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

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in the case with magnetic field. This can be made

rigorous (see Jerrard and Soner (2002) and Sandier

and Serfaty (to appear)), that is, one can expres s that

curl hiu; rui2



! 0as" ! 0 ½10

(or respectively curlhi , r

iþcurlA  2



! 0) in some weak functional norm, thus giving a

rigorous use of [8]. The quantity

ðuÞ¼curl hiu; rui½11

ð ; AÞ¼curl hi ; r

iþcurl A ¼ curl j þ h ½12

in the case with magnetic field, will thu s be called

the vorticity and be used to trace the vortices, in this

limit " ! 0. The relation

  2



! 0as" ! 0 ½13

states that it is close to being a measure.

This is also called the Jacobian determinant if

written (with differential forms) Ju = dhiu,dui=

hidu,dui= 2(u

 u

)dx

^ dx

, and unde r this

form it can be used in higher dimensions.

The Cost of Each Vortex

Here we investigate informally the cost of a vortex

of degree d. We know already that the characteristic

length scale of variation of u is ", and that (1 

juj

)

is strongly penalized. Thus, we may expect

that juj is close to 1 at a distance " of the zeros.

Assuming that x

is a zero of u, and taking formally

juj= 1 for jx  x

j", we may write u = e

i’

and

jruj= jr’j for jx  x

j".

Then, we have

Rjxx

j"

jruj



@Bðx

;rÞ

@’

@





@Bðx

;rÞ

@’

@

2r

dr ½14



4

2

¼ d

log

½15

where we have used the Cauchy–Schwarz inequality

for [14], and the characterization of the degree [7].

We may also observe that this lower bound is sharp

if @’=@ is constant, that is, if the phase is d (and

the vortex radial). The cost associated to juj in the

energy imposes the length scale " and is generally of

order 1 (jrujC="), thus negligible compared to

the cost associated to the phase, which blows up as

log 1=" as " ! 0.

The above estimate is only valid as long as

B(x

, R) does not contain any other zero of u.If

vortices get close to each other or become numer-

ous, one needs refined techniques to estimate their

cost. This can be done through a ‘‘ball-construction

method’’ introduced independently by Jerrard and

Sandier.

Evaluating the Total Interaction Cost of Vortices

In a first approach, one studies configurations which

satisfy the upper bound E

(u)  Cjlog "j. Then,

lower bounds of the type [15] show that the total

sum of the degrees (hence the total number of

vortices of nonzero degree) remains bounded as " ! 0.

Up to extraction, we may assume these zeros a

converge as " ! 0 to a finite set of points p

,witha

total degree still denoted d

. This can also be expressed

as (u

) ! 2



as " ! 0.

This is not the only case of interest, since

unbounded numbers of vortices do arise, especially

in the physical situation of the energy with magnetic

field, as we will see in the next section. However,

this hypothesis, which was made in the work of

Bethuel–Brezis–He´lein, makes the analysis easier

and already allows us to exhibit the main

phenomena.

Vortices in superconductors are generated by the

presence of the external magnetic field h

. For the

energy without magnetic field, this has to be

replaced by some boundary condition which forces

some degree. Bethuel–Brezis–He´lein considered the

fixed Dirichlet boundary condition u

= g on @,

where g is a fixed unit-valued map on @, of degree

d >0. This forces u to have a total degree d in .

However, the Neumann boundary condition, for

instance, can also be considered (the minimizers of

are then simply con stants, they are trivial, but

one can still look for other critical points).

Let us return to lower bounds in order t o look

for the next order term in the energy (still with

formal arguments). Cutting out holes [

B(p

, )of

fixed size  around the limiting vortices p

,wemay

assume that u = e

i’

in n[

B(p

, ) =



,with’ a

real-valued function, defined modulo 2. Minimiz-

ing the energy outside of the holes amounts to

solving

min

u:



u¼g on @

degðu;@Bðp

;ÞÞ¼d





jruj

358 Variational Techniques for Ginzburg–Landau Energies

This is a harmonic map proble m, whose solution is

given in terms of ’ by

’ ¼ 0in



@’

@

¼ ig;

@



on @

@Bðp

;Þ

@’

@

¼ 2d

and in terms of the harmonic conjugate  which is

the function (up to a constant) such that

r’ = r

,

 ¼ 0in



@

@

¼ ig;

@



on @

@Bðp

;Þ

@

@

¼ 2d

½16

As  ! 0,  behaves like the solution of



¼ 2



in 

@

@

¼ ig;

@



on @

½17

Hence, we have





jr’j





jrj

’





jr

¼ 

log



þ W

ðp

; ...; p

þ oð1Þ as  ! 0 ½18

where

ða

; ...; a

Þ¼

i6¼j

log jp

 p

 

Rða

@



ig;

@



½19

and R(x) =

(x) 

log jx  p

j. The function

W was introduced by Bethuel–Brezis–He´lein and

called the renormalized energy, since it consists in

the part of the energy that is left after subtracting

the ‘‘infinite part’’ in jlog "j from E

. It contains the

(logarithmic) interaction energy between the vor-

tices: we see that vortices with degrees of same sign

repel one another while vortices with degrees of

opposite signs attract one another. The d

log 1=

term corresponds to the self-interaction, or cost of

the vortex of core of size ; it is what replaces the

infinite term in the formal calculation.

Now [18] is a good estimate for the optimal

energy outside of the holes, while the energy in holes

of size  can be bounded below by [15]. Given the

degree d

on the boundary @B(p

, ) of the small

hole, B(p

, ) contains one or several zeros of u of

degrees 

with total degree



= d

. In view of

[15], since the cost of a vortex of degree d grows like

d

jlog "j, and since the infimum of



under the

constraint



= d

is 

= sign(d

), the least costly

way to achieve this is to have jd

j vortices of degree

sign(d

). The smallest lower bound possible is thus

Bðp

;Þ

jruj

ð1 juj

 j d

jlog



þ C ½20

where the constant C can be described explicitly.

Adding up the results of [20] and [18], we find

ðuÞ

log



þ 

jlog



þ W

ðp

; ...; p

þ nC þ o



ð1Þþo

ð1Þ

 

jlog

þ W

ðp

; ...; p

þ nC þ o

ð1Þ½21

with equality only if u has jd

j zeros of degree

sign(d

) in each B(p

, ).

This provides a lower bound of the energy in

terms of the vortices. Moreover, this bound is sharp:

one can construct test confi gurations which have the

given limiting vortices (p

, d

), and an energy equal

to the right-hand side of [21].

One can thus deduce the behavior of global

minimizers of the energy. Give n the total degree

d = deg(g)>0on@ ,weneed

= d,andthe

lowest value achievable under this constraint in

the right-hand side of [21] is to have d

= 1for

every i, and thus to have exactly d vortices of

degree 1. Moreover, the limiting points p

’s

should minimize W. We thus are led to the first

main result.

Theorem 1 (Bethuel–Brezis–He´lein). Minimizers of

under the boundary condition u = g,deg(g) = d > 0,

have d zeros of degree 1, which converge as " ! 0

to a minimizer of W.

This result can be rephrased as a result of

-convergence of E

 djlog "j. It reduces the

minimization of E

to one of W, which is a finite-

dimensional problem (interaction of point charges).

Variational Techniques for Ginzburg–Landau Energies 359

Thus, we see again the interest of studying this

asymptotic limit " ! 0 because the vortices become

pointlike and the problem reduces to a finite-

dimensional one, or one of minimizing the vortex

interaction.

Further Results

A nonvariational approach also allowe d Bethuel–

Brezis–He´lein to prove a further correspondence

between E

and W: they obtained that critical points

of E

, under the upper bound E

 Cjlog "j, have

vortices which con verge to a critical point of W.

Other important resul ts are the study of the blow-up

profiles or solutions in the whole plane, by Brezis–

Merle–Rivie` re and Mironescu.

In two dimensions, the variational approach is

also used to solve inverse problems (construct

solutions) and study variants of the energy with

pinning (or weighted) terms.

The variational approach is also fruitful in higher

dimensions. In dimension 3, for example, vortices are

not points but vortex lines, and the Jacobian

Ju = d(iu,du) can be seen as a current carried by the

vortex line, with kJuktotal mass of the current equal to

 times the length of the line, and it was established by

Jerrard and Soner that Ju

is compact in some weak

sense, and converges, up to extraction, to some  times

integer-multiplicity rectifiable current J,with

lim inf

"!0

ðu

jlog "j

kJ k

In fact, a compl ete -convergence result of

=jlog "j can be proved, see the work of Alberti–

Baldo–Orlandi, and thus minimizing E

reduces at

the limit to minimizing the length of the line, leading

to straight lines, or in higher dimensions, to

codimension-2 minimal currents. This is a nontrivial

problem, contrarily to dimension 2, where the -

limit of E

=jlog "j is trivial, which required to go to

the lower-order term to find the nontrivial renorma-

lized energy limit W.

The Functional with Magnetic Field

The aim here is to achieve the same objective:

express or bound from below the energy by terms

which depend only on the vortices and their degrees.

The method consists in transposing the type of

analysis above taking into account the magnetic

field contribution to see how the external field

triggers the sudden appearance of vortices, and for

what values they appear (thus retrieving the critical

fields, etc.). One of the main difficulties consists in the

fact that the number of vortices becomes divergent,

which requires more delicate estimates. Also, it is then

no longer possible to study the convergence of the

individual zeros of , so one studies instead the limit of

rescalings of the vorticity measures ( , A).

Splitting of the Energy and Main Results

Let us recall that in the case with magnetic field, the

vorticity is given by [12]. In addition, we may

assume that the second set of equations in [5]

r

h ¼ j in ; h ¼ h

on @ ½22

is satisfied (if not, keeping fixed and choosing A

which satisfies this equation always decreases the

energy). Taking the curl of this equation, we find

exactly

h þ h ¼ ð ; AÞ in 

h ¼ h

on @

½23

Thus, the vorti city and the induced magnetic field

are in one-to-one correspondence with each other.

Combining it to the relation [13], we are led to the

approximate relation

 h þ h ’ 2



in 

h ¼ h

on @

½24

where again the a

’s are the vortex centers and d

’s

their degrees, well known in physics as the

‘‘London equation.’’ It shows how the magnetic

field is induced by the vortices which act like

‘‘charges,’’ and how the magnetic field ‘‘penetrates

the sample’’ around the positive vortex locations.

Of course this equation is only an approximation,

because the singularities at the a

’s, where h would

become infinite, are really smoothed out in ( , A);

however, the approximation is good far f rom

the vortex cores, just as [17] is an approximation

for [16].

It is then natural to introduce the field corre-

sponding to the vortex-free situation, which is h

where h

solves

h

þ h

¼ 0in

¼ 1on@

½25

is thus a fixed smooth function, depending only

on , and when there are no vortices, we expect h to

be approximately h

. Moreover, h

:= h  h

then solves

h

þ h

¼ ð ; AÞ’2



in 

¼ 0on@

½26

360 Variational Techniques for Ginzburg–Landau Energies

Defining the Green kernel G(. , y)by

 G þ G ¼ 

in 

G ¼ 0on@

½27

and S by S(x, y) = 2G(x, y) þ log jx  yj,forx far

enough from the a

’s, we may approximate h

ðxÞ¼2

Gðx; a

Þ½28

Using the second Ginzburg–Landau equation [22]

and the fact that j j1, we have jr

jjjj= jrhj,

thus G

( , A)  (1=2)



jrhj

þjh  h

. Plugging

in the decomposition h = h

þ h

and using an

integration by parts and [26], one finds

ð ; AÞ¼



jrh

þjh

 1j

þ h



rh

þðh

 1Þh



jrh

þjh

¼ h

þ h



ðh

 1Þð ; AÞ



jrh

þjh

½29

where J

is the constant (1=2)



jrh

þjh

 1j

The right-hand side of eqn [29] can be expressed

in terms of the vortices. First, using [26],we

have



1)( , A) ’2

1)(a

). Second,

the expression



jrh

þjh

can be treated exactly

like E

(u) in the previous section, using lower bounds for

the cost of vortices provided by the Jerrard–Sandier

method, we are led to the (approximate) relation



jrh

þjh



jlog

 

i6¼j

log ja

 a

þ 

i;j

Sða

; a

Þ½30

Combining this to [29] we find the decomposition

ð ; AÞh

þ 

jjlog "j

þ 2h

ðh

 1Þða

 

i6¼j

log ja

 a

þ 

i;j

Sða

; a

Þ½31

On the other hand, this inequality is sharp: as

before, given vortices a

, one can construct a

configuration ( , A) for which this is an equality,

at leading order.

In that relation, h

is a fixed energy, the energy

of the vortex-free configuration. To it are added the

intrinsic cost of each vortex jd

jjlog "j , the interac-

tion cost between vortices, and the interaction

between the vortices and the external field

2h

 1)(a

It is then simple, by minimizing the right-hand

side with respect to the vortices for a given h

, and

observing that h

 1  0, to deduce a few basic

facts about vortices: vortices of positive degree (and

of degree þ1) are preferred, each vo rtex costs

jlog "j, an d allows to gain at best an energy

2h

max jh

 1j when placed at the minimum of

 1. Therefore, vortices become favorable when

their cost becomes smaller than the gain, that is,

when h

becomes larger than the ‘‘first critical field’’



jlog "j

2jminðh

 1Þj

½32

We have the first main result.

Theorem 2 (Sandier–Serfaty). When " is small

enough and h

 H

, then minimizers of G

have

no vortices.

On the other hand, if h

 H

, the vortices

cannot all be located at the same minimum point of

 1, because their repulsion 

i6¼j

log ja

 a

would be infinite. There is thus a trade-off between

their repulsion and the cost for being far from the

minimum of h

 1. Only if n, the number of

vortices, is small compared to h

do the vortices

tend to concentrate near the minimum of h

 1. If

so, then, assuming for simplicity that the minimum

of h

 1 is achieved at a unique point p, and

denoting by Q the Hessian of h

 1atp,inthe

relation above (h

 1)(a

) can be approximated by

min (h

 1) þ (1=2)Q(a

 p) and thus G

( , A)by

ð ;AÞh

þnjlog "jþ2nh

minðh

1Þ

þh

Qða

pÞ



i6¼j

logja

a

jþn

Sðp;pÞ½33

From this relation, optimizing on ‘, the character-

istic distance to p and characteristic distance

between the vortices, we find that ‘ =

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

n=h

optimal.

Moreover, optimizing with respect to n, we find

that n should remain bounde d (as " ! 0) when

 H

þ O( log jlog "j). In that regime, rescaling

by setting x

= ((a

 p)=‘), we have the following

result:

Variational Techniques for Ginzburg–Landau Energies 361

Theorem 3 (Sandier–Serfaty). There exist fields

 H

þ C(n  1) log jlog "j such that when

 h

< H

nþ1

, minimizers of G

have n vortices

of degree 1, and the rescaled vortices x

’s tend to

minimize:

ðx

; ...; x

Þ¼

i6¼j

log jx

 x

þ n

i¼1

Qðx

Þ½34

If h

 H

 log jlog "j, then the optimal number

of vortices n becomes unbounded as " ! 0. The

analysis above still holds, but in order to get a

convergence of the vortices, one needs to rescale the

vorticity measure by n. There is an intermediate

regime, for log jlog "jh

 H

jlog "j for

which n should be 1 but still n  h

,so‘  1:

vortices are numerous, but still concentrate around p.

Rescaling by the scale ‘ as above, we prove that the

density of vortices (after dividing it by n)convergesto

a probability measure, minimizer of the energy

IðÞ¼

R

log jx  yjdðxÞdðyÞ

þ 

QðxÞdðxÞ½35

This is an averaged/continuous form of [34].

If h

 H

is of order jlog "j, then the optimal

number n becomes of order h

and the vortices no

longer concentrate around a single point.

The simplest approach is then to simply consider

the vorticity measure ( , A) and to rescale it by the

order n, hence by h

. Then (1=h

)( , A) con-

verges, after extraction, to some measure 



continuous version of [31] can thus be written, using

[12],as

ð ;AÞ



jlog "j



j



jþ



jrh





þjh





½36

where h





solves

 h





þ h





¼ 



in 





¼ 1on@

Again, this inequality can be proved to be sharp (by

a construction) and allows to show that minimizers

of G

have a vorticity ( , A) such that ( , A)=h

converges to a minimizer of

Gð



Þ¼

lim

"!0

jlog "j





j



jþ



jrh





þjh





In fact the stronger result holds, in that sense:

Theorem 4 (Sandier–S erfaty). G

-converges

to G.

The limit pr oblem of minimizing G turns out to

have a simple solution in terms of an obstacle

problem: the optimal 



is a uniform density of

vortices on a subdomain of  determined through a

free boundary problem (and depending on h

which is nonzero.

In all these regimes, we have thus been able to

identify the optimal number and repartition of

vortices through a  -convergence-type approach,

that is, by reducing the minimization of the energy

to the minimization of a limiting problem: w

or I or G,

according to the regime.

Further Results

Concerning vortices, in the same spirit as what was

done for E

, we can obtain necessary conditions

characterizing limiting vorticities obtained from

sequences of (nonminimizing) critical points of

the energy G

. They consist in passing to the limit

in the conservative form of the Ginzburg–Landau

equations [5].

Most of the results concerning the phase transi-

tions at the next critical fields H

and H

are also

obtained by nonvariational methods, and often by

linear analysis.

The study of the Ginzburg–Landau energy in non-

simply-connected domains is also very interesting

because it leads to nontrivial topological effects, since

in such domains there exist unit-valued maps with

nonzero degree (corresponding to permanent currents).

See also: Abelian Higgs Vortices; Aharonov–Bohm Effect;

Bose–Einstein Condensates; Gamma-Convergence and

Homogenization; Gauge Theory: Mathematical

Applications; Ginzburg–Landau Equation; High T

Superconductor Theory; Image Processing:

Mathematics; Superfluids; Topological Defects and Their

Homotopy Classification; Variational Techniques for

Microstructures.

Further Reading

Bethuel F, Brezis H, and He´lein F (1994) Ginzburg–Landau

Vortices. Boston: Birkha¨ user.

DeGennes PG (1966) Superconductivity of Metal and Alloys.

New York: Benjamin.

Jerrard RL and Soner HM (2002) The Jacobian and the

Ginzburg–Landau energy. Calculus of Variations and Partial

Differential Equations 14(2): 151–191.

Sandier E and Serfaty S Vortices in the Magnetic Ginzburg–

Landau Model. Birkha¨ user (monograph to appear).

Tinkham M (1996) Introduction to Superconductivity, 2nd edn.

McGraw-Hill.

362 Variational Techniques for Ginzburg–Landau Energies

Variational Techniques for Microstructures

G Dolzmann, University of Maryland, College Park,

MD, USA

Austenite–Martensite Transformations

and the Shape Memory Effect

Microstructures in materials that typically form in

response to phase transformations in the solid state,

and their impact on the elastic properties of these

materials have been known for centuries. The

discovery of the complex phase diagram of iron

revolutionized the production of steels at the end of

the nineteenth century. Starting in the 1980s, the

mathematical description of microstructures in the

framework of nonlinear elasticity has led to deep

analytical questions and surprising developments in

the calculus of variations and in nonlinear partial

differential equations.

The mathematical approach outlined here is based

on the following fundamental assumptions:

1. The observed configurations correspond to mini-

mizers of or elements of minimizing sequences

for an energy functional.

2. The qualitative properties of low energy states

are determined from the set of minima of the free

energy density.

Under these assumptions one aims at explaining

experimental observations and to predict material

properties based on minimizing an energy functional

of the form

IðuÞ¼



WðDuÞdx

Here  is an ideal, unstressed reference configura-

tion in R

, u :  !R

is an elastic deformation, and

W : M

mn

!R is the stored energy density. In the

case of physical interest, m = n = 2orm = n = 3. For

applications in elasticity we assume that m = n, but

this assumption is not needed in the general theory.

The energy density W and its structure depend

critically on the temperature. However, since we are

interested in the analysis of the material at a given

temperature, we do not include this dependence

explicitly.

The key ingredient of this model is the stored

energy density W which has to reflect the properties

of the specific material one wants to model.

Frequently these are alloys, in particular shape

memory alloys that undergo an austenite–martensite

transformation. For most materials a closed analytic

expression for W is not available. In the spirit of the

fundamental assumption (2) one therefore focuses

on the structure of the set of minima of W which is

determined from general invariance and symmetry

principles. We may assume that W 0 and that

K = {X: W(X) = 0} 6¼;. The principle of material

frame indifference then asserts that

WðRFÞ¼WðFÞ for all R 2SOðnÞ

Here SO(n) is the group of proper rotations, that is,

the set of all matrices R 2M

nn

with R

R = Id and

det R = 1.

The symmetry of the austenitic (high-temperature)

phase implies that the energy density in the

martensitic (low-temperature) phase is invariant

under all changes of basis that leave the underlying

lattice in the austenitic phase invariant. Therefore,

WðR

FRÞ¼WðFÞ for all R 2P

where P

is the point group of the austenite. In the

case of a cubic to tetragonal phase transformation,

this leads to K = SO(3) in the austenitic phase and to

K ¼ SOð3ÞU

[ SOð3ÞU

½1

with

¼ 

 e



I  e

 e

ðÞ½2

in the martensitic phase (see Figure 1). A set of the

form SO(n)U

is often referred to as an energy well.

The origin of the shape memory effect is the

availability of a rich class of geometric patterns in

which the martensitic phases can be arranged, thus

leading to a great flexibility of the material to

accommodate macroscopic deformations. Upon heat-

ing of the material above the transformation tem-

perature, the martensitic phases lose their stability

and the material returns to its unique shape in the

(a) (b) (c)

Figure 1 Two-dimensional cartoon of a cubic to tetragonal

phase transformation in a single crystal: (a) a cubic lattice, (b)

and (c) tetragonal variants which are stretched in directions e

and e

, respectively. (Sketch not to scale.)

Variational Techniques for Microstructures 363

austenitic phase. The two solutions of Hadamard’s

compatibility condition

 U

¼ a  b; Q 2 SOð3Þ

are given by



þ 1

2



 10

1  

2

00

þ 1

½3

and Q

= Q

(see Figure 2). The normals (in the

reference configuration) are given by (1, 1, 0)=

ﬃﬃﬃ

It is one of the successes of the theory that it

provides an analytical derivation of the normals to

the twinning planes.

The Direct Method in the Calculus

of Variations

The mathematical interest in the variational prob-

lems described in the previous section lies in the fact

that existence of minimizers cannot in general be

obtained by a straightforward application of the

direct methods in the calculus of variations. This

approach is based on the idea to (1) choose a

minimizing sequence for the functional I, (2) show

that this sequence is bounded and precompact,

and (3) prove that the functional is lower semicon-

tinuous with respect to the notion of convergence,

IðuÞlim inf

j!1

Iðu

Þ if u

! u

The typical choice is to seek u

in a suitable Sobolev

space W

1, p

(; R

) with 1 < p 1which is related

to growth and coercivity conditions for the energy

density W,

jFj

 c

 WðFÞc

jF j

þ 1



for all F 2M

mn

½4

This leads to weak compactness in W

1, p

(; R

)

(weak- compactness in W

1, 1

(; R

)) and to the

requirement of sequential weak lower semicontinu-

ity of the functional,

IðuÞlim inf

j!1

Iðu

Þ if u

* u in W

1;p

ð; R

(sequential weak- lower semicontinuity for p = 1).

Morrey’s fundamental work establishes a link

between convexity conditions for the energy density

and lower semicontinuity of the variational integral:

under suitable growth and coercivity conditions,

sequential weak- lower semicontinuity is equivalent

to quasiconvexity of the integrand.

Definition 1 A function W : M

mn

!R is said to be

quasiconvex at F if



WðFÞdx 



WFþ DðÞdx

for all  2W

1;1

ð; R

and for all open and bounded domains   R

with

(@) = 0. It is said to be quasiconvex if it is

quasiconvex at all F.

In the language of nonlinear elasticity, W is

quasiconvex if affine functions are minimizers of

the energy functional subject to their own boundary

conditions. The direct method implies the following

classical existence theorem.

Theorem 1 Suppose that W : M

mn

!R is quasi-

convex and satisf ies the growth and coercivity

condition [4]. Let u

1, p

(; R

). Then the varia-

tional problem: minimize I(u) in

A¼ u 2W

1;p

ð; R

Þ: u  u

1;p

ð; R

has a minimizer.

The remarkable fact is that the structure of the

zero set of a typical energy W modeling a phase-

transforming material in its low-temperature phase

prevents W from being quasiconvex. In order to see

this, let Q  R

be a cube with two of its sides

perpendicular to b = (1, 1, 0)=

ﬃﬃ

(

2) and let h be the

1-periodic function with h

= 0 on (0,  )andh

= 1on

(, 1) with  2(0, 1). Define v

(x) = U

x þ ah( jx  b)=j

and

ðxÞ¼min v

ðxÞ; distðx;@QÞ



¼ min U

x þ ahðjx  bÞ=j; distðx;@QÞ

where dist(x, @Q) = inf {kx  yk

, y 2 @Q }. Then

!u, u(x) = Cx strongly in L

(Q; R

)andweakly-

in W

1, 1

(Q; R

)withC = U

þ (1  )Q

=2K

where K is the zero set of W, see the previous section.

(a) (b) (c)

Figure 2 Formation of phase boundaries in a single crystal.

(a) The upper right half of the lattice deforms into phase I with

the constant deformation gradient U

, the lower left half of the

lattice deforms into phase II with constant deformation gradient

: (b) An additional rotation is needed to accomplish a

continuous deformation, see formula [3]. (c) A different config-

uration with a different orientation of the interface. (Sketch not

to scale.)

364 Variational Techniques for Microstructures

Moreover, Du

2{U

, Q

} except in a small transi-

tion layer of volume O(1=j)closeto@Q and

IðuÞ¼



WðCÞdx > lim inf

j!1

Iðu

Þ¼0

This inequality shows that the functional is not

weakly- lower semicontinuous and therefore W

fails to be quasiconvex. The oscillations of u

on a

scale 1=j are part of the mathematical model for the

microstructures frequently observed in shape mem-

ory alloys. More generally, whenever u is a Sobolev

function on a domain  such that Du takes only two

values, say Du 2{A, B}, on open sets which are not

empty and whose union is  (up to a set of measure

zero), then the tangential continuity of the deriva-

tives implies that the difference A  B is a matrix of

rank 1, A  B = a  b, and that the interfaces

between the regions with Du = A and Du = B are

hyperplanes with normal parallel to b. This state-

ment is usually referred to as ‘‘Hadamard’s compat-

ibility condition.’’ Moreover, the pattern in Figure 3

is known as a ‘‘simple laminate’’ and the matrices A

and B are said to be rank-1 connected.

Relaxation

The discussion in the previous section shows that the

variational problems related to models in materials

science typically fail to be weakly lower semicon-

tinuous. One approach which allows us to recover

the macroscopic energy of the system and the macro-

scopic stress–strain relation is to pass to the relaxed

variational problem which involves the quasiconvex

envelope of the energy density W.

Definition 2 Let W : M

mn

!R be given. The

function

¼ sup f : f  W; f quasiconvex

is called the quasiconvex envelope of W. Equivalently,

ðFÞ¼ inf

 2W

1;1

ð;R

jj



WðF þ DÞdx

This formula implies that W

is the macroscopic

energy of the system in the sense that it characterizes

the smallest energy per unit volume that is required

to subject a volume element to a deformation with

affine boundary conditions. Here the system is

allowed to minimize its energy with microstructures

at any scale, a mechanism which was already

explored in the previous section. The arguments in

this section prove that W

that the zero set of W

can be strictly larger than

the zero set of K, see Definition 4. The relaxed

functional is given by

ðuÞ¼



ðDuÞdx

Since W

satisfies the growth and coercivity

conditions [4] if they are satisfied by W,the

functional I

attains its minimum subject to given

boundary conditions. The functional I

is the

weakly lower semicontinuous envelope of I in the

sense that minimizing sequences for I contain

subsequences that converge to minimizers of I

and for all u there exists a sequence u

which

converges in W

1, p

(; R

)tou such that the

energies converge, I(u

) !I(u). However, a lot of

information in particular about oscillation patterns

might be lost in the passage from I to I

since the

knowledge of a minimizer u for I

does not

provide any immediate information about the

behavior of any minimizing sequence for I that

converges to u. Moreover, the minimization pro-

blem required in the definition of the relaxed

energy has been solved explicitly only for very

special energy densities.

In this context, one often relies on two related

notions of convexity, one sufficient and the other

necessary for quasiconvexity. For F 2M

mn

let

M(F) 2 R

d(m, n)

be the vector of all minors (sub-

determinants) of F. In the special case m = n = 2

we have M(F) = (F, det F) 2R

and for m = n = 3

we find M(F ) = (F, cof F, det F) 2 R

where cof

F is the 3  3 matrix of all 2  2 subdeterminants

of F.

Definition 3 Let W : M

mn

!R be given. The

function W is said to be polyconvex if there exists

a convex function g : R

d(m, n)

!R such that

W(F) = g (M(F)). The function W isrank-1convexif

it is convex along all rank-1 lines in M

mn

,thatis,the

function t 7!W(F þ tR)isconvexforallF 2 M

mn

and all R 2 M

mn

with rank(R) = 1.

λ /j

(1–λ)/j

Figure 3 Construction of a minimizing sequence u

with Du

fA, Bg in measure and affine boundary conditions u(x) = A þ

(1  )B Hadamard’s compatibility condition requires that A 

B = a  b is a rank-1 matrix and that the planar interfaces are

perpendicular to b.

Variational Techniques for Microstructures 365

All notions of convexity reduce to classical

convexity if m = 1orn = 1. In the vector-valued

case m, n > 1 the following implications are true:

f convex ) f polyconvex ) f quasiconvex

) f rank-1 convex

The reverse statements for the first two implications

are not true. Rank-1 convexity does not imply

quasiconvexity for m 3 and it is a fundamental

open problem with deep connections to harmonic

analysis to decide whether rank-1 convexity and

quasiconvexity are equivalent for m = n = 2.

The polyconvex and the rank-1 convex envelope

of an energy density W are defined analogously to

Definition 2. In view of the implications between the

different notions of convexity, one has W



 W

and essentially all explicitly known

relaxation formulas are based on the approach to

construct a candidate W



for W

and to verify that



is polyconvex. Then the inequalities become

equalities and one obtains a characterization for the

relaxed energy. This approach does not work for

extended-valued functions which are used in models

for incompressible materials since quasiconvexity

does not imply rank-1 convexity in this case.

However, for a model system of particular interest,

nematic elastomers, a complete characterization of

the relaxed energy, the macroscopic stress–strain

relation, and the macroscopic phase diagram have

been obtained.

Classical and Generalized Minimizers

The discussion of observed configurations as ele-

ments of minimizing sequences {u

} in the section

‘‘The direct method in the calculus of variations’’

leaves the question of the existence of minimizers

open. The answer cannot be obtained via the direct

methods since minimizing sequences do not need to

converge strongly to minimizers. One approach to

obtain the existence of solutions u with I(u) = 0isto

solve the differential relation Du 2K, u(x) = Fx on

@ by constructing special minimizing sequences

that converge strongly so that one can pass to

the limit in the energy integral. This idea has led

to surprising solutions u with affine boundary

conditions for the two-well problem where K =

SO(2)diag(,1=) [ SO(2)diag(1=,  ). However, the

structure of the solutions is intrinsically complicated

in the sense that the phase boundary has infinite

length unless the boundary conditions are given by

u(x) = Fx with F 2K.

More generally, the right tool to pass to the limit in

nonlinear functions of z

= Du

like the energy is the

‘‘Young measure’’ generated by a subsequence. It is

given by a family of probability measures 

that

provide statistical information about the distribution of

the values of z

close to a given point x. The existence

and the fundamental properties of Young measures are

described in the following theorem. For simplicity we

assume that the sequence z

is uniformly bounded.

Theorem 2 (Fundamental theorem on Young

measures). Let E  R

be measurable, L

(E) < 1,

and let z

: E !R

be a measurable and bounded

sequence. Then there exists a subsequence z

and a

weakly- measurable map  : E !M(R

) such that

the following assertions are true:

(i) The measures 

are non-negative probability

measures.

(ii) If there exists a compact set K such that u

in measure, then supp 

 K for a.e. x 2E.

(iii) If f 2C(R

) and if f(z

) is relatively weakly

compact in L

(E), then f (z

) * finL

(E)

where

f (x) = h

, f i.

Here h

, f i denotes the integration of the func-

tion f with respect to the measure 

. For example,

the Young measure generated by the sequence Du

constructed in the section ‘‘The direct method in the

calculus of variations’’ generates the Young measure



= (1=2)

þ (1=2)

(see Figure 3) and

Iðu

Þ¼



WðDu

Þdx



mn

WðYÞd

ðYÞdx ¼ 0

A Young measure generated by a sequence of

gradients is called a gradient Young measure

(GYM). It is said to be homogeneous if 

=  is

independent of x. We restrict our attention in the

following to homogeneous GYMs generated by

sequences that are bounded in L

. The importance

of quasiconvexity is also reflected in the following

characterization of homogeneous GYMs.

Theorem 3 A non-negative probability measure 

is a GYM if and only if there exists a compact set

K  M

mn

with supp  K and Jensen’s inequality

h, f if (h,idi) holds for all quasiconvex functions

f : M

mn

!R.

This motivates to characterize the generalized

limits of minimizing sequences as

ðKÞ¼  2MðKÞ: f ðh; idiÞ  h; f i

for all f : M

mn

! R quasiconvexg

where M(K) is the set of all probability measures

supported on K.If is generated by a sequence of

366 Variational Techniques for Microstructures

functions with affine boundary conditions

(x) = Fx, then h,idi= F. The set of all affine

deformations of the material that can be recovered

by heating (shape memory effect) is therefore given

as the set of all centers of mass of homogeneous

GYMs supported on K, the so-called ‘‘quasiconvex

hull’’ K

of K.

Definition 4 Suppose that K  M

mn

is compact.

We define the quasiconvex hull of K by

¼ F ¼h; idi:  2M

ðKÞ

There are several equivalent definitions of K

The foregoing definition corresponds to the defini-

tion of the convex hull of a set as the set of all

centers of mass of probability measures supported

on K (which satisfy Jensen’s inequality for all

convex f ). The set K

can also be defined as the

set of all points that cannot be separated by

quasiconvex functions from K or as the zero set of

the quasiconvex envelope of the distance function to

K. The ‘‘polyconvex hull’’ K

and the ‘‘rank-1

convex hull’’ K

are defined analogously by replac-

ing quasiconvexity with polyconvexity and rank-1

convexity in the foregoing definitions. It follows that

 K

and all of these inclusions can be

strict.

A particularly useful set of conditions are the

minors conditions

h; Mi¼M h; idiðÞ

for all minors M which follow from the weak

continuity of the minors. For example, if

K = {A, B}  M

22

, then any probability measure

supported on K is given by  = 

þ (1  )

. The

minors condition with M = det implies that

det A þð1  ÞBðÞ¼deth;idi¼h; deti

¼  det A þð1  Þdet B

This identity is equivalent to

ð1  ÞdetðA  BÞ¼0

and therefore the quasiconvex hull is equal to K if

and only if det (A  B) 6¼ 0. A very instructive

example is the set K = {(1, 3), (1, 3), (3, 1),

(3, 1)} viewed as a subset of the space of all

diagonal matrices in M

22

. It is frequently referred

to as a T

configuration. The rank-1 convex hull is

equal to the quasiconvex hull and given by the four

points, the line segments, and the square in the

center, the polyconvex hull is bounded by four

hyperbolic arcs, and the convex hull is the square

with the points as corners, see Figure 4.Itis

remarkable that the rank-1 convex hull is strictly

larger than the set K itself despite the fact that the

set K does not contain any rank-1 connections.

There are only a few examples in which explicit

characterizations of the convex hulls for sets

invariant under SO(n) have been obtained. For

K = SO(3)U

[ SO(3)U

(see [2]), one finds

¼ F 2M

33

: F

F ¼

ac 0

cb 0

001=

;

ab  c

¼ 

; a þ b þ 2jcj



)

The quasiconvex hull of the three-well problem [1]

is not known. In two dimensions one finds for

K ¼ SOð2ÞU

[[SOð2ÞU

;

det U

¼ 1; i ¼ 1; ...; n

that

¼ F 2 M

22

: det F ¼ 1; jFej

 max

i¼1;...;n



All examples in which envelopes of functions or hulls

of sets have been obtained explicitly are based on the

exceptional property that the polyconvex envelope

coincides with the rank-1 convex envelope. The T

configuration in Figure 4 is one of the few cases where

the quasiconvex hull is known to be different from the

polyconvex hull. The construction of quasiconvex

functions and the understanding of their properties is

one of the challenges left for the future.

Bibliographical Remarks

This article can only review some of the highlights

of mathematical developments related to models in

Figure 4 The four-point subset K in the space of all diagonal

matrices and its convex hulls: K

= K

are given by K, the line

segments and the shade square, K

is bounded by the dashed

hyperbolic arcs, and the convex hull is the outer square.

Variational Techniques for Microstructures 367