Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics

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Example G(x, y) = c

=jx  yj

D2

; then the only

requirement on the function u in [128] is



2½’

uðrÞ¼c

; ½r¼1 ½131

All objects defined by the scaled covariance [128]

are labeled with the interval [l

, l[. For instance,

a Gaussian volume element 

, l[

is abbreviated

to 

, l[

A Coarse-Graining Operator

The following coarse-graining operator has been

used for constructing a parabolic semigroup equa-

tion in the scaling variable (Brydges et al. 1998):

F :¼ S

l=l

 

½l

;l½

 F ½132

where the convolution product is by definition



½l

;l½

 F



ð’Þ¼



d

½l

;l½

ð ÞFð’ þ Þ

The coarse-graining operator P

rescales the con-

volution of a Gaussian volume element 

, l[

so that

all volume elements entering the construction of the

semigroup renormalization equation are scale

independent.

Some properties of the coarse-gr aining operator:



= P



The scaled eigenfunctions of the coarse-graining

operator are Wick-ordered monomials (Wurm

and Berg 2002)

: ’

ðxÞ:

½l

;1½



n½’

: ’



½l

;1½

½133

Note that P

preserves the scale range.



Let H be the generator of the coarse-graining

operator

H :¼





l¼l

;



¼ l

½134

The semigroup renormalization equation (a.k.a.

the flow equation)



Fð’Þ¼HP

Fð’Þ

Fð’Þ¼Fð’Þ

½135

Brydges et al. have applied the coarse-graining

operator to the quantum field theory known as

‘‘’

’’ (more precisely the Wick-ordered Lagrangian

of ’

). The flow equation [135] plays the role of

the ‘‘-function’’ equation in perturbative quantum

field theory.

Functional Integrals in Quantum Field Theory

Functional integrals in quantum field theory have

been modeled to some extent on path integrals in

quantum mechanics: mutatis mutandis, the defini-

tion [23] of Gaussian volume elements, the diagram

expansion [30], the property [36] of linear maps,

semiclassical expansions [87], the homotopy theo-

rem [114], and the scaling eqns [135 ] apply to

functional integrals in quantum field theory. The

time ordering encoded in a path integral becom es a

chronological ordering dictated by light cones in

functional integrals of fields on Minkowski fields.

The fundamental difference betw een quantum

mechanics (systems with a finite number of degrees

of freedom) and quantum field theory (systems with

an infinite number of degrees of freedom) can be

said to be ‘‘radiative corrections.’’ In quantum field

theory, the concept of ‘‘particle’’ is intrinsically

associated to the concept of ‘‘field.’’ A particle is

affected by its field. Its mass and charge are

modified by the surrounding fields, namely its own

and other fields interacting with it. One speaks of

‘‘bare mass’’ and ‘‘renormalized mass’’ when the

bare mass is renormalized by surrounding fields.

Computing radiative corrections is a delicate proce-

dure because the Green functions G defined by [25]

are singular. Reg ularization techniques have been

developed for handling singular Green functions.

Particles in quantum mechanics are simply particles,

and bosons and fermions can be treated separately.

Not so in quantum field theory. Therefore, the

configuration space in quantum field theory is a

supermanifold. For functional integrals in this theory,

we refer the reader to the ‘‘Further reading’’ section, in

particular to the book of A Das for an introduction, to

the book of B DeWitt for an in-depth study, and to the

book of K Fujikawa and H Suzuki for applications to

quantum anomalies.

Concluding Remarks

The key issue in functional integration is the domain

of integration, that is, a function space. This infinite-

dimensional space, say X, cannot be considered as

the limit n = 1 of R

Concepts of R

stated without reference to D are

likely to be meaningful on X. Other approaches

which have been used for exploring X are



projective system of finite-dimensional spaces

coherently defined on X (DeWitt-Morette et al.

1979),



one-parameter curves on X (Figure 1), and



projecting X on finite-dimensional spaces (cylind-

rical integrals).

446 Functional Integration in Quantum Physics

Functional integration has advanced our under-

standing of infinite-dimensional spaces, and like all

good mathematical tools, it improves with usage.

See also: BRST Quantization; Euclidean Field Theory;

Feynman Path Integrals; Infinite-Dimensional

Hamiltonian Systems; Knot Theory and Physics;

Malliavin Calculus; Path Integrals in Noncommutative

Geometry; Quantum Mechanics: Foundations; Stationary

Phase Approximation; Topological Sigma Models.

Bibliography/Further Reading

The book (Cartier and DeWitt-Morette 2006) was nearing

completion while this article was being written; many results

and brief comments in this article are fully developed in the book.

The book (Grosche and Steiner 1998) includes a bibliography

with 945 references.

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Brydges DC, Dimock J, and Hurd TR (1998) Estimates on

renormalization group transformations. Can. J. Math. 50:

756–793. A non-gaussian fixed point for O

in 4-dimensions.

Communications in Mathematical Physics 198: 111–156.

Cartier P and DeWitt-Morette C (2006) Functional Integration;

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Plenum.

Das A (1993) Field Theory, a Path Integral Approach. Singapore:

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Functional Integration in Quantum Physics 447

G-Convergence and Homogenization

G Dal Maso, SISSA, Trieste, Italy

Introduction

Several asymptotic problems in the calculus of

variations lead to the following question: given a

sequence F

of functionals, defined on a suitable

function space, does there exist a functional F such

that the solutions of the minimum problems for F

converge to the solutions of the corresponding

minimum problems for F? -convergence, introduced

by Ennio De Giorgi and his collaborators in 1975, and

developed as a powerful tool to attack a wide range of

applied problems, provides a unified answer to this

kind of question.

Definition and Main Properties

Let U be a topological space with a countable

base and let F

be a sequence of functions defined

on U with values in the extende d real line

R := R [ {1, þ1}. We say that F

-converges

to a function F : U!

R,orthatF is the -lim it of

, if for every u 2Uthe following conditions are

satisfied:

1. For every sequence u

converging to u in U we

have

FðuÞlim inf

k!1

ðu

2. There exists a sequence u

converging to u in U

such that

FðuÞ¼lim

k!1

ðu

Property (1) appears to be a variant of the usual

definition of lower semicontinuity. Property (2)

requires the existence, for every u 2U, of a ‘‘recovery

sequence,’’ which provides an approximation of the

value of F at u by means of values attained by F

near u.

It follows immediately from the definition that,

if F

-converges to F, then F

þG-conve rges to

FþGfor every continuous function G: U!R.

The first general property of -limits is lower

semicontinuity: if F

-converges to F, then F is

lower semicontinuous on U; that is,

FðuÞlim inf

k!1

Fðu

for every u 2Uand for every sequence u

converg-

ing to u in U.

Another important property of -convergence is

compactness: every sequence F

has a -convergent

subsequence.

For every k assume that the function F

has a

minimum point u

. The following property is the

link between -convergence and convergence of

minimizers: if F

-converges to F and u

con-

verges to u, then u is a minimu m point of F and

) converges to F(u), hence

min

v2U

FðvÞ¼ lim

k!1

min

v2U

ðvÞ½1

Under suitable coerciveness assumptions, the

convergence of u

is obtained by a compactness

argument. We recall that a sequence of functions F

is said to be equicoercive if for every t 2 R there

exists a compact set K

(independent of k) such that

fu 2U:F

ðuÞtgK

½2

for every k.

If F

is equicoercive and -converges to F, the

previous result implies that [1] holds. If, in addition,

F is not identically þ1, then the sequence u

minimizers considered above has a subsequence u

which converges to a minimizer u of F. The whole

sequence u

converges to u whenever F has a unique

minimizer u.

In many applications to the calculus of variations,

U is the Lebesgue space L

(; R

), with  a

bounded open subset of R

and 1  p < þ1, but

the effective domains of the functionals F

, defined

as {u 2U: F

(u) 2 R}, are often contained in the

Sobolev space W

1,p

(; R

), composed of all func-

tions u 2 L

(; R

) whose distributional gradient

ru belongs to L

(; R

mn

). When one considers

homogeneous Dirichlet boundary conditions, the

effective domains of the functionals F

are often

contained in the smaller Sobolev space W

1,p

(; R

composed of all functions of W

1,p

(; R

) which

vanish on the bounda ry @, technically defined as

the closure of C

(; R

)inW

1,p

(; R

In this case, the equicoerciveness condition [2] can

be obtai ned by using Rellich’s theorem, which

asserts that the natural embedding of W

1,p

(; R

)

into L

(; R

) is compact. Therefore, a sequence of

functionals F

defined on L

(; R

) is equicoercive

if there exists a constant >0 such that

ðuÞ



jruj

for every u 2 W

1,p

(; R

), while F

(u) = þ1 for

every u =2W

1,p

(; R

Homogenization Problems

Many problems for composite materials (fibered or

stratified materials, porous media, materials with

many small holes or fissures, etc.) lead to the study

of mathematical models with many interacting scales,

which may differ by several orders of magnitude.

From a microscopic viewpoint, the systems considered

are highly inhomogeneous. Typically, in such com-

posite materials, the physical parameters (such as

electric and thermal conductivity, elasticity coeffi-

cients, etc.) are discontinuous and oscillate between

the different values characterizing each component.

When these components are intimately mixed,

these parameters oscillate very rapidly and the

microscopic structure becomes more and more com-

plex. On the other hand, the material becomes quite

simple from a macroscopic point of view, and it tends

to behave like an ideal homogeneous material, called

‘‘homogenized material.’’ The purpose of the mathe-

matical theory of homogenization is to describe this

limit process when the parameters which describe the

fineness of the microscopic structure tend to zero.

Homogenization problems are often treated by

studying the partial differential equations that

govern the physical properties under investigation.

Due to the small scale of the microscopic structure,

these equations contain some small parameters. The

mathematical problem consists then in the study of

the limit of the solutions of these equations when the

parameters tend to zero. -convergence is a very

useful tool to obtain homogenization results for

systems governed by variational principles, which

are the only ones described in this article.

Let Q := (1=2, 1=2)

be the open unit cube in R

centered at 0. We say that a function u defined on R

Q-periodic if, for every z 2 R

with integer coordi-

nates, we have u(x þ z) = u(x) for every x 2 R

Let f : R

 R

mn

! [0, þ1) be a function such

that x 7!f (x, ) is measurable and Q-periodic on R

for every  2 R

mn

and  7!f (x, ) is convex on R

mn

for every x 2 R

. Given a bounded open set   R

and a constant p > 1, let F

: L

(; R

) ! [0, þ1]

be the family of functionals defined by

ðuÞ :¼



fx="; ruðÞdx if u 2 W

1;p

ð; R

þ1 otherwise

(

In the applications to composite materials, the func-

tional F

represents the energy of the portion of the

material occupying the domain  . The fact that the

energy density depends on x=" reflects the "-periodic

structure of the material, which implies that the energy

density oscillates faster and faster as " ! 0.

Assume that there exist two constants   >0

such that

jj

 f ðx;Þð1 þjj

Þ½3

for every x 2  and every  2 R

mn

. Then for every

sequence "

! 0 the functionals F

-converge to

the functional F

hom

: L

(; R

) ! [0, þ1] defined by

hom

ðuÞ :¼



hom

ðruÞdx if u 2 W

1;p

ð; R

þ1 otherwise

(

½4

The integrand f

hom

: R

mn

! [0, þ1) is obtained by

solving the cell problem

hom

ðÞ :¼ min

w2W

1;p

per

ðQ;R

f ðx;þrwÞdx ½5

where W

1,p

per

(Q; R

) denotes the space of functions

w 2 W

1,p

loc

; R

) which are Q-periodic.

The function f

hom

is always convex and satisfies

[3]. If it is strictly convex, the basic properties of

-convergence imply that for every g 2 L

(; R

with 1=p þ 1=q = 1, the solutions u

of the minimum

problems

min

v2W

1;p

ð;R



; rv



 gðxÞv

dx ½ 6

converge in L

(; R

), as " ! 0, to the solut ion u of

the minimum problem

min

v2W

1;p

ð;R



hom

ðrvÞgðxÞv½dx ½7

450 G-Convergence and Homogenization

Similar results can be proved for nonhomogeneous

Dirichlet boundary conditions, as well as for

Neumann boundary conditions.

In the special case m = 1, p = 2, and

f ðx;Þ¼

i;j¼1

ðxÞ



½8

with a

(x) Q-periodic, the function f

hom

takes the

form

hom

ðÞ¼

i;j¼1

hom



for suitable constant coeffic ients a

hom

By considering the Euler equations of the prob-

lems [6] and [7] in this special case, from the

previous result we obtain the homogenization

theorem for symmetric elliptic operators in diver-

gence form, which asserts that for every g 2 L

()

the solutions u

of the Dirichlet problems



i;j¼1



ðxÞ



¼ gðxÞ on 

ðxÞ¼0on@

converge in L

() to the solution u of the Dirichlet

problem



i;j¼1

hom

uðxÞ¼gðxÞ on 

uðxÞ¼0on@

An extensive literature is devoted to precise

estimates of the homogenized coefficients a

hom

depending on various structure conditions on the

periodic coefficients a

(x). Some of these esti-

mates are based on a clever use of the variational

formula [5].

Explicit formulas for a

hom

are known in the case

of layered materials, which correspond to the case

where R

is periodically partitioned into parallel

layers on whi ch the coefficients a

(x) take consta nt

values.

Easy examples show that, even if the composite

material is isotropic at a microscopic layer (i.e.,

(x) = a(x)

for some scalar function a(x)), the

homogenized materi al can be anisotropic (i.e.,

hom

6¼ a

), due to the anisotro py of the periodic

function a(x), which describes the microscopic

distribution of the different components of the

composite material.

In the vector case m > 1, the convexity hypothesis

on  7!f (x, ) is not satisfied by the most interesting

functionals related to nonlinear elasticity. If  7!f ( x, )

is not convex, one can still prove that F

-converges

to a functional F

hom

: L

(; R

) ! [0, þ1]ofthe

form [4], but this time f

hom

: R

mn

! [0, þ1) cannot

be obtained by solving a problem in the unit cell.

Instead, it is given by the asymptotic formula

hom

ðÞ :¼ lim

R!1

min

w2W

1;p

ðQ

f ðx;þrwÞdx

where Q

:= (R=2, R=2)

is the open cube of side

R centered at 0. Similar formulas can be obtained

for quasiperiodic integrands f and for stochastic

homogenization problems.

In the nonperiodic case one can prove that, if

: R

 R

mn

! [0, þ1) are arbitrary Borel func-

tions satisfying [3], with constants independent of ",

and G

: L

(; R

) ! [0, þ1] are defined by

ðuÞ :¼



ðx; ruÞdx if u 2 W

1;p

ð; R

þ1 otherwise

(

then there exists a sequence "

! 0 such that the

functionals G

-converge to a funct ional G of the

form

GðuÞ :¼



gðx; ruÞdx if u 2 W

1;p

ð; R

þ1 otherwise

(

with g satisfying [3].

In this case, no easy formula provides the integrand

g(x, ) in terms of simple operations on the integrands

(x, ). The indirect connection between these

integrands can be obtained by introducing the

functions M

(x, , ) defined, for x 2 ,  2 R

mn

and 0 <<dist(x, @), by

ðx;;Þ :¼ min

w2W

1;p

ðBðx;ÞÞ

Bðx;Þ

ðy;þrwÞdy

where B(x, ) is the open ball with center x and radius

. These functions describe the local behavior of the

integrands g

in some special minimum problems. The

sequence G

-converges to G if and only if

gðx;Þ¼lim inf

!0

lim inf

k!1

ðx;;Þ

jBðx;Þj

¼ lim sup

!0

lim sup

k!1

ðx;;Þ

jBðx;Þj

for almost every x 2  and every  2 R

mn

Similar results have also been proved for integral

functionals of the form

ðuÞ :¼



ðx; u; ruÞdx if u 2 W

1;p

ð; R

þ1 otherwise

(

under suitable structure conditions for the inte-

grands g

G-Convergence and Homogenization 451

Perforated Domains

In some homogenization problems, the integrand is

fixed, but the domain depends on a small parameter "

and its boundary becomes more and more fragmented

as " ! 0. A typical example is given by periodically

perforated domains with small holes. Given a

bounded open set   R

and a compact set K  Q,

both with smooth boundaries, for every ">0we

consider the perforated sets



:¼ n

[

z2Z



ð"z þ "KÞ½9

where Z



is the set of vectors z 2 R

with integer

coordinates such that "z þ "Q  .

Given g 2 L

(), let F

: L

() ! [0, þ1] be the

functionals defined by

ðuÞ :¼



jruj

 gu

dx if u 2 W

1;2

ðÞ

þ1 otherwise

(

½10

Minimizing [10] is equivalent to solving the mixed

problems

u

¼ g on 

¼ 0on@ ½11

@

¼ 0on@

n@

The homogenization formula [5] is still valid, with

minor modifications. It leads to a matrix of

coefficients a

hom

such that

i;j¼1

hom



:¼ min

w2W

1;2

per

ðQÞ

QnK

j þrwj

for every  2 R

. For every sequence "

! 0the-limit

of the functionals F

is the functional F : L

() !

[0, þ1] defined by

FðuÞ :¼



i;j¼1

hom

u  mgu

if u 2 W

1;2

ðÞ

þ1 otherwise

where m := jQnKj is the volume fraction of the

sets 

Since a slight modification of the functionals F

satisfies an equicoerciveness condition, it follows

from the basic properties of -convergence that the

solutions u

of the mixed problems [11] in the

perforated domains [9], extended to the holes so

that u

are harmonic on n





and u

2 W

1,2

(),

converge in L

() to the solution u of the Dirichlet

problem



i;j¼1

hom

u ¼ mg on 

u ¼ 0on@

Therefore, the asymptotic effect of the small holes

with Neumann boundary condition is a change in

the coefficients of the elliptic equation.

In the case of Dirichlet boundary conditions, it is

interesting to consider perforated domains with

holes of a different size, namely



:¼ n

[

z2Z



ð"z þ "

n=ðn2Þ

KÞ½12

with "

n=(n2)

replaced by exp (1="

)ifn = 2, while

the case n = 1 gives only trivial results.

Given g 2 L

(), let G

: L

() ! [0, þ1] be the

functionals defined by

ðuÞ:¼



jruj

gu

dx if u 2W

1;2

ð

þ1 otherwise

(

½13

Minimizing [13] is equivalent to solving the Dirichlet

problems

u

¼ g on 

¼ 0on@



½14

For every sequence "

! 0the-limit of the

functionals G

is the functional G: L

() ! [0, þ1]

defined by

GðuÞ: ¼



jruj

gu

dx if u 2W

1;2

ðÞ

þ1 otherwise

(

where, for n 3,

c :¼ capðKÞ :¼ inf

w2C

ðR

w¼1onK

jrwj

Since a slight modification of the functionals G

satisfies an equicoerciveness condition, it follows

from the basic properties of -convergence that the

solutions u

of the Dirichlet problems [14] in the

perforated domains [12], extended as zero on

 n 

, converge in L

() to the solution u if the

Dirichlet problem

 u þ cu ¼ g on 

u ¼ 0on@

½15

In the electrostatic interpretation of these problems,

the boundary @

is a conductor kept at potential

452 G-Convergence and Homogenization

zero. The extra term cu in [15] is due to the electric

charges induced on @

by the charge distribution g.

These results on Dirichlet and Neumann boundary

conditions have been extended to more general

functionals and also to a wide class of nonperiodic

distributions of small holes.

Dimension Reduction Problems

In the study of thin elastic structures, like plates,

membranes, rods, and strings, it is customary to

approximate the mechanical behavior of a thin three-

dimensional body by an effective theory for two- or

one-dimensional elastic bodies. -convergence provides

a useful tool for a rigorous deduction of the lower-

dimensional theory.

Let us focus on the derivation of plate theory from

three-dimensional finite elasticity. The reference

configuration of the thin three-dimensional elastic

body is a cylinder of the form



:¼ S

;



where ">0andS is a bounded open subset of R

with smooth boundary. We assume that the body is

hyperelastic, with stored elastic energy



WðruÞdx

where u : 

! R

is the deformation. The energy

density W : R

33

![0, þ1], depending on the

material, is continuous and f rame indifferent; that

is, W(QF) = W(F) for every rotation Q and every

F 2 R

33

,whereQF denotes the usual product of

33 matrices. We assume that W vanishes on the

set SO(3) of rotations, is of class C

in a

neighborhood SO(3), and satisfies the inequality

WðFÞ dist

ðF; SOð3ÞÞ for every F 2 R

33

½16

with a constant >0.

Plate theory is obtained in the limit as " ! 0 when

the densities of the volume forces applied to the

body have the form "

f (x

, x

), with f 2 L

(S; R

We assume that f is balanced; that is,



f dx ¼ 0;



x ^ f dx ¼ 0

Stable equilibria are then obtained by minimizing

the functionals



WðruÞ"

f  u



dx ½17

on W

1,2

(

; R

To study the behavior of [17] as " ! 0, it is

convenient to change variables, so that the scaled

deformations v(x

, x

):= u(x

, x

, "x

) are

defined on the same domain

:¼ S

;



The scaled energy density W

: R

33

! [0, þ1]is

then defined as

ðF

Þ :¼ WF



where (F

) denotes the 33 matrix with

columns F

, F

, and F

. This implies that



WðruÞ"

f  u



¼ "



ðrvÞ"

f  v



The asymptotic behavior of the minimizers of

these functionals can be obtained from the knowl-

edge of the -limit of the functionals

: L

(; R

) ! [0, þ1] defined by

ðvÞ :¼



ðrvÞdx if v 2 W

1;2

ð; R

þ1 otherwise

Let us fix a sequence "

! 0. The -limit of F

turns out to be finite on the set (S; R

) of all

isometric embeddings of S into R

of class W

2,2

; that

is, v 2 (S; R

) if and only if v 2 W

2, 2

(S; R

) and

(rv)

rv = I a.e. on S. The elements of (S; R

) will

be often regarded as maps from  into R

independent of x

To describe the -limit, we introduce the quad-

ratic form Q

defined on R

33

ðFÞ :¼

WðIÞ½F; F

which is the density of the linearized energy for the

three-dimensional problem, and the quadratic form Q

defined on the space of symmetric 2  2matricesby



:¼ min

ðb

Þ2R

The -limit of F

is the functional F : L

(; R

) ! [0, þ1] defined by

FðvÞ :¼



ðAÞdx if v 2 ðS; R

þ1 otherwise



G-Convergence and Homogenization 453

where A(x

, x

) denotes the second fundamental

form of v; that is,

:¼D

v   ½18

with normal vector  := D

v ^ D

The equicoerciveness of the functionals F

(; R

) is not trivial for this problem: it follows

from [16] through a very deep geometric rigidity

estimate which generalizes Korn’s inequality

(see Friesecke et al. (2002)). The basic properties of

-convergence imply that

min

u2W

1;2

ð



WðruÞ"

f  u



¼ "

min

v2ðS;R

ðAÞf  v



þ oð"

with x

:= (x

, x

) and A defined by [18].

For every ">0letu

be a minimizer of [17] and let

, x

):= u

, x

, "x

). Then the basic proper-

ties of -convergence imply that there exists a sequence

! 0 such that v

, x

) converges in L

(; R

)

to a solution v(x

, x

) of the minimum problem

min

v2ðS;R

ðAÞf  v



½19

These results provide a sound mathematical justification

of the reduced two-dimensional theory of plates based

on the minimum problem [19].

Similar results have been proved for shells,

membranes, rods, and strings.

Phase Transition Problems

The Cahn–Hilliard gradient theory of phase transi-

tions deals with a fluid with mass m, under

isothermal cond itions, confined in a bounded open

subset  of R

with smooth boundary, whose Gibbs

free energy, per unit volume, is a prescribed function

W of the density distribution u. Given a small

parameter ">0, the energy functional F

: L

() !

[0, þ1] has the form

ðuÞ:¼



WðuÞþ"

jruj

dx if u 2AðmÞ

þ1 otherwise

½20

where A(m) is the set of all functions u 2W

1,2

()

with



u=m.

We assume that W : R ! [0, þ1) is continuous

and that there exist ,  2 R,withjj < m <jj,

such that W(t) = 0 if and only t =  or t = .

Moreover, we assume that W(t) !þ1 as t !1.

In the minimization of F

, the Gibbs free energy

W(u) favors the functions whose values are close to 

and , which represent the pure phases, while the

gradient term penalizes the transitions between

different phases.

It is easy to see that for every sequence "

the sequence F

-converges to the functional

F : L

() ![0, þ1] defined by

FðuÞ :¼



WðuÞdx if



u ¼ m

þ1 otherwise



The set M(, , m) of minimum points of F is

composed of all measurable functions u on  which

take only the values  and  (on E



and E



respectively), and satisfy the mass constraint jE



jþ

jE



j= m, which is equivalent to



j¼

 jjm

  

½21

From the basic properties of -convergence, we

deduce that

min

u2AðmÞ



WðuÞþ"

jruj

dx ! 0 ½22

and that there exists a sequence "

!0 such that the

minimizers u

of F

converge in L

()toa

function u which takes only the values  and 

and satisfies [21].

This result can be improved by considering the

rescaled functionals

ðuÞ :¼

ðuÞ½23

where F

is defined by [20]. Then for every

sequence "

!0 the sequence G

-converges to

the functional G: L

() ![0, þ1] defined by

GðuÞ :¼

2cPðE



; Þ if u 2 Mð; ; mÞ

þ1 otherwise



where

c :¼





ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

WðtÞ

and

PðE; Þ

:¼ sup

div ’ dx : ’ 2 C

ð; R

Þ; j’j1



is the Caccioppoli–De Giorgi perimeter of E in ,

which coincides with the (n  1)-dimensional mea-

sure of  \ @E when E is smooth enough.

Note that the effective domain A(m) of the

functionals G

is disjoint from the effective domain

of the limit functional G, which is the set of all

functions u 2 M(, , m) with P(E



, ) < þ1.

454 G-Convergence and Homogenization

As the functionals [20] and [23] have the same

minimizers, we deduce that there exists a sequence

!0 such that the minimizers u

of F

converge in

() to a function u which takes only the values

 and , satisfies [21], and fulfills the minimal

interface criterion

PðE



; ÞPðE; Þ

for every measurable set E   with jEj= jE



Moreover, [22] can be improved, and we obtain

min

u2W

1;2

ðÞ

ðuÞ¼"2cPðE



; Þþoð"Þ

Similar results have been proved when the term

jruj

in [20] is replaced by a general quadratic form

like [8], which leads to an anisotropic notion of

perimeter.

Free-Discontinuity Problems

Free-discontinuity problems are minimum problems

for functionals composed of two terms of different

nature: a bulk energy, typically given by a volume

integral depending on the gradient of an unknown

function u; and a surface energy, given by an

integral on the unknown discontinuity surface of u.

These problems arise in many different fields of

science and technology, such as liquid crystals,

fracture mechanics, and computer vision.

The prototype of free-discontinuity problems is

the minimum problem proposed by David Mumford

and Jayant Shah:

min

ðu;KÞ2A

nK

jruj

dx þH

n1

ðK \ Þ

(

nK

ju  gj

)

½24

where  is a bounded open subset of R

, H

n1

denotes the (n  1)-dimensional Hausdorff measure,

g 2 L

(), and A is the set of all pairs (u, K) with K

compact, K  R

,andu 2 C

( n K).

In the applications to image segmentation problems

the dimension n is 2 and the function g represents the

grey level of an image. Given a solution (u, K)ofthe

minimum problem [24],thesetK is interpreted as

the set of the relevant boundaries of the objects in

the image, while u provides a smoothed version of the

image. The first term in [24] has a regularizing effect,

the purpose of the second term is to avoid over-

segmentat ion, while the last term, called ‘ ‘fidelity term,’ ’

forces u to be close to g. Of course, in the applications

these terms are multiplied by different coefficients,

whose relative values are very important for image

segmentation problems, since they determine the

strength of the effect of each term. However, the

mathematical analysis of the problem can be easily

reduced to the case where all coefficients are equal to 1.

To solve [24], it is convenient to introduce a weak

formulation of the problem based on the space

GSBV() of generalized special functions with

bounded variation (see Ambrosio et al. (2000)).

Without entering into details, here it is enough to

say that every u 2 GSBV() has, at almost every

point, an approximate gradient ru in the sense of

geometric measure theory. This is a measurable map

from  into R

which coincides with the usual

gradient in the sense of distributions on every open

subset U of  such that u 2 W

1,1

(U).

The functional F : L

() ![0, þ1] used for the

weak formulation of [24] is defined by

FðuÞ:¼



jruj

dx þH

n1

ðJ

Þ if u 2GSBVðÞ

þ1 otherwise

(

½25

where J

is the jump set of u, defined in a measure-

theoretical way as the set of points x 2 such that

lim sup

!0

jBðx;Þj

Bðx;Þ

juðyÞajdy > 0

for every a 2 R.

For every g 2 L

(), the functional

FðuÞþ



ju  gj

is lower semicontinuous and coercive on L

();

therefore, the minimum problem

min

u2L

ðÞ

FðuÞþ



ju  gj



½26

has a solution. The connection with the Mumford–

Shah problem is given by the following regularity

result, proved by Ennio De Giorgi and his colla-

borators: if u is a solution of [26] and

is its

closure, then H

n1

( \ (J

)) = 0, u 2 C

(nJ

and (u,

) is a solution of [24].

Since the numerical treatment of [24] and [26] is

quite difficult, -convergence has been used to

approximate [26] by means of minimum problems

for integral functionals, whose minimizers can be

obtained by standard numerical techniques.

Let us consider the nonlocal functionals

: L

() ![0, þ1] defined by

ðuÞ :¼



f " Avðjruj

; x;"Þ



dx if u 2 W

1;2

ðÞ

þ1 otherwise

G-Convergence and Homogenization 455