G-Convergence and Homogenization

G G-Convergence and Homogenization G Dal Maso, SISSA, Trieste, Italy ª 2006 Elsevier Ltd. All rights reserved.

Introduction Several asymptotic problems in the calculus of variations lead to the following question: given a sequence F k of functionals, defined on a suitable function space, does there exist a functional F such that the solutions of the minimum problems for F k 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

It follows immediately from the definition that, if F k -converges to F , then F k þ G -converges to F þ G for every continuous function G : U ! R. The first general property of -limits is lower semicontinuity: if F k -converges to F , then F is lower semicontinuous on U; that is, F ðuÞ  lim inf F ðuk Þ k!1

for every u 2 U and for every sequence uk converging to u in U. Another important property of -convergence is compactness: every sequence F k has a -convergent subsequence. For every k assume that the function F k has a minimum point uk . The following property is the link between -convergence and convergence of minimizers: if F k -converges to F and uk converges to u, then u is a minimum point of F and F k (uk ) converges to F (u), hence min F ðvÞ ¼ lim min F k ðvÞ v2U

Let U be a topological space with a countable base and let F k be a sequence of functions defined on U with values in the extended real line R := R [ {1, þ1}. We say that F k -converges to a function F : U ! R, or that F is the -limit of F k , if for every u 2 U the following conditions are satisfied: 1. For every sequence uk converging to u in U we have F ðuÞ  lim inf F k ðuk Þ k!1

2. There exists a sequence uk converging to u in U such that F ðuÞ ¼ lim F k ðuk Þ k!1

Property (1) appears to be a variant of the usual definition of lower semicontinuity. Property (2) requires the existence, for every u 2 U, of a ‘‘recovery sequence,’’ which provides an approximation of the value of F at u by means of values attained by F k near u.

k!1 v2U

½1

Under suitable coerciveness assumptions, the convergence of uk is obtained by a compactness argument. We recall that a sequence of functions F k is said to be equicoercive if for every t 2 R there exists a compact set Kt (independent of k) such that fu 2 U: F k ðuÞ  tg  Kt

½2

for every k. If F k is equicoercive and -converges to F , the previous result implies that [1] holds. If, in addition, F is not identically þ1, then the sequence uk of minimizers considered above has a subsequence ukj which converges to a minimizer u of F . The whole sequence uk converges to u whenever F has a unique minimizer u. In many applications to the calculus of variations, U is the Lebesgue space Lp (; Rm ), with  a bounded open subset of Rn and 1  p < þ1, but the effective domains of the functionals F k , defined as {u 2 U: F k (u) 2 R}, are often contained in the Sobolev space W 1,p (; Rm ), composed of all functions u 2 Lp (; Rm ) whose distributional gradient

450 G-Convergence and Homogenization

ru belongs to Lp (; Rmn ). When one considers homogeneous Dirichlet boundary conditions, the effective domains of the functionals F k are often contained in the smaller Sobolev space W01,p (; Rm ), composed of all functions of W 1,p (; Rm ) which vanish on the boundary @, technically defined as m 1,p the closure of C1 (; Rm ). 0 (; R ) in W In this case, the equicoerciveness condition [2] can be obtained by using Rellich’s theorem, which 1,p asserts that the natural embedding of W0 (; Rm ) m p into L (; R ) is compact. Therefore, a sequence of functionals F k defined on Lp (; Rm ) is equicoercive if there exists a constant  > 0 such that F k ðuÞ  

Z

jrujp dx

 1,p

for every u 2 W0 (; Rm ), while F k (u) = þ1 for every u 2 = W01,p (; Rm ).

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 composite materials, the physical parameters (such as electric and thermal conductivity, elasticity coefficients, 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 complex. 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 mathematical 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)n be the open unit cube in Rn centered at 0. We say that a function u defined on Rn is Q-periodic if, for every z 2 Rn with integer coordinates, we have u(x þ z) = u(x) for every x 2 Rn . Let f : Rn  Rmn ! [0, þ1) be a function such that x 7! f (x, ) is measurable and Q-periodic on Rn for every  2 Rmn and  7! f (x, ) is convex on Rmn for every x 2 Rn . Given a bounded open set   Rn and a constant p > 1, let F " : Lp (; Rm ) ! [0, þ1] be the family of functionals defined by (R F " ðuÞ :¼



1;p

if u 2 W0 ð; Rm Þ

f ðx="; ruÞ dx

otherwise

þ1

In the applications to composite materials, the functional 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 jjp  f ðx; Þ  ð1 þ jjp Þ

½3

for every x 2  and every  2 Rmn . Then for every sequence "k ! 0 the functionals F "k -converge to the functional F hom : Lp (; Rm ) ! [0, þ1] defined by (R 1;p f ðruÞ dx if u 2 W0 ð; Rm Þ F hom ðuÞ :¼  hom þ1 otherwise ½4 The integrand fhom : Rmn ! [0, þ1) is obtained by solving the cell problem Z fhom ðÞ :¼ min f ðx;  þ rwÞ dx ½5 1;p w2Wper ðQ;Rm Þ

Q

1,p where Wper (Q; Rm ) denotes the space of functions 1,p n w 2 Wloc (R ; Rm ) which are Q-periodic. The function fhom is always convex and satisfies [3]. If it is strictly convex, the basic properties of -convergence imply that for every g 2 Lq (; Rm ), with 1=p þ 1=q = 1, the solutions u" of the minimum problems Z h   i x ½6 min f ; rv  gðxÞv dx 1;p " v2W0 ð;Rm Þ 

converge in Lp (; Rm ), as " ! 0, to the solution u of the minimum problem Z min ½ fhom ðrvÞ  gðxÞv dx ½7 v2W01;p ð;Rm Þ



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; Þ ¼

n 1X aij ðxÞj i 2 i;j¼1

½8

with aij (x) Q-periodic, the function fhom takes the form fhom ðÞ ¼

n 1X ahom j i 2 i;j¼1 ij

for suitable constant coefficients ahom ij . By considering the Euler equations of the problems [6] and [7] in this special case, from the previous result we obtain the homogenization theorem for symmetric elliptic operators in divergence form, which asserts that for every g 2 L2 () the solutions u" of the Dirichlet problems n  x  X  Di aij Dj u" ðxÞ ¼ gðxÞ on  " i;j¼1 u" ðxÞ ¼ 0

on @

converge in L2 () to the solution u of the Dirichlet problem 

n X

ahom Di Dj uðxÞ ¼ gðxÞ ij

on 

i;j¼1

uðxÞ ¼ 0

on @

An extensive literature is devoted to precise estimates of the homogenized coefficients ahom ij , depending on various structure conditions on the periodic coefficients aij (x). Some of these estimates are based on a clever use of the variational formula [5]. Explicit formulas for ahom are known in the case ij of layered materials, which correspond to the case where Rn is periodically partitioned into parallel layers on which the coefficients aij (x) take constant values. Easy examples show that, even if the composite material is isotropic at a microscopic layer (i.e., aij (x) = a(x)ij for some scalar function a(x)), the homogenized material can be anisotropic (i.e., ahom 6¼ aij ), due to the anisotropy of the periodic ij 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 "k -converges

451

to a functional F hom : Lp (; Rm ) ! [0, þ1] of the form [4], but this time fhom : Rmn ! [0, þ1) cannot be obtained by solving a problem in the unit cell. Instead, it is given by the asymptotic formula Z 1 fhom ðÞ :¼ lim n min f ðx;  þ rwÞ dx R!1 R w2W 1;p ðQR ;Rm Þ QR 0 where QR := (R=2, R=2)n 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 g" : Rn  Rmn ! [0, þ1) are arbitrary Borel functions satisfying [3], with constants independent of ", and G" : Lp (; Rm ) ! [0, þ1] are defined by (R g ðx; ruÞ dx if u 2 W01;p ð; Rm Þ G" ðuÞ :¼  " þ1 otherwise then there exists a sequence "k ! 0 such that the functionals G"k -converge to a functional G of the form (R 1;p gðx; ruÞ dx if u 2 W0 ð; Rm Þ GðuÞ :¼  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 g"k (x, ). The indirect connection between these integrands can be obtained by introducing the functions M" (x, , ) defined, for x 2 ,  2 Rmn , and 0 <  < dist(x, @), by Z M" ðx; ; Þ :¼ min g" ðy;  þ rwÞ dy w2W01;p ðBðx;ÞÞ

Bðx;Þ

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"k -converges to G if and only if M"k ðx; ; Þ jBðx; Þj k!1 M"k ðx; ; Þ ¼ lim sup lim sup jBðx; Þj !0 k!1

gðx; Þ ¼ lim inf lim inf !0

for almost every x 2  and every  2 Rmn . Similar results have also been proved for integral functionals of the form (R g ðx; u; ruÞ dx if u 2 W01;p ð; Rm Þ G" ðuÞ :¼  " þ1 otherwise under suitable structure conditions for the integrands g" .

452 G-Convergence and Homogenization

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   Rn and a compact set K  Q, both with smooth boundaries, for every " > 0 we consider the perforated sets [ ð"z þ "KÞ ½9 " :¼ n z2Z"

where Z" is the set of vectors z 2 Rn with integer coordinates such that "z þ "Q  . Given g 2 L2 (), let F " : L2 () ! [0, þ1] be the functionals defined by (R h i 2 1 jruj  gu dx if u 2 W01;2 ðÞ F ðuÞ :¼ " 2 "

otherwise

þ1

½10 Minimizing [10] is equivalent to solving the mixed problems u" ¼ g u" ¼ 0 @u" ¼0 @

on " on @

½11

on @" n@

The homogenization formula [5] is still valid, with minor modifications. It leads to a matrix of coefficients ahom such that ij Z n X ahom   :¼ min j þ rwj2 dx j i ij i;j¼1

1;2 w2Wper ðQÞ

QnK

for every  2 Rn . For every sequence "k ! 0 the -limit of the functionals F "k is the functional F : L2 () ! [0, þ1] defined by # 8 " n R 1X > > > ahom Dj u Di u  mgu dx > ij <  2 i;j¼1 F ðuÞ :¼ > if u 2 W01;2 ðÞ > > > : 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  " and u" 2 W 1,2 (), that u" are harmonic on n  0

converge in L2 () to the solution u of the Dirichlet problem 

n X

ahom ij Di Dj u ¼ mg

on 

i;j¼1

u¼0

on @

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 [ ð"z þ "n=ðn2Þ KÞ ½12 " :¼ n z2Z"

with "n=(n2) replaced by exp (1="2 ) if n = 2, while the case n = 1 gives only trivial results. Given g 2 L2 (), let G" : L2 () ! [0, þ1] be the functionals defined by i (R h 2 1 jruj  gu dx if u 2 W01;2 ð" Þ  2 " G" ðuÞ :¼ ½13 þ1 otherwise Minimizing [13] is equivalent to solving the Dirichlet problems  u" ¼ g on " ½14 u" ¼ 0 on @" For every sequence "k ! 0 the -limit of the functionals G"k is the functional G : L2 () ! [0, þ1] defined by (R h i 2 1 c 2 jruj þ u  gu dx if u 2 W01;2 ðÞ  2 2 GðuÞ :¼ otherwise

þ1 where, for n  3, c :¼ capðKÞ :¼

inf 1

w2Cc ðRn Þ w¼1 on K

Z n

jrwj2 dx

R

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 L2 () to the solution u if the Dirichlet problem  u þ cu ¼ g u¼0

on  on @

½15

In the electrostatic interpretation of these problems, the boundary @" is a conductor kept at potential

G-Convergence and Homogenization

453

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.

To study the behavior of [17] as " ! 0, it is convenient to change variables, so that the scaled are deformations v(x1 , x2 , x3 ) := u(x1 , x2 , "x3 ) defined on the same domain   1 1  :¼ S  ; 2 2

Dimension Reduction Problems

The scaled energy density W" : R33 ! [0, þ1] is then defined as   1 W" ðF1 jF2 jF3 Þ :¼ W F1 jF2 j F3 "

In the study of thin elastic structures, like plates, membranes, rods, and strings, it is customary to approximate the mechanical behavior of a thin threedimensional body by an effective theory for two- or one-dimensional elastic bodies. -convergence provides a useful tool for a rigorous deduction of the lowerdimensional 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  ; 2 2 where " > 0 and S is a bounded open subset of R2 with smooth boundary. We assume that the body is hyperelastic, with stored elastic energy Z WðruÞ dx "

where u : " ! R3 is the deformation. The energy density W : R33 ! [0, þ1], depending on the material, is continuous and frame indifferent; that is, W(QF) = W(F) for every rotation Q and every F 2 R33 , where QF denotes the usual product of 33 matrices. We assume that W vanishes on the set SO(3) of rotations, is of class C2 in a neighborhood SO(3), and satisfies the inequality 2

WðFÞ   dist ðF; SOð3ÞÞ for every F 2 R

33

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 "2 f (x1 , x2 ), with f 2 L2 (S; R3 ). We assume that f is balanced; that is, Z Z f dx ¼ 0; x ^ f dx ¼ 0 "

"

Stable equilibria are then obtained by minimizing the functionals Z   WðruÞ  "2 f  u dx ½17 " 3

on W 1,2 (" ; R ).

where (F1 jF2 jF3 ) denotes the 33 matrix with columns F1 , F2 , and F3 . This implies that Z   WðruÞ  "2 f  u dx " Z   ¼" W" ðrvÞ  "2 f  v dx 

The asymptotic behavior of the minimizers of these functionals can be obtained from the knowledge of the -limit of the functionals F " : L2 (; R3 ) ! [0, þ1] defined by 8 Z <1 W" ðrvÞ dx if v 2 W 1;2 ð; R3 Þ F " ðvÞ :¼ "2  : þ1 otherwise Let us fix a sequence "k ! 0. The -limit of F "k turns out to be finite on the set (S; R3 ) of all isometric embeddings of S into R3 of class W 2,2 ; that is, v 2 (S; R3 ) if and only if v 2 W 2, 2 (S; R3 ) and (rv)T rv = I a.e. on S. The elements of (S; R3 ) will be often regarded as maps from  into R3 , independent of x3 . To describe the -limit, we introduce the quadratic form Q3 defined on R33 by Q3 ðFÞ :¼ 12 D2 WðIÞ½F; F which is the density of the linearized energy for the three-dimensional problem, and the quadratic form Q2 defined on the space of symmetric 2  2 matrices by   a11 a12 Q2 a12 a22 0 1 a11 a12 b1 B C :¼ min Q3 @ a12 a22 b2 A ðb1 ;b2 ;b3 Þ2R3

b1

b2

b3

The -limit of F "k is the functional F : L2 (; R3 ) ! [0, þ1] defined by 1R Q ðAÞ dx if v 2 ðS; R3 Þ F ðvÞ :¼ 12  2 þ1 otherwise

454 G-Convergence and Homogenization

where A(x1 , x2 ) denotes the second fundamental form of v; that is, Aij :¼ Di Dj v  

½18

with normal vector  := D1 v ^ D2 v. The equicoerciveness of the functionals F " in L2 (; R3 ) 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 Z   min WðruÞ  "2 f  u dx u2W 1;2 ð" ;R3 Þ

¼"

3

"

min

v2ðS;R3 Þ

Z S

1 Q2 ðAÞ  f  v dx0 þ oð"3 Þ 12

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 "k ! 0 the sequence F "k -converges to the functional F : L1 () ! [0, þ1] defined by R R if  u ¼ m  WðuÞ dx F ðuÞ :¼ 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 j þ jE j = m, which is equivalent to jE j ¼

0

with x := (x1 , x2 ) and A defined by [18]. For every " > 0 let u" be a minimizer of [17] and let v" (x1 , x2 , x3 ) := u" (x1 , x2 , "x3 ). Then the basic properties of -convergence imply that there exists a sequence "k ! 0 such that v"k (x1 , x2 , x3 ) converges in L2 (; R3 ) to a solution v(x1 , x2 ) of the minimum problem

Z 1 Q2 ðAÞ  f  v dx0 ½19 min v2ðS;R3 Þ S 12 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 transitions deals with a fluid with mass m, under isothermal conditions, confined in a bounded open subset  of Rn 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 " : L1 () ! [0, þ1] has the form 8R h i 2 2 < WðuÞ þ " jruj dx if u 2 AðmÞ  ½20 F " ðuÞ :¼ : þ1 otherwise whereR A(m) is the set of all functions u 2 W 1, 2 () with  u = m. We assume that W : R ! [0, þ 1) is continuous and that there exist ,  2 R, with jj < m < jj, 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 

 jj  m 

½21

From the basic properties of -convergence, we deduce that Z h i min WðuÞ þ "2 jruj2 dx ! 0 ½22 u2AðmÞ



and that there exists a sequence "k ! 0 such that the minimizers u"k of F "k converge in L1 () to a function u which takes only the values  and  and satisfies [21]. This result can be improved by considering the rescaled functionals 1 G" ðuÞ :¼ F " ðuÞ "

½23

where F " is defined by [20]. Then for every sequence "k ! 0 the sequence G"k -converges to the functional G : L1 () ! [0, þ1] defined by  2cPðE ; Þ if u 2 Mð; ; mÞ GðuÞ :¼ þ1 otherwise where c :¼

Z



pffiffiffiffiffiffiffiffiffiffiffi WðtÞ dt



and PðE; Þ :¼ sup

Z E

div ’ dx : ’ 2

C1c ð; Rn Þ; j’j

1

is the Caccioppoli–De Giorgi perimeter of E in , which coincides with the (n  1)-dimensional measure 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.

G-Convergence and Homogenization

As the functionals [20] and [23] have the same minimizers, we deduce that there exists a sequence "k ! 0 such that the minimizers u"k of F "k converge in L1 () 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 j. Moreover, [22] can be improved, and we obtain min

u2W 1;2 ðÞ

F " ðuÞ ¼ "2cPðE ; Þ þ oð"Þ

Similar results have been proved when the term jruj2 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: (Z min jruj2 dx þ Hn1 ðK \ Þ ðu;KÞ2A

nK

þ

Z

) 2

ju  gj dx

½24

nK

where  is a bounded open subset of Rn , Hn1 denotes the (n  1)-dimensional Hausdorff measure, g 2 L1 (), and A is the set of all pairs (u, K) with K compact, K  Rn , and u 2 C1 ( 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) of the minimum problem [24], the set K 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 oversegmentation, 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

455

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 Rn 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 : L1 () ! [0, þ1] used for the weak formulation of [24] is defined by (R 2 n1 ðJu Þ if u 2 GSBVðÞ  jruj dx þ H F ðuÞ :¼ þ1 otherwise ½25 where Ju is the jump set of u, defined in a measuretheoretical way as the set of points x 2  such that Z 1 lim sup juðyÞ  aj dy > 0 !0 jBðx; Þj Bðx;Þ for every a 2 R. For every g 2 L1 (), the functional Z F ðuÞ þ ju  gj2 dx 

is lower semicontinuous and coercive on L1 (); therefore, the minimum problem  Z 2 min F ðuÞ þ ju  gj dx ½26 u2L1 ðÞ



has a solution. The connection with the Mumford– Shah problem is given by the following regularity result, proved by Ennio De Giorgi and his collaborators: if u is a solution of [26] and Ju is its closure, then Hn1 ( \ (Ju nJu )) = 0, u 2 C1 (n Ju ), and (u, Ju ) 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 F " : L1 () ! [0, þ1] defined by 8 Z   <1 f " Avðjruj2 ; x; "Þ dx if u 2 W 1;2 ðÞ F " ðuÞ :¼ "  : þ1 otherwise

456 G-Convergence and Homogenization

where 2

Avðjruj ; x; "Þ Z 1 jruðyÞj2 dy :¼ jBðx; "Þ \ j Bðx;"Þ\ and f : [0, þ1) ! [0, þ1) is any increasing continuous function with f (0) = 0, f 0 (0) = 1, and f (t) ! 1=2 as t ! þ1. Then for every sequence "k ! 0 the sequence F "k -converges to F . Given g 2 L1 (), for every " > 0 let u" be a solution of the minimum problem  Z   1 min f " Avðjruj2 ; x; "Þ dx u2W 1;2 ðÞ "  Z 2 þ ju  gj dx 

From the basic properties of -convergence it follows that there exists a sequence "k ! 0 such that u"k converges in L1 () to a solution u of [26], so that (u, Ju ) is a solution of [24]. Other approximations by nonlocal functionals use finite differences instead of averages of gradients. A different approximation can be obtained by using the local functionals G" : (L1 ())2 ! [0, þ1] defined by

8Z " 1 2 2 > > jrvj hðvÞ dx g ðvÞjruj þ þ > <  " 2 2" G" ðu; vÞ :¼ > if ðu; vÞ 2 ðW 1;2 ðÞÞ2 > > : þ1 otherwise where g" (t) := " þ t2 , 0 < " << ", and h(t) := (1  t)2 for 0  t  1, while h(t) := þ1 otherwise. Let G : (L1 ())2 ! [0, þ1] be the functional defined by  F ðuÞ if v ¼ 1 a:e: on  Gðu; vÞ :¼ þ1 otherwise where F is defined [25]. Then for every sequence "k ! 0 the sequence G"k -converges to G. Given g 2 L1 (), for every " > 0 let (u" , v" ) be a solution of the minimum problem Z h " min g" ðvÞjruj2 þ jrvj2 2 ðu;vÞ2ðW 1;2 ðÞÞ2 

1 2 þ hðvÞ þ ju  gj dx ½27 2" From the basic properties of -convergence it follows that there exists a sequence "k ! 0 such that u"k converges in L1 () to a solution u of [26], so that (u, Ju ) is a solution of [24]. The approximation of the solutions of [24] based on [27] has been used to construct numerical algorithms for image segmentation.

Free discontinuity problems similar to [24] appear in the mathematical treatment of Griffith’s model in fracture mechanics. In this case, u is a vector-valued function, which represents the deformation of an elastic body, the first term in [24] is replaced by a more general integral functional which represents the energy stored in the elastic region nK, while the second term is interpreted as the energy dissipated to produce the crack K. An approximation based on minimum problems similar to [27] has been used to construct numerical algorithms to study the process of crack growth in brittle materials. An important research line, connected with these problems, has been developed in the last years to derive the macroscopic theories of fracture mechanics from the microscopic theories of interatomic interactions. Using -convergence, some theories expressed in the language of continuum mechanics can be obtained as limits of discrete variational models on lattices, as the distance between neighboring points tends to zero. See also: Convex Analysis and Duality Methods; Elliptic Differential Equations: Linear Theory; Free Interfaces and Free Discontinuities: Variational Problems; Geometric Measure Theory; Image Processing: Mathematics; Variational Techniques for Ginzburg– Landau Energies; Variational Techniques for Microstructures.

Further Reading Allaire G (2002) Shape Optimization by the Homogenization Method. Berlin: Springer. Ambrosio L, Fusco N, and Pallara D (2000) Functions of Bounded Variation and Free Discontinuity Problems. Oxford: Oxford University Press. Bakhvalov N and Panasenko G (1989) Homogenisation: Averaging Processes in Periodic Media. Mathematical Problems in the Mechanics of Composite Materials (translated from the Russian by D. Leıtes). Dordrecht: Kluwer. Bensoussan A, Lions JL, and Papanicolaou GC (1978) Asymptotic Analysis for Periodic Structures. Amsterdam: North-Holland. Braides A (1998) Approximation of Free-Discontinuity Problems, Lecture Notes in Mathematics, vol. 1694. Berlin: Springer. Braides A (2002) -Convergence for Beginners. Oxford: Oxford University Press. Braides A and Defranceschi A (1998) Homogenization of Multiple Integrals. Oxford: Oxford University Press. Cherkaev A and Kohn RV (eds.) (1997) Topics in the Mathematical Modelling of Composite Materials. Boston: Birkha¨user. Christensen RM (1979) Mechanics of Composite Materials. New York: Wiley. Cioranescu D and Donato P (1999) An Introduction to Homogenization. New York: The Clarendon Press, Oxford University Press. Dal Maso G (1993) An Introduction to -Convergence. Boston: Birkha¨user. Friesecke G, James RD, and Mu¨ller S (2002) A theorem on geometric rigidity and the derivation of nonlinear plate theory

Gauge Theoretic Invariants of 4-Manifolds from three-dimensional elasticity. Communications on Pure and Applied Mathematics 55: 1461–1506. Jikov VV, Kozlov SM, and Oleınik OA (1994) Homogenization of Differential Operators and Integral Functionals (translated from the Russian by G. A. Yosifian). Berlin: Springer. Milton GW (2002) The Theory of Composites. Cambridge: Cambridge University Press.

457

Oleınik OA, Shamaev AS, and Yosifian GA (1992) Mathematical Problems in Elasticity and Homogenization. Amsterdam: North-Holland. Panasenko G (2005) Multi-scale Modelling for Structures and Composites. Dordrecht: Springer. Sanchez-Palencia E (1980) Nonhomogeneous Media and Vibration Theory, Lecture Notes in Physics, vol. 127. Berlin: Springer.

Gauge Theoretic Invariants of 4-Manifolds S Bauer, Universita¨t Bielefeld, Bielefeld, Germany ª 2006 Elsevier Ltd. All rights reserved.

Introduction Poincare´ duality is fundamental in the study of manifolds. In the case of an orientable closed manifold X, this duality appears as an isomorphism : H k ðX; ZÞ ! Hnk ðX; ZÞ between integral cohomology and homology. The map is defined by cap product with a chosen orientation class. This article focuses on dimension n = 4, where Poincare´ duality induces a bilinear form Q on H2 (X; Z) by use of the Kronecker pairing Qð; 0 Þ ¼ h

1

ðÞ; 0 i 2 Z

One of the outstanding achievements of modern topology, the classification of simply connected topological 4-manifolds by Freedman (1982), can be phrased in terms of the intersection pairing Q. Indeed, two simply connected differentiable 4-manifolds X and X0 are orientation preservingly homeomorphic if and only if the associated pairings Q and Q0 are equivalent. Freedman’s classification scheme has been extended to also cover a wide range of fundamental groups, resulting in a fair understanding of topological 4-manifolds (Freedman and Quinn 1990). When it comes to differentiable 4-manifolds, the situation changes drastically. On the one hand, there is an abundance of topological 4-manifolds which do not admit a differentiable structure at all. On the other hand, there also are topological 4-manifolds supporting infinitely many distinct differentiable structures. A classification of differentiable 4-manifolds up to differentiable equivalence seems out of reach of current technology, even in the most simple cases. The discrepancy between topological and differentiable 4-manifolds was uncovered by gauge-theoretic methods, applying the concepts of instantons and of monopoles. In order to study these, one has to equip a 4-manifold both with a Riemannian metric and some

additional structure: a Hermitian rank-2 bundle in the case of instantons and a spinc -structure in the case of monopoles. Given such data, instantons and monopoles arise as solutions to partial differential equations the gauge equivalence classes of which form finite-dimensional moduli spaces. As it turns out, these moduli spaces encode significant information about the differentiable structures of the underlying 4-manifolds. A decoding of such information contained in the instanton moduli and in the monopole moduli is achieved through Donaldson invariants and Seiberg– Witten invariants, respectively. This article outlines these theories from a mathematical point of view.

Instantons and Donaldson Invariants Let X denote a closed, connected, oriented differentiable Riemannian 4-manifold. We will consider a principal bundle P over X with fiber a compact Lie group G with Lie algebra g. Connections on P form an infinite-dimensional affine space A(P) = A0 þ 1 (X; gP ) modeled on the vector space of 1-forms with values in the adjoint bundle gP ¼ P AdðGÞ g The curvature FA 2 2 (X, gP ) of a connection A is a gP -valued 2-form satisfying the Bianchi identity DA FA = 0. The group G of principal bundle automorphisms of P acts in a natural way on the space of connections with quotient space BðPÞ ¼ AðPÞ=G The Yang–Mills functional YM : AðPÞ ! R 0 associates to a connection A the norm square Z 2 kFA k ¼  trðFA ^ FA Þ X

of its curvature. Here denotes the Hodge star operator defined by the metric on X and the orientation. The metric tr: g  g ! R is Ad(G)invariant and hence YM is invariant under the