsilon^{4}\sum_{k=1}^{K}\sum_{i,j=1}^{2}\frac{\gamma_{ij}w_{i}^{k}w_{j}^{k}}{4\pi}+\epsilon^{4}\sum_{k=1}^{K}\sum_{i,j=1}^{2}\frac{\gamma_{vj}}{2}\int_{B_{7}^{k}}\int_{B_{\supset}^{k}}\frac{1}{2\pi}\log\frac{1}{|\hat{x}-\hat{y}|}d\hat{x}d\hat{y}$$+ \epsilon^{4}\sum_{k=1}^{K}\sum_{i,j=1}^{2}\frac{\gamma_{ij}}{2}w_{i}^{k}w_{j}^{k}R(\xi^{k}, \xi^{k})+\epsilon^{4}\sum_{k\neq l}\sum_{i,j=1}^{2}\frac{\gamma_{ij}}{2}w_{i}^{k}w_{j}^{l}G(\xi^{k}, \xi^{l})+O(|\gamma|\epsilon^{5})$

.

(5.8)

In the second step, perturb eachin a special way to define a restricted class of perturbed double
bubble assemblies. There are actually two parts in the perturbation. First move the two triple points ofvertically in opposite directions by the same amount. Connect the new triple points by three circular arcs.
The two sets bounded by the new arcs still have the areas
respectively and the radii
andof the newarcs satisfy the condition
However the 120 degree angle condition at triple points
no longer holds for the new arcs. In the second part of the restricted perturbation, the arcs are changed
to more general curves, while the areas of the two enclosed sets remain to be
and the triple points areunchanged. This perturbed double bubble is denoted
It is scaled down by and mapped into $D$ by thesame
The collection $T(P)=(T^{1}(P^{1}), T^{2}(P^{2}), T^{K}(P^{K}))$ is an assembly of perturbed double bubbles.
All assemblies obtained this way form a restricted class of perturbed double bubble assemblies. This class is
specified by, and $w.$
It turns out that each assembly in a restricted class is identified by an element of a Hilbert space. The
element consists of $3K$ functions
and $K$ numbers
for $k=1$ , 2, , $K$ and $i=1$ , 2, O. Collectively they
are denoted by,where. Within the, andrestricted classbecomes a functional on the Hilbert space.
In each restricted class there is an element) that locally minimizes
within the restricted class.
This third step is most technical, involving an error estimate of the exact double bubble assembly $T(B)$ ,proving the positivity of the second variation of
at $T(B)$ , and a fixed point argument. It is shownthatsatisfies a weakened version of $(5.3)-(5.5)$ where the constants
may vary from one
andperturbed double bubble to another perturbed double bubble in the assembly.
To fix this problem and also to have the 120 degree angle condition (5.6) satisfied, revisit the restrictedclass of perturbed double bubble assemblies. Since this class is specified by, the energy minimizing$B^{k}$$B^{k}$$w_{1}^{k}$$w_{2}^{k}$$\rho_{i}^{k}$$(\rho_{1}^{k})^{-1}-(\rho_{2}^{k})^{1}=(\rho_{0}^{k})^{-1}$$w_{i}^{k}$$P^{k}$$\epsilon$$T^{k}$$\xi,$$\theta$$\eta^{k}$$\phi_{i}^{k}$$\cdots$$(\phi, \eta)$$\phi=(\phi^{1}, \phi^{2}, \ldots, \phi^{K})$$\phi^{k}=(\phi_{1}^{k}, \phi_{2}^{k}, \phi_{0}^{k})$$\eta=(\eta^{1}, \eta^{2}, \cdots, \eta^{K})$$\mathcal{J}_{T}$$(\phi^{*})\eta^{*}$$\mathcal{J}_{T}$$\mathcal{J}_{T}$$(\phi^{*}, \eta^{*})$$\lambda_{1}$$(\xi, \theta, w)$$\lambda_{2}$94Figure 2: First row: two dimensional symmetric and asymmetric double bubbles. Second row: three dimensional symmetric and asymmetric double bubbles.
element
in this class should be denoted by,
In the fourth step one finds thatthe energy) of this element is close to that of,given in (5.8). Treating, and , one minimizes it with respect tothis quantity as a function ofand finds a minimum. If one uses the restricted class of assemblies specified by this particular, then thelocally energy minimizing element,) solves $(5.3)-(5.5)$ exactly and also satisfies
the 120 degree angle condition (5.6) at triple points. This completes the sketch of the proof.
$(\phi^{*} \xi, \theta, w)$$(\phi^{*}, \eta^{*})$$\mathcal{J}(\phi^{*} \xi, \theta, w)$$\eta^{*}(\xi,$$\theta,$$w$$\eta^{*}(\xi, \theta, w)$$\xi,$$\theta$$\mathcal{J}_{T}(T(B))$$(\xi, \theta, w)$$w$$(\xi^{*}, \theta^{*}, w^{*})$$(\xi^{*}, \theta^{*}, w^{*})$$(\phi^{*} \xi^{*}, \theta^{*}, w^{*})$

6$\eta^{*}(\xi^{*}, \theta^{*}, w^{*})$

Conclusions

I have mainly discussed stationary assemblies of the functionals
andin this article. But implications
go beyond these two functionals.
In the study of biological processes of organism development, the Gierer-Meinhardt system [14, 15, 18]
is an often used model for morphogenesis patterns on animal coats and skin pigmentation. This is anactivator-inhibitor type system of two partial diﬀerential equations on a bounded domain $D$ :$\mathcal{J}_{B}$$u_{t}= \epsilon^{2}\Delta u-u+\frac{u^{p}}{(1+\kappa u^{p})v^{q}}$;$\mathcal{J}\tau$$\tau v_{t}=d\triangle v-v+\frac{u^{r}}{v^{s}}$

,

(6.1)

with the zero Neumann boundary condition for both variables and . One can show formally that if is of
the order , the component of a stationary solution has a limit as. This limit is a positive constanton a subset of $D$ and is on
Lebesgue measure
is determined by the parameters in (6.1),
and most interestingly satisfies the same equation (2.20); see [36] for detail.
Despite the lack of a free energy functional, the Gierer-Meinhardt system contains mechanisms that
produce growth and inhibition. The Euler-Lagrange equation (2.20) of
can be regarded as a singular
limit of (6.1), a simplification that retains the key properties of (6.1).
This article should have convi\’{n}ced the reader that the functionals, although minimalist, areandpowerful enough to model pattern formation. I hope that more sophisticated patterns of interest will emerge
from deep analysis of these inhibitory systems.
$u$

$d$

$v$

$\epsilonarrow 0$$u$$\frac{1}{\epsilon}$$0$$\Omega$$D\backslash \Omega.$$\prime The$$|\Omega|$$\Omega$$\mathcal{J}_{B}$$\mathcal{J}_{B}$$\mathcal{J}_{T}$95
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