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

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Everything is gauge invariant, contrary to the

perturbative formulation, where a gauge fixing is

required to define the vector meson propagator.

By Weierstrass theorem, the integral is finite for any

finite number of links, the gauge group being compact.

Any other choice of the lattice action differing from

theWilsonactionofeqn[14] by terms of higher order in

a will have the same continuum limit: there is significant

freedom in the choice of the action.

In the language of statistical mechanics, the

Euclidean lattice formulation is a spin model.

Different choices of the action correspond to different

spin models. In the vicinity of a second-order phase

transition, however, the correlation length becomes

large with respect to the lattice spacing and all the

irrelevant terms become negligible. All the spin

models at the critical point belong to the same

universality class and define the same field theory.

This is what happens for QCD because of

asymptotic freedom. By renormalization group

arguments, the lattice spacing behaves as

aðÞ



expðb

Þ½15

at sufficiently large ,whereb

is the coefficient of

lowest-order term of the -function, b

is positive and 

is a physical scale. As  !1, a tends exponentially to

zero in physical units and the coarse structure of the

lattice becomes unimportant, indicating that the short-

distance limit in the definition of the Feynman integral

exists. The theory also develops a mass scale  which

insures the existence of a finite correlation length and

hence of the thermodynamical limit. In practice, when 

is increased, the lattice space becomes exponentially

small in physical units. As a consequence, however, the

physical scale becomes exponentially large in lattice

units, and an exponentially large lattice is needed to

insure the large-distance convergence. This makes life

difficult if the Feynman integral has to be computed

numerically.

Quarks

Fermion fields are defined on lattice sites. The

naive lattice transcription of the fermion term

in eqn [1] consists in replacing the covariant

derivatives by finite differences with parallel

transports to make the result gauge covariant. In

principle, D



(x) = U

(x) (x þ

)  (x) is a correct

definition. In practice, a more symmetric difference

is used which is correct O(a

), namely



ðxÞ

½UðxÞ ðx þ

ÞU

ðx 

Þ ðx 

Þ ½16

The fermionic Lagrangian then reads



ðxÞ½i6D

 m ð xÞ



x;x







ðxÞM

1

ðx ;x



ðx

Þ½17

It is convenient to indicate this expression in the

form S



1

, where is a large column whose

elements are labeled by the site x and by the

component . The functional integral over can

explicitly be done by using the standard rules of

integration on Grassman variables, since the action

is bilinear,

Z ¼



ðxÞd ðxÞd



ðxÞ

 expðS

½U



M Þ½18

The result is

Z ¼



ðxÞexpðS

½UÞdet M ½19

The effect of fermions is to multiply the weight by a

functional determinant which depends on the gauge

field configuration.

A problem exists, however, in this procedure

already at the level of free fermions, that is, putting

U = 1 in the action and in the determinant of

eqn [18] . The equation of motion reads, in Fourier

transform,







sin 2





 m



ðkÞ¼0 ½20

With respect to the continuum, the momentum



= 2k



=L has been replaced by its sinus. At

small values of p



, eqn [20] coincides with the

Dirac equation. However, an alternative solution

exists at p



 , for each  independently. The new

equation differs from the other by a change of sign

of 



. Changing sign of one of the gammas means

changing sign to 

 



, which is the

chirality of the fermion. Instead of one fermion,

we then have 2

= 16 fermion species, organized in

pairs with opposite chiralities. It is impossible to

have a single fermion with a given chirality. A

number of recipes have been proposed to circum-

vent this artifact of the lattice regulation, for

example, introduce by ha nd a term in the action

which removes the spurious particles in the limit of

zero lattice spacing (Wilson’s fermions); double the

lattice spacing by constructing two sublattices on

even and odd sites, respectively, which propagate

fermions of opposite chirality (staggered fermions),

Lattice Gauge Theory 277

so that the argument of the sinus in the derivative is

doubled. More recently, an idea which goes back to

Ginsparg and Wilson has been implemented, which

consists in replacing a strictly local equation of

motion like eqn [20] by an equation with the

same continuum limit which is nonlocal, but with a

nonlocality falling off exponentially at large

distances, a recipe which makes propagation of

chiral fermions possible. This is an important

improvement, even if very demanding in computer

power.

Numerical Simulations

Solving analytically the lattice version of QCD

would allow one to follow constructively all the

steps which bring to the definition of Z, that is, the

ultraviolet and the infrared limit, as explai ned

earlier. Presently that is out of reach. Also an

attempt by Wilson to solve the lattice renormaliza-

tion group equations by techniques of decimation is

not conclusive.

The pro blem can be attacked num erically. One

way would be to compute the integral numerically.

That is, however, prohibitive: it would be like

solving exactly the equations of motion for the

molecules of a gas. The lattice theory is in fact a

four-dimensional statistical mechanics with the

Boltzmann factor  = 2N

and Hamiltonian

equal to the Euclidean action. As in statistical

mechanics the way out is to create a significant

sample of configurations with weight exp (S

)

and to determine the field correlators which describe

physics by an average on this ensemble. This is done

by Monte Carlo techniques.

The basic principle is to start from an arbit-

rary field configuration and make a sequence of

random changes, normally on a single link at a

time, with uniform probability in the group

measure so as to converge toward the equilibrium

distribution exp (S

). For that purpose, the

probability P

to change from a configuration C

to another C

is constrained to obey the detailed

balance relation

expðS½CÞ ¼ P

expðS½C

Þ ½ 21

A common algorithm is known as metropolis. The

way to implement the condition (eqn [21]) is to accept

the new trial configuration C

if S[C

]  S[C], and to

accept it with probability exp ( [S(C

)  S(C)]) if

S[C

]  S[C]. An alternative method is known as

‘ ‘heat-bath’’. If the probability of the configuration for

one link at a fixed value of the other variables is

explicitly known, the change can be accepted with that

probability.

In the presence of dynamical quarks, the integral

eqn [18] is converted into an integral on bosonic

variables by inverting the matrix M:

Z ¼



ðxÞdðxÞdðxÞ

 expðS

½U

½M

M

1

Þ½22

The property has been used such that

d(x)d

(x) exp (

1

) = jdet Mj.A

metropolis updating is then performed on the

combined U



and  vari ables. To have a choice

of the trial uniform in the measure, an algorithm is

commonly used which is based on ergodicity,

known a s hybrid molecular dynamics. A fictitious

conjugate momentum is associated with all

variables, and a fictitious Hamiltonian is defined

by adding to the action, considered as a potential

energy, the sum of the squares of the conjugate

momenta. A classical evolution is then performed in

time by small steps which should displace the state

in phase space ergodically: the evolution is called a

trajectory. After a number of steps, a metropolis test

is made as explained above.

Typically, the computer time needed to produce a

significant configuration is proportional to the

volume V of the lattice for pure gauge systems, to

5=4

in the hybrid algorithm for full QCD.

As explained before, in order to have a good

approximation to the Feynman integral the lattice

spacing has to be small compared to the physical

scales, for example, with respect to the Compton

wavelength of the heaviest quark. On the other

hand, to control volume effects it has to be large

compared to the bigge st physical length, for

example, with respect to the Compton wavelength

of the lightest quark. Since there is a factor

top

 3  10

between these two lengths, the

lattice size needed would be prohibitive from

numerical point of view. In practice, lat tices of

size L

are affordable with L  64  128. For

this reason, only the light quarks u, d, s are ke pt,

which have mass smaller than the typical scale of

the theory, which can be identified as the square

root of the string tension. In the limit in which light

quark masses are small compared to QCD scale,

the Lagrangian is invariant under any unitary

mixing of them. A global SU(3) invariance exists,

which is known as flavor symmetry, and is broken

by the difference of quark masses. Heavier

quarks can be described by an effective theory,

since they have negligible dyn amical effects at low

energies.

278 Lattice Gauge Theory

A Selection of Physics Results

String Tension

A big excitement followed the first numerical

calculations by M Creutz at the beginning of the

1980s in which the static potential V(r) between a

quark and an antiquark was computed in pure-

gauge theory on the lattice. One way to measure it is

to measure the correlator of two Polyakov lines at a

distance r on a significant ensemble of field config-

urations. The Polyakov line is the parallel transport

in the fundamental representation along the time

axis across the lattice: with periodic boundary

conditions it is a closed loop, and hence it is gauge

invariant. It can be proved that the log of this

correlator is equal to V(r)aL

with L

the extension

of the lattice in the time direction. It was found that

VðrÞ¼r ½ 23

The parameter  is known as string tension. A

potential of the form eqn [23] means confinement:

an infini te amount of energy is required to pull apart

the particles at infinite distance. The parameter 

can be determined phenomenologically from the

mass spectrum of the mesons and 2  1 GeV.

What is measured on the lattice is

aðÞ

½24

where n is the distance of the two Polyakov lines in

lattice spacings and a() the lattice spacing in

physical units. In fact, the computer only produces

pure numbers. If the lattice QCD belongs to the

same universality class of QCD at the critical point,

that is, if the lattice really defines QCD, the

dependence of a()on is dictated by the

-function of the renormalization group. At suffi-

ciently large  = 6=g

aðÞ



latt

expðb

Þ½25

with b

= (11=3)N

=16

. 

latt

is the energy scale of

the theory. The measurement of the potential gives

indeed a dependence of the lattice spacing on 

consistent with eqn [25] and allows one to deter-

mine =

latt

. The absolute value of the lattice

spacing can be determined by comparison with the

physical value of the string tension. The theory is

able to produce a physical scale. The correlation

length is finite and as a consequence the infrared

limit of the Feynman integral exists.

Mass Spectrum

Any operator with the quantum numbers of a

particle can be used as interpolating field for it.

The correlator of the operator at large distances

behaves like a sum of exponentials exp (mr) with

m the masses of the particles with the same quantum

numbers. At large distances the lightest particle

dominates, especially if the operat or has a good

overlap, that is, if its matrix element between

vacuum and the state of the particle is the biggest.

From the correlators mr can be determined. On the

lattice r = na() so that, by eqn [25] what is really

determined is the ratio m=

latt

.If

latt

has been

determined, for example, from the string tension,

the mass of the particle results in physical units.

Alternatively, the ratios of any two masses can be

determined and the scale fixed by the value of one of

them. A good agreement is obtained already in pure

gauge (quenched approximation) indicating that the

quark loops are relevant at the level of 10%

typically. This fact supports the idea that the large

-limit is a good app roximation to reality, quark

loops being nonleading in that limit. The light

particle masses are more difficult to compute,

being sensitive to the masses of light quarks which

cannot be taken at realistic values due to computa-

tional difficulties: large lattices are required and big

fluctuations are present near the chiral point. The

spectrum of particles made of heavy quarks can be

computed using effective theories, and nicely fits

experiment. A byproduct is a precise determination

of the gauge coupling constant, competitive with

phenomenological determinations from short dis-

tance perturbative QCD.

Weak Interaction Matrix Elements

There exist matrix elements of currents (or products

thereof) entering in weak amplitudes which involve

large distances and are not computable in perturba-

tion theory. Lattice can be used to evaluate them.

Renormalization problems can appear in this

approach when the cutoff is removed, which,

however, are not difficulties of principle but only

of technical nature. This activity is of fundamental

importance to have precise predictions in order to

understand the limits of the standard model.

Finite-Temperature QCD and the Deconfinement

Transition

The static thermodynamics of a system of fields is

described by the partition function

¼ tr½exp ð H=TÞ ½26

It is easy to show that Z

is equal to the Euclidean

Feynman integral on the imaginary time interval

(0, 1=T) with boundary conditions in time periodic

for bosons and an tiperiodic for fermions. Indeed, the

Lattice Gauge Theory 279

Boltzmann factor is formally an imaginary time

evolution by 1/T. A lattice of extension L

with

 L

provides the partition function at a tem-

perature T = 1=aL

,ifa is the lattice spacing in

physical units.

Finite-temperature simulations are important to

investigate the transition from the phase in which

color is confined to a phase in which quarks and

gluons can propagate as free particles. This phase is

called deconfined phase or quark gluon plasma.

Big experiments at Brookhaven and at CERN are

looking for this phase transition in high-energy

collisions between heavy nuclei, but no definite

evidence has yet been produced for it. Lattice

simulations instead definitely prove that such a

transition exists. For pure SU(3) gau ge theory

(quenched) at T  270 MeV, a first-order phase

transition is observed, at which the string tension

vanishes. In a more realistic theory with

dynamical qua rks, a transition is also observed at

T  160 MeV, where chiral symmetry, which is

spontaneously broken at zero temperature, is

restored. This transition is also associated to decon-

finement even if, in the presence of light quarks, the

string tension does not exist. Indeed, when pulling

apart a quark and an antiquark, an inst ability for

production of quark–antiquark pairs sets in when

the potential energy becomes large enough, which

physically manifests itself as a production of light

mesons. An alternative order parameter is needed.

The possibility of defining alternative order para-

meters is discussed in next section.

The equation of state can also be studied relating

internal energy to pressure, which is useful to

understand heavy ion collisions.

From the features of the deconfinement transition,

information can be extracted on the mechanisms by

which QCD confines color.

A connected issue is the behavior of QC D at

nonzero baryon density or chemical potential. The

corresponding thermodynamics is described by a

grand canonical ensemble



¼ tr½exp ½ðH þ NÞ=T ½27

where N =

is the baryon number operator

and  the chemical potential. In the process of

converting the partition function Z



into a Feynman

integral, the term H at the exponent of eqn [27]

generates the Euclidean action, which is real. The

term proportional to N becom es imaginary. The

integral is well defined, but the analogy with a four-

dimensional statistical mechanics is broken, the

effective Hamiltonian being non-Hermitian and no

sampling can be made. Approximate methods have

been developed, but the problem is open. Exploring

numerically the region of phase space with  6¼ 0

would be interesting, since a rich structure is

expected, which could be relevant to dense systems

such as neutron stars.

Mechanisms of Color Confinement

Understanding how QCD manages to confine color

is one of the most fascinating problems in field

theory.

To prove con finement, one should, in principle,

prove that, at zero temperature, no gauge-invariant

quasilocal operator exists, carrying nontrivial color

and obeying cluster property at large distances. This

proof is not known. There exists evidence form

lattice simulations that a string tension exists, as

discussed before. In any case, a guess can be made of

the physical mechanism of confinement. If confine-

ment is an absolute property reflecting a symmetry

property of the vacuum, an order parameter should

exist which discriminates between confined and

deconfined phase, and the transition between the

two phases has to be a true transition. Obse rving a

crossover in some part of the boundary between the

two phases would disprove this view. A lattice

determination of the order of the deconfining

transition is therefore of fundamental importance.

A possible mechanism of confinement proposed by

G ’t Hooft is dual superconductivity of the vacuum:

dual means interchange of electric with magnetic

with respect to ordinary superconductors. In the same

way as the magnetic field is constrained into

Abrikosov flux tubes in an ordinary superconductor,

the chromoelectric field acting between a quark and

an antiquark would be constrained into flux tubes by

a dual Meissner effect producing an energy propor-

tional to the distance, or a string tension.

This mechanism can be investigated by lattice

simulations, by checking if any magnetically charged

operator exists whose vacuum expectation value is

nonzero in the confined phase signaling condensation

of magnetic charges and zero in the deconfined phase.

Progress has been made in this direction which,

however, is not yet conclusive. Chromoelectric flux

tubes between q–

q pairs are observed in lattice field

configurations.

Topology

Euclidean QCD admits classical solutions with finite

action and with a nontrivial topology which makes

them stable. These solutions, known as instantons

or multi-instantons, realize a mapping of the three-

dimensional sphere at infinity on the gauge group, and

the topological charge is the winding number of this

mapping. The Jacobian of this mapping is the Chern

280 Lattice Gauge Theory

current K



and its divergence @



(x)  Q(x)isthe

density of topological charge. Q =

xQ(x)isthe

topological charge which has integer values.

Explicitly,

QðxÞ¼

16

tr½G







½28

with G





= (1=2)





the dual field strength tensor.

Q(x) plays an important role in hadron physic s,

being related to the anomaly of the flavor singlet

axial current J







. J



is conserved at the

classical level in the chiral limit m

= 0, but this

symmetry does not survive quantization. In fact,



¼ 2N

QðxÞ½29

A consequence of eqn [29] is the high mass m





1 GeV of the flavor singlet partner 

of the

pseudoscalar flavor octet. An N

!1 argument by

Witten and Veneziano relates m



to the response of

the quenched (no quark) vacuum to topological

excitation, the topological susceptibility  

x <

0jTQ(x)Q(0)j0 >. The relation is



 ¼½m



þ m



 2m

½1 þ Oð1=N

Þ ½30

This approximate relation has been checked on the

lattice.  has been determined by different methods

which agree in confirming it. This is an important

verification of QCD.

Instantons are stable solutions in the continuum,

approximately stable in the lattice discretized ver-

sion. A cooling procedure which locally freezes

short-distance quantum fluctuations would leave

the instantons untouched if they were stable. On

the lattice the instanton is stable anyhow if the

distance in correlation reached by the local cooling

procedure is small compared to the size of the

instanton: cooling is indeed a diffusion process and

the distance involved grows as the square root of the

number of cooling iterations. Instanton configura-

tions can nicely be exposed by cooling.

See also: Anomalies; Quantum Chromodynamics;

Renormalization: General Theory; Spin Foams;

Symmetry Breaking in Field Theory.

Further Reading

Creutz M (1983) Quarks, Gluons and Lattices. Cambridge:

Cambridge University Press.

Feynman RP (1948) Space–time approach to nonrelativistic

quantum mechanics. Reviews of Modern Physics 20: 367.

Feynman RP and Hibbs AR (1965) Quantum Mechanics and Path

Integral. New York: McGraw-Hill.

Ginsparg PH and Wilson KG (1982) A remnant of chiral

symmetry on the lattice. Physical Review D 25: 2649.

Gottlieb S, Liu W, Toussaint D, Renken RL, and Sugar RL (1987)

Hybrid-molecular-dynamics algorithms for the numerical

simulation of QCD. Physical Review D 35: 2531.

Kogut J and Susskind L (1975) Hamiltonian formulation of

Wilson’s lattice gauge theories. Physical Review D 11: 395.

’t Hooft G (1981) Topology of the gauge condition and new

confinement phases in non-abelian gauge theories. Nuclear

Physics B 190: 455.

Veneziano G (1979) U(1) without instantons. Nuclear Physics B

159: 213.

Wilson KG (1974) Confinement of quarks. Physical Review D 10:

2445–2459.

Wilson KG (1983) The renormalization group and critical

phenomena (Nobel Lecture). Reviews of Modern Physics

55: 583.

Witten E (1979) Current algebra theorems for the U(1) goldstone

boson. Nuclear Physics B 156: 269.

Leray–Schauder Theory and Mapping Degree

J Mawhin, Universite

Catholique de Louvain,

Louvain-la-Neuve, Belgium

Introduction

The Leray–Schauder theory gives a powerful and

versatile continuation method for proving the

existence, multiplicity, and bifurcation of solutions

of nonlinear operator, differential and integral

equations. Let X and Y be topological spaces, A  X,

f : X !Y, a continuous mapping, and y 2 Y.The

fundamental idea of a continuation method to solve

the equation f (x) = y in A consists in embedding it into

a one-parameter family of equations

Fðx;Þ¼zðÞ½1

where the continuous functions F : X  [0, 1] ! Y,

z : [0, 1] ! Y arechoseninsuchawaythatF(  ,1)=

f , z(1) = y and

1. equation F(x,0)= z(0) has a nonempty set of

solutions in A;

2. one of those solutions at least can be continued

into a solution in A of [1] for each  2 [0, 1].

Simple examples show that Assertion 2 can be

violated when all solutions of [1] leave A after some

Leray–Schauder Theory and Mapping Degree 281





2 ]0, 1[. A way to avoid such a situation consists

in ‘‘closing the boundary,’’ through the ‘‘boundary

condition’’:

Fðx;Þ 6¼ zðÞ for each ðx;Þ2@A ½0; 1

When this condition is satisfied, Assertion 2 can

still fail when two existing solutions for  small

disappear after coalescing at some 

< 1. Losing all

solutions through this process can be eliminated by

reinforcing Assumption 1 into

. Equation F(x,0)= z(0) has a ‘‘robust’’ nonempty

set of solutions in A.

This statement can be made precise through the

concept of topological degree of a mapping, an

‘‘algebraic’’ count of the number of its zeros. In a

finite-dimensional setting, this concept was intro-

duced by Kronecker for smooth mappings and

by Brouwer for continuous mappings. Its extension

by Leray and Schauder to some classes of mappings

in Banach spaces made much wider applications

to nonlinear differential and integral equations

possible.

Topological Degree of a Mapping

If U  R

is a bounded open set, z 2 R

and

F :



U ! R

is a C

mapping su ch that z 62 F(@U)

and det F

(x) 6¼ 0onF

1

(z), the Brouwer degree

deg

[F, U, z] is defined (analytically) by

deg

½F; U; z :¼

x2F

1

ðzÞ

sign det F

ðxÞ

x2F

1

ðzÞ

ð1Þ

ðxÞ

where (x) is the sum of the multiplicities of the

negative eigenvalues of F

(x). The case of a

continuous F such that z 62 F(@U) is treated by

approximating F through mappings of the above

type, and showing that the corresponding degrees

stabilize to an unique value, defining deg

[F, U, z]in

the general case. This number remains the same

under sufficiently small pertur bations of F and/or z,

which expresses the ‘‘robustness’’ mentioned above.

When n = 2 and U is bounded by a closed Jordan

curve, then deg

[F, U, 0] is nothing but the winding

number of F=kFk along @U.

Leray and Schauder have extended Brouwer

degree to the important class of compact perturba-

tions of identity in a normed space. A compac t

mapping f : A !B between metric spaces is a

continuous mapping on A such that f(A) is relatively

compact. If f : A !B is continuous and compact on

each bounded B  A, f is called ‘‘completely con-

tinuous’’ on A.

If X is a real normed space, U  X an open bounded

set, f :



U !X compact, and z 62 (I  f )(@U), the

Leray–Schauder degree deg

[I  f , U, z]ofI  f in

U over z is constructed from Brouwer degree by

approximating the compact mapping f over



U by

mappings f



with range in a finite-dimensional sub-

space X



of X containing z. One shows that the values

of the Brouwer degrees deg

[(I  f



, U \ X



, z]

stabilize for sufficiently small positive  to a common

value which defines deg

[I  f , U, z].

Again, this topological degree is an algebraic

count of the number of elements of (I  f )

1

(z),

equal to 0 when z 62 (I  f )(U). When f is of class

,andI  f

(x) invertible at each fixed point x 2

(I  f )

1

(z), (I  f )

1

(z) is finite and the

Leray–Schauder formula holds:

deg

½I  f ; U; z¼

x2ðIf Þ

1

ðzÞ

ð1Þ

ðxÞ

½2

where (x) is the sum of the algebraic multiplicities

of the eigenvalues of f

(x) contained in [1, þ1].

Let I = [0, 1]. For A  X  I, and  2 I , we write



= {x 2 X :(x,) 2 A}. The Leray–Schauder degree

inherits the basic properties of Brouwer degree:

1. A dditivi ty.IfU = U

[ U

,whereU

and U

are open and disjoint, and if z =2(I  f )(@U

) [

(I  f )(@U

), then

deg

½I  f ; U; z¼deg

½I  f ; U

; z

þ deg

½I  f ; U

; z

2. Existence. If deg

[I  f , U, z] 6¼ 0, then

z 2 (I  f )(U).

3. Homotopy invariance. Let   X  I be a

bounded open set, and let F :



 !X be compact.

If x  F(x, ) 6¼ z for each (x, ) 2 @, then

deg

[I  F(  , ), 



, z] is independent of .

In particul ar, if a is an isolated fixed point of f,

and B(a, r) denotes the open ball of center a and

radius r, deg

[I  f , B(a, r), 0] is defined and inde-

pendent of r for sufficiently small r > 0. Its value is

called the ‘‘Leray–Schauder index’’ of I  f at a, and

denoted by ind

[I  f , a].

Fixed-Point Theorems for Compact

Perturbations of Identity in a Normed

Space

An important application of Leray–Schauder degree

is the obtention of general fixed point theorems for

compact mappings in normed spaces based on

282 Leray–Schauder Theory and Mapping Degree

continuation along a parameter. If F : A 

X  I !X, we denote by 

the (possibly empty)

solution set defined by



¼fðx;Þ2A : x ¼ Fðx;Þg

Let   X  I be a bounded open set and

F :



 !X be a compact mapping. The general

Leray–Schauder fixed-point theorem goes as follows:

Theorem If the following conditions hold:

(i) 





\ @=; (a priori estimate )

(ii) deg

[I  F(  , 0), 

,0]6¼ 0(degree condition),

then 





contains a continuum C along which

 takes all values in I. In other words, 





contains a compact connected subset C connect-

ing 





to 

. If one refines Assumption (ii) into

(iii) 





is a finite nonempty set {a

, ..., a



} and

ind

[I  F(  , 0), a

] 6¼ 0, the conclusion takes

the form of an ‘‘alternative’’: if assumptions

(i) and (iii) hold, then (a

,0) belongs either to

a continuum in 





containing one of the points

, 0), ...,(a



, 0), or to a continuum in 





along which  takes all the values in I.

Condition (iii) automatically holds in the following

important special case: If 





\ @=;, F(  ,0)= 0,

and 0 2 

,then





contains a continuum C3(0, 0)

along which  takes all values in I. When dealing with

the fixed-point problem x = f (x)withf :



U  X !X

compact, U open and bounded, a natural choice is

F(x , ) = f (x), =U  I, giving the statement: If

0 2 U and if x 6¼ f (x) for each (x, ) 2 @U  I,then

{(x, ) 2



U  I : x = f (x)} contains a continuum C3

(0, 0) along which  takes all values in I.

Condition (i) requires the a priori knowledge of

the localization of the solution set 





and is in

general very difficult to check. An important special

case occurs when 

is a priori bounded: if F is

completely continuous on X  I, F( ,0)= 0, and



 B(r )  I for some r > 0, then 

contains a

continuum C3(0, 0) along which  takes all values

in I. Its special case with F (, x) = f (x) can be

stated as Schaefer’s alternative: Let f : X !X be

completely continuous. Then either there exists, for

each  2 [0, 1], at least one x 2 X such that

x = f (x), or the fixed point set {x 2 X :

x = f (x), 0 <<1} is unbounded in X. Schaefer’s

alternative is equivalent to the following Schauder

fixed-point theorem:

Theorem Any compact mapping f :

B(r) !B(r) has

a fixed point.

A simple consequence of Schauder’s theorem is

that, for any continuous and bounded g : R ! R,

any open bounded D  R

, any  different from an

eigenvalue of   on D with Dirichlet boundary

conditions, the nonlinear Dirichlet problem

u þ u þ gðuÞ¼hðxÞ in D

u ¼ 0on@D

has a weak solution for each h 2 L

(D).

An interesting consequence of Leray–Schauder

theorem with 

a priori bounded is that, for any

bounded domain D  R

with @D of class C

, the

Dirichlet problem for the equation of surfaces with

constant mean curvature 

ð1 þkruk

Þu 

i;j¼1

u @

¼ nð1 þkruk

3=2

has a unique solution for arbitrary smooth boundary

data if and only if the mean curvature of the boundary

@D is everywhere greater than [n=(n  1)]jj.

The use of auxiliary continuous functionals gives

a fixed-point theorem in the absence of a priori

bounds:

Theorem (Capietto–Mawhin–Zanolin). Let  

X  I be an open set and F :



 !X be completely

continuous. If 





is bounded, deg

[I  F(  , 0),

,0]6¼ 0 for some open bounded neighborhood

of 





, and if there exists a continuous

mapping ’ : X  I !R

, proper on 





, and c



min







’( ,0) max







’( ,0) < c

such that 





, c

} and 

@

62 [c



, c

], then 





contains a

continuum C along which  takes all values in I .

This result implies, for example, that for g : R !R

continuous, odd and superlinear (lim

juj!1

g(u)=

u = þ1), and p : [0, 1]  R

with at most linear

growth in u and u

at infinity, the two-point

boundary-value problem

þ gðuÞ¼pðt; u; u

Þ; uð0Þ¼uð1Þ¼0

has, for all sufficiently large j, at least one solution

having exactly j þ 1 zeros on [0, 1], and

!1 if j !1.

Extensions of Leray–Schauder degree

Fixed-point theorems for operators between suitable

nonlinear spaces can also be proved using topologi-

cal continuation arguments. For example, if C  X

is a nonempty convex set, one has the following

extension of a result of the previous section to

mappings in C:ifU  C is open and bounded,

F :cl

U  I !C compact and such that x 6¼ F(x, )

for each (x, ) 2 @

U  I, F(,0)= x

2 U, then

Leray–Schauder Theory and Mapping Degree 283

F( , ) has a fixed point in U for each  2 I. The

special case where C is a wedge is useful in finding

positive solutions of nonlinear differential or inte-

gral equations. For nonlinear spaces, the degree has

to be replaced by the fixed-point index,

which generalizes both the ‘‘Hopf–Lefschetz num-

ber’’ and Leray–Schauder degree.

The Leray–Schauder degree also has been

extended to other classes of operators. Compact

operators can be replaced by k-set-contractive or

condensing mappings f, with respect to various

measures of noncompactness , and fixed-point pro-

blems can be replaced by problems of the form x 2

F(x ) for multivalued mappings F. Equivariant degree

theories have been developed when U is invariant

and f equivariant with respect to the action of some

compact Lie group G on X. The special case of

G = S

is of special importance in the study of

periodic solutions of autonomous differential sys-

tems. Deg ree theories have also been constructed for

various classes of mappings between two different

Banach spaces or manifolds, which include mono-

tone-like and nonlinear Fredholm operators. We just

describe a simple but useful situation in this

direction.

Many differential equations, when expressed as

equations in an abstract space, do not have the

fixed-point form but can be written as Lx = Nx with

L : D(L)  X ! Z linear, N :



U !Z, X and Z real

normed spaces. If L is invertible, the equation is

trivially equivalent to the fixed-point problem

x = L

1

Nx, to which Leray–Schauder theory can be

applied when L

1

N is compact. The situation is

more delicate when L has no inverse. If L is a linear

Fredholm mapping of index zero (its range R(L)is

closed and has a finite codimension equal to the

dimension of its null space N(L)), the set F(L)of

linear continuous mappi ngs of finite rank A : X !Z

such that L þ A : D(L) !Z is a bijection is none-

mpty and the compactness of (L þ A)

1

G does not

depend upon the choice of A 2F(L). G is then called

‘‘L-compact’’ on E,and‘‘L-completely continuous’’

on E when compact on each bounded set of E.

The following continuation theorem for perturbed

Fredholm mapping of index zero holds.

Theorem Let   X  I be open and bounded,

L : D(L)  X !Z linear Fredholm of index zero,

N :



 ! ZL-compact, and let ={(x,) 2(D(L) I)



:Lx=N(x,)}. If

(i)  \ @ 6¼;(a priori estimate),

(ii) N(





 {0})  Y, with Y  R(L) = Z (transvers-

ality condition), and

(iii) deg

[N( ,0)j

kerL

, 

\ kerL,0] 6¼ 0(degree

condition)

then  contains a continuum C along which  takes

all values in I.

When dealing with equation Lx = f (x) with f

L-completely continuous, an interesting special case

of the above result follows from the choice

N(x, ) = f (x) þ (1  )Qf (x), with Q : Z !Z a

projector such that N(Q) = R(L). In this case, the

homotopy is equivalent to

Lx ¼ f ðxÞð 20; 1Þ

Qf ðxÞ¼0; x 2 NðL Þð ¼ 0Þ

An application (among many) of this result,

for g : R !

R continuous such that 1 <

lim sup

u !1

g(u) < lim inf

u !þ1

g(u) < þ1, D  R

open, bounded, 

an eigenvalue of the Dirichlet

problem for  on D, is the weak solvability of the

nonlinear problem

u þ 

u þ gðuÞ¼hðxÞ in D

u ¼ 0on@D

for each h 2 L

(D) such that

hðxÞ’ðxÞdx <

lim sup

u!1

gðuÞ



’

ðxÞdx 

lim inf

u!þ1

gðuÞ

’



ðxÞdx

for all eigenfunctions ’ associated to 

. The

addition of the nonlinearity g ‘‘widens’’ the range

{h 2 L

(D):

h’ = 0} of the corresponding linear

problem.

Bifurcation Theory

Leray–Schauder degree is a powerful tool in bifurca-

tion theory, where, given a family F of solutions,

one tries to detect and analyze other ones branching

or bifurcating from F. Consider the equation

x ¼ Lx þ Rðx;Þ½3

in a real normed space X, where L : X !X, linear,

and R : X  R ! X are completely continuous, and

R(0, ) = 0 for each  2 R. Thus, {(0, ): 2 R}is

the trivial solution set of [3]. A bifurcation point

(



, 0) for [3] is the limit of a sequence (

, x

)of

solutions of [3] in Rn{0}.

lim

x!0

kRðx;Þk

kxk

¼ 0

uniformly on bounded -sets ½4

it is easy to prove that if (



, 0) is a bifurcation point for

[3],then



is a characteristic value (reciprocal of an

eigenvalue) of L. Leray–Schauder theory gives a partial

284 Leray–Schauder Theory and Mapping Degree

converse to this result known as Krasnosel’skii’s

bifurcation theorem:

Theorem For each real characteristic value 



of L

with odd algebraic multiplicity,(



,0) is a bifurcation

point of [3]. Of fundamental importance in the proof is

the special case of [2] with f = L and N(I  L) = {0}.

Another fruitful concept is Krasnosel’skii’s bifur-

cation from infinity. We say (



, 1) is a bifurcation

point for [3] if there exists a sequence (

, x

)of

solutions of [3] such that 

!



and kx

k!1.

The corresponding bifurcation result goes as follows

(Krasnosel’skii): if

lim

kxk!1

kRðx;Þk

kxk

¼ 0

uniformly on bounded -sets ½5

then, for each real characteristic value 



of L with

odd algebraic multiplicity, (



, 1) is a bifurcation

point of [3].

Global versions of Krasnosel’skii’s theorems can be

given, whose statements are reminiscent of Leray–

Schauder’s alternative theorem. Let S denote the

closure in R  X of the set of (, x) 2 R  (X n {0})

satisfying [3]. For bifurcation from zero, one has

Rabinowitz global bifurcation theorem:

Theorem If [4] holds and 



is a real charac teristic

value of L with odd algebraic multiplicity, then S

contains a component C which either is unbounded,

or contains (



, 0), where 



6¼ 



is a character-

istic value of L.

As an application, one can show that the non-

linear Sturm–Liouville prob lem

ðpðxÞu

þqðxÞu ¼aðxÞ u þhðx;u;u

;Þðx 20;1½Þ

uð0Þþb

ð0Þ¼a

uð1Þþb

ð1Þ¼0

with p 2C

positive, q, a, h continuous, a positive,

þb

)(a

þb

) 6¼0 and h(x,u,v)=o(jujþjvj)if

jujþjvj!0 uniformly on compac t -intervals, has,

for each k 2N, an unbounded component of

solution C

in R C

([0,1]) emanating from (

,0),

with 

an eigenvalue of the problem with h 0

(Rabinowitz).

One has also global bifurcation from infinity: if

[5] holds and if 



is a real characteristic value of L

with odd algebraic multiplicity, then [3] has an

unbounded component of solutions D which con-

tains (



, 1).

See also: Bifurcation Theory; Bifurcations in Fluid

Dynamics; Bifurcations of Periodic Orbits; Minimal

Submanifolds; Minimax Principle in the Calculus of

Variations; Partial Differential Equations: Some

Examples; Riemann–Hilbert Problem; Topological

Defects and Their Homotopy Classification; Viscous

Incompressible Fluids: Mathematical Theory.