56 Pages
English
Gain access to the library to view online
Learn more

On the Hausdorff Dimension of the Mather Quotient

-

Gain access to the library to view online
Learn more
56 Pages
English

Description

On the Hausdorff Dimension of the Mather Quotient Albert Fathi ?, Alessio Figalli †, Ludovic Rifford ‡ 8 November, 2007 Abstract Under appropriate assumptions on the dimension of the ambient man- ifold and the regularity of the Hamiltonian, we show that the Mather quo- tient is small in term of Hausdorff dimension. Then, we present applications in dynamics. 1 Introduction Let M be a smooth manifold without boundary. We denote by TM the tangent bundle and by pi : TM ? M the canonical projection. A point in TM will be denoted by (x, v) with x ? M and v ? TxM = pi?1(x). In the same way a point of the cotangent bundle T ?M will be denoted by (x, p) with x ?M and p ? T ?xM a linear form on the vector space TxM . We will suppose that g is a complete Riemannian metric on M . For v ? TxM , the norm ?v?x is gx(v, v)1/2. We will denote by ?·?x the dual norm on T ?M . Moreover, for every pair x, y ?M , d(x, y) will denote the Riemannian distance from x to y. We will assume in the whole paper that H : T ?M ? R is an Hamiltonian of class Ck,?, with k ≥ 2, ? ? [0, 1], which satisfies the three following conditions: (H1) C2-strict convexity: ?(x, p) ? T ?M , the

  • critical viscosity subsolution

  • euler-lagrange flow

  • hausdorff dimension

  • positive definite

  • ?m

  • mather quotient

  • global viscosity

  • hamilton-jacobi equation does

  • following mather


Subjects

Informations

Published by
Reads 10
Language English

Exrait

OntheHausdorffDimensionoftheMather
Quotient

AlbertFathi,

AlessioFigalli,

LudovicRifford

8November,2007

Abstract
Underappropriateassumptionsonthedimensionoftheambientman-
ifoldandtheregularityoftheHamiltonian,weshowthattheMatherquo-
tientissmallintermofHausdorffdimension.Then,wepresentapplications
indynamics.

1Introduction
Let
M
beasmoothmanifoldwithoutboundary.Wedenoteby
TM
thetangent
bundleandby
π
:
TM

M
thecanonicalprojection.Apointin
TM
willbe
denotedby(
x,v
)with
x

M
and
v

T
x
M
=
π

1
(
x
).Inthesamewayapoint
ofthecotangentbundle
T

M
willbedenotedby(
x,p
)with
x

M
and
p

T
x

M
alinearformonthevectorspace
T
x
M
.Wewillsupposethat
g
isacomplete
Riemannianmetricon
M
.For
v

T
x
M
,thenorm
k
v
k
x
is
g
x
(
v,v
)
1
/
2
.Wewill
denoteby
k∙k
x
thedualnormon
T

M
.Moreover,foreverypair
x,y

M
,
d
(
x,y
)
willdenotetheRiemanniandistancefrom
x
to
y
.
Wewillassumeinthewholepaperthat
H
:
T

M

R
isanHamiltonianof
class
C
k,α
,with
k

2


[0
,
1],whichsatisfiesthethreefollowingconditions:
(H1)C
2
-strictconvexity:

(
x,p
)

T

M
,thesecondderivativealongthefibers

2
H/∂p
2
(
x,p
)isstrictlypositivedefinite;
(H2)
uniformsuperlinearity:
forevery
K

0thereexistsafiniteconstant
C
(
K
)suchthat

(
x,p
)

T

M,H
(
x,p
)

K
k
p
k
x
+
C
(
K
);

UMPA,ENSLyon,46Alle´ed’Italie,69007Lyon,France.
e-mail:albert.fathi@umpa.ens-
lyon.fr

Universite´deNice-SophiaAntipolis,ParcValrose,06100Nice,France.
e-mail:fi-
galli@unice.fr

Universite´deNice-SophiaAntipolis,ParcValrose,06100Nice,France.
e-mail:rif-
ford@unice.fr

1

(H3)
uniformboundednessinthefibers:
forevery
R

0,wehave
sup
{
H
(
x,p
)
|k
p
k
x

R
}
<
+

.
M∈xBytheWeakKAMTheoremweknowthat,undertheaboveconditions,there
is
c
(
H
)

R
suchthattheHamilton-Jacobiequation
H
(
x,d
x
u
)=
c
(HJ
c
)
admitsaglobalviscositysolution
u
:
M

R
for
c
=
c
(
H
)anddoesnotadmit
suchsolutionfor
c<c
(
H
),see[22,9,6,11,15].Infact,for
c<c
(
H
),the
Hamilton-Jacobiequationdoesnotadmitanyviscositysubsolution(forthetheory
ofviscositysolutions,wereferthereadertothemonographs[1,2,11]).Moreover,
if
M
isassumedtobecompact,then
c
(
H
)istheonlyvalueof
c
forwhichthe
Hamilton-Jacobiequationaboveadmitsaviscositysolution.Theconstant
c
(
H
)is
calledthe
criticalvalue
,orthe
Man˜e´criticalvalue
of
H
.Inthesequel,aviscosity
solution
u
:
M

R
of
H
(
x,d
x
u
)=
c
(
H
)willbecalleda
criticalviscositysolution
ora
weakKAMsolution
,whileaviscositysubsolution
u
of
H
(
x,d
x
u
)=
c
(
H
)will
becalleda
criticalviscositysubsolution
(or
criticalsubsolution
if
u
isatleastC
1
).
TheLagrangian
L
:
TM

R
associatedtotheHamiltonian
H
isdefinedby

(
x,v
)

TM,L
(
x,v
)=
p

m
T
a

x
M
{
p
(
v
)

H
(
x,p
)
}
.
xSince
H
isofclassC
k
,with
k

2,andsatisfiesthethreeconditions(H1)-(H3),it
iswell-known(seeforinstance[11]or[15,Lemma2.1]))that
L
isfiniteeverywhere
ofclassC
k
,andisaTonelliLagrangian,i.e.satisfiestheanalogousofconditions
(H1)-(H3).Moreover,theHamiltonian
H
canberecoveredfrom
L
by

(
x,p
)

T
x

M,H
(
x,p
)=max
{
p
(
v
)

L
(
x,v
)
}
.
MT∈vxThereforethefollowinginequalityisalwayssatisfied
p
(
v
)

L
(
x,v
)+
H
(
x,p
)
.
ThisinequalityiscalledtheFenchelinequality.Moreover,duetothestrictcon-
vexityof
L
,wehaveequalityintheFenchelinequalityifandonlyif
(
x,p
)=
L
(
x,v
)
,
where
L
:
TM

T

M
denotestheLegendretransformdefinedas
L∂L
(
x,v
)=
x,
(
x,v
)
.
v∂Underourassumption
L
isadiffeomorphismofclassatleast
C
1
.Wewilldenote
by
φ
tL
theEuler-Lagrangeflowof
L
,andby
X
L
thevectorfieldon
TM
that
2

generatestheflow
φ
tL
.Ifwedenoteby
φ
tH
theHamiltonianflowof
H
on
T

M
,
thenasiswell-known,seeforexample[11],thisflow
φ
tH
isconjugateto
φ
tL
by
theLegendretransform
L
.
AsdonebyMatherin[26],itisconvenienttointroducefor
t>
0fixed,the
function
h
t
:
M
×
M

R
definedby
tZ∀
x,y

M,h
t
(
x,y
)=inf
L
(
γ
(
s
)

˙(
s
))
ds,
0wheretheinfimumistakenoveralltheabsolutelycontinuouspaths
γ
:[0
,t
]

M
with
γ
(0)=
x
and
γ
(
t
)=
y
.The
Peierlsbarrier
isthefunction
h
:
M
×
M

R
definedby
h
(
x,y
)=li
t
m

i

nf
{
h
t
(
x,y
)+
c
(
H
)
t
}
.
Itisclearthatthisfunctionsatisfies

x,y,z

M,h
(
x,z
)

h
(
x,y
)+
h
t
(
y,z
)+
c
(
H
)
t
h
(
x,z
)

h
t
(
x,y
)+
c
(
H
)
t
+
h
(
y,z
)
,
andthereforeitalsosatisfiesthetriangleinequality

x,y,z

M,h
(
x,z
)

h
(
x,y
)+
h
(
y,z
)
.
Moreover,givenaweakKAMsolution
u
,wehave

x,y

M,u
(
y
)

u
(
x
)

h
(
x,y
)
.
Inparticular,wehave
h>
−∞
everywhere.Itfollows,fromthetriangleinequal-
ity,thatthefunction
h
iseitheridentically+

oritisfiniteeverywhere.If
M
iscompact,
h
isfiniteeverywhere.Inaddition,if
h
isfinite,thenforeach
x

M
thefunction
h
x
(

)=
h
(
x,

)isacriticalviscositysolution(see[11]or[16]).The
projectedAubryset
A
isdefinedby
A
=
{
x

M
|
h
(
x,x
)=0
}
.
FollowingMather,see[26,page1370],wesymmetrize
h
todefinethefunction
δ
M
:
M
×
M

R
by

x,y

M,δ
M
(
x,y
)=
h
(
x,y
)+
h
(
y,x
)
.
Since
h
satisfiesthetriangleinequalityand
h
(
x,x
)

0everywhere,thefunc-
tion
δ
M
issymmetric,everywherenonnegativeandsatisfiesthetriangleinequality.
Therestriction
δ
M
:
A×A→
R
isagenuinesemi-distanceontheprojectedAubry
set.Wewillcallthisfunction
δ
M
the
Mathersemi-distance
(evenwhenwecon-
sideriton
M
ratherthanon
A
).Wedefinethe
Matherquotient
(
A
M

M
)tobe
themetricspaceobtainedbyidentifyingtwopoints
x,y
∈A
iftheirsemi-distance

3

δ
M
(
x,y
)vanishes.Whenweconsider
δ
M
onthequotientspace
A
M
wewillcall
itthe
Matherdistance
.
In[29],Matherformulatedthefollowingproblem:
Mather’sProblem
.If
L
isC

,istheset
A
M
totallydisconnectedforthetopol-
ogyof
δ
M
,i.e.iseachconnectedcomponentof
A
M
reducedtoasinglepoint?

In[28],Matherbroughtapositiveanswertothatprobleminlowdimension.
Moreprecisely,heprovedthatif
M
hasdimensiontwo,oriftheLagrangianisthe
kineticenergyassociatedtoaRiemannianmetricon
M
indimension

3,then
thequotientAubrysetistotallydisconnected.Noticethatonecaneasilyshow
thatforadensesetofHamiltonians,theset(
A
M

M
)isreducedtoonepoint.
Mathermentionedin[29,page1668]thatitwouldbeevenmoreinterestingtobe
abletoprovethatthequotientAubrysethasvanishingone-dimensionalHausdorff
measure,becausethisimpliestheuppersemi-continuityofthemapping
H
7→A
.
HealsostatedthatforArnold’sdiffusionaresultgenericintheLagrangianbut
trueforeverycohomologyclasswasmorerelevant.Thiswasobtainedrecently
byBernardandContreras[5].
Theaimofthepresentpaperistoshowthatthevanishingoftheone-
dimensionalHausdorffmeasureoftheMatherquotientissatisfiedundervarious
assumptions.Letusstateourresults.
Theorem1.1.
If
dim
M
=1
,
2
and
H
ofclass
C
2
or
dim
M
=3
and
H
of
class
C
k,
1
with
k

3
,thentheMatherquotient
(
A
M

M
)
hasvanishingone-
dimensionalHausdorffmeasure.
AbovetheprojectedAubry
A
,thereisacompactsubset
A
˜

TM
calledthe
Aubryset(seeSection2.1).Theprojection
π
:
TM

M
inducesahomeo-
morphism
π
|A
˜
from
A
˜onto
A
(whoseinverseisLipschitzbyatheoremdueto
Mather).TheAubrysetcanbedefinedasthesetof(
x,v
)

TM
suchthat
x
∈A
and
v
istheuniqueelementin
T
x
M
suchthat
d
x
u
=
∂L/∂v
(
x,v
)foranycritical
viscositysubsolution
u
.TheAubrysetisinvariantundertheEuler-Lagrange
flow
φ
tL
:
TM

TM
.Therefore,foreach
x
∈A
,thereisonlyoneorbitof
φ
tL
in
A
˜whoseprojectionpassesthrough
x
.Wedefinethe
stationaryAubryset
A
˜
0
⊂A
˜asthesetofpointsin
A
˜whicharefixedpointsoftheEuler-Lagrange
flow
φ
t
(
x,v
),i.e.
A
˜
0
=
{
(
x,v
)
∈A
˜
|∀
t

R

tL
(
x,v
)=(
x,v
)
}
.
Infact,seeProposition3.2,itcanbeshown,that
A
˜
0
istheintersectionof
A
˜with
thezerosectionof
TM
A
˜
0
=
{
(
x,
0)
|
(
x,
0)
∈A
˜
}
.
Wedefinethe
projectedstationaryAubryset
A
0
astheprojectionon
M
of
A
˜
0
A
0
=
{
x
|
(
x,
0)
∈A
˜
}
.

4

Attheveryendofhispaper[28],Mathernoticedthattheargumentheusedinthe
casewhere
L
isakineticenergyindimension3provesthetotaldisconnectedness
ofthequotientAubrysetindimension3aslongas
A
0
M
isempty.Infact,ifwe
considertherestrictionof
δ
M
to
A
0
,wehavethefollowingresultonthequotient
metricspace(
A
0
M

M
).
Theorem1.2.
Supposethat
L
isatleast
C
2
,andthattherestriction
x
7→
L
(
x,
0)
of
L
tothezerosectionof
TM
isofclass
C
k,
1
.Then
(
A
0
M

M
)
hasvanishing
Hausdorffmeasureindimension
2dim
M/
(
k
+3)
.Inparticular,if
k

2dim
M

3
then
H
1
(
A
0
M

M
)=0
,andif
x
7→
L
(
x,
0)
is
C

then
(
A
0
M

M
)
haszero
Hausdorffdimension.
Asacorollary,wehavethefollowingresultwhichwasmoreorlessalready
mentionedbyMatherin[29,
§
19page1722],andprovedbySorrentino[34].
Corollary1.3.
Assumethat
H
isofclass
C
2
andthatitsassociatedLagrangian
L
satisfiesthefollowingconditions:
1.

x

M,
min
v

T
x
M
L
(
x,v
)=
L
(
x,
0)
;
2.themapping
x

M
7→
L
(
x,
0)
isofclass
C
l,
1
(
M
)
with
l

1
.
If
dim
M
=1
,
2
,or
dim
M

3
and
l

2dim
M

3
,then
(
A
M

M
)
istotally
disconnected.Inparticular,if
L
(
x,v
)=
21
k
v
k
x
2

V
(
x
)
,with
V

C
l,
1
(
M
)
and
l

2dim
M

3(
V

C
2
(
M
)
if
dim
M
=1
,
2)
,then
(
A
M

M
)
istotallydisconnected.
Since
A
0
istheprojectionofthesubset
A
˜
0
⊂A
˜consistingofpointsin
A
˜
whicharefixedunderthetheEuler-Lagrangeflow
φ
tL
,itisnaturaltoconsider
A
p
thesetof
x
∈A
whichareprojectionofapoint(
x,v
)
∈A
˜whoseorbitunder
thetheEuler-Lagrangeflow
φ
tL
isperiodicwithstrictlypositiveperiod.Wecall
thissetthe
projectedperiodicAubryset
.Wehavethefollowingresult:
Theorem1.4.
If
dim
M

2
and
H
ofclass
C
k,
1
with
k

2
,then
(
A
pM

M
)
hasvanishingHausdorffmeasureindimension
8dim
M/
(
k
+8)
.Inparticular,if
ppk

8dim
M

8
then
H
1
(
A
M

M
)=0
,andif
H
is
C

then
(
A
M

M
)
haszero
Hausdorffdimension.
Inthecaseofcompactsurfaces,usingthefinitenessofexceptionalminimal
setsofflows,wehave:
Theorem1.5.
If
M
isacompactsurfaceofclass
C

and
H
isofclass
C

,
then
(
A
M

M
)
haszeroHausdorffdimension.
Inthelastsection,wepresentapplicationsindynamicwhoseTheorem1.6
belowisacorollary.If
X
isaC
k
vectorfieldon
M
,with
k

2,theMan˜e´
Lagrangian
L
X
:
TM

R
associatedto
X
isdefinedby
1L
X
(
x,v
)=
k
v

X
(
x
)
k
x
2
,

(
x,v
)

TM.
25

Wewilldenoteby
A
X
theprojectedAubrysetoftheLagrangian
L
X
.
Thefirstauthorhasraisedthefollowingproblem,comparewiththelistof
questions
http://www.aimath.org/WWN/dynpde/articles/html/20a/
.
Problem
.Let
L
X
:
TM

R
betheMan˜e´LagrangianassociatedtotheC
k
vectorfield
X
(
k

2)onthecompactconnectedmanifold
M
.
(1)Isthesetofchain-recurrentpointsoftheflowof
X
on
M
equaltothe
projectedAubryset
A
X
?
(2)Giveaconditiononthedynamicsof
X
thatinsuresthattheonlyweak
KAMsolutionsaretheconstants.

Thetheoremsobtainedinthefirstpartofthepapertogetherwiththeappli-
cationsindynamicsdevelopedinSection6giveananswertothisquestionwhen
dim
M

3.
Theorem1.6.
Let
X
bea
C
k
vectorfield,with
k

2
,onthecompactconnected
C

manifold
M
.Assumethatoneoftheconditionshold:
(1)Thedimensionof
M
is
1
or
2
.
(2)Thedimensionof
M
is
3
,andthevectorfield
X
nevervanishes.
(3)Thedimensionof
M
is
3
,and
X
isofclass
C
3
,
1
.
ThentheprojectedAubryset
A
X
oftheMan˜e´Lagrangian
L
X
:
TM

R
associ-
atedto
X
isthesetofchain-recurrentpointsoftheflowof
X
on
M
.Moreover,
theconstantsaretheonlyweakKAMsolutionsfor
L
X
ifandonlyifeverypoint
of
M
ischain-recurrentundertheflowof
X
.
Theoutlineofthepaperisthefollowing:Sections2and3aredevotedto
preparatoryresults.Section4isdevotedtotheproofsofTheorems1.1,1.2and
1.4.Sections5and6presentapplicationsindynamics.

2Preliminaryresults
Throughoutthissection,
M
isassumedtobeacompleteRiemannianmanifold.
Asbefore,
H
:
T

M

R
isanHamiltonianofclassatleastC
2
satisfyingthethree
usualconditions(H1)-(H3),and
L
istheTonelliLagrangianwhichisassociated
toitbyFenchel’sduality.

6

2.1SomefactsabouttheAubryset
WerecalltheresultsofMatherontheAubryset,andalsoanimportantcomple-
mentduetoDiasCarneiro.
ThefollowingresultsareduetoMather,see[25,26]fortheproofinthe
compactcase.
Theorem2.1
(Mather)
.
Thereexistsaclosedsubset
A
˜

TM
suchthat:
(1)Theset
A
˜
isinvariantundertheEuler-Lagrangeflow.
(2)Theprojection
π
:
TM

M
isinjectiveon
A
˜
.Moreover,wehave
π
(
A
˜)=
A
,andtheinversemap
(
π
|A
˜)

1
:
A→A
˜
islocallyLipschitz.
(3)Let
(
x,v
)
bein
A
˜
,andcall
γ
(
x,v
)
thecurvewhichistheprojectionofthe
orbit
φ
tL
(
x,v
)
oftheEuler-Lagrangeflowthrough
(
x,v
)
γ
(
x,v
)
(
t
)=
πφ
t
(
x,v
)
.
Thiscurveisentirelycontainedin
A
,anditisan
L
-minimizer.Moreover,
wehave

t,t
0

R

M
(
γ
(
t
)

(
t
0
))=0
,
thereforethewholecurve
γ
(
x,v
)
projectstothesamepointas
x
intheMather
quotient.
(4)If
x
∈A
and
γ
n
:[0
,t
n
]

M
isa
R
t
s
n
equenceof
L
-minimizersuchthat
t
n

+


n
(0)=
γ
n
(
t
n
)=
x
,and
0
L
(
γ
n
(
s
)

˙
n
(
s
))
ds
+
c
(
H
)
t
n

0
,
thenbothsequences
γ
˙
n
(0)

˙
n
(
t
n
)
convergein
T
x
M
totheunique
v

T
x
M
suchthat
(
x,v
)
∈A
˜
.
ThefollowingtheoremofDiasCarneiro[7]isanicecomplementtotheThe-
oremabove:
Theorem2.2.
Forevery
(
x,v
)
∈A
˜
,wehave
Hx,∂L
(
x,v
)=
c
(
H
)
.
v∂WeendthissubsectionbythefollowingimportantestimationoftheMather
semi-distance(duetoMather),see[26,page1375].
Proposition2.3.
Foreverycompactsubset
K

M
,wecanfindafiniteconstant
C
K
,suchthat

x
∈A∩
K,

y

K,δ
M
(
x,y
)

C
K
d
(
x,y
)
2
,
where
d
istheRiemanniandistanceon
M
.
Notethatonecanprovedirectlythispropositionfromthefactthat
h
is
locallysemi-concaveon
M
×
M
,usingthat
δ
M

0,togetherwiththefactthat
δ
M
(
x,x
)=0forevery
x
∈A
.

7

2.2AubrysetandHamilton-Jacobiequation
Inthissectionwerecasttheaboveresultsintermsofviscositysolutionsofthe
Hamilton-Jacobiasisdonein[11,16,15].
Wefirstrecallthenotionofdomination.If
c

R
,afunction
u
:
M

R
issaidtobedominatedby
L
+
c
(whichwedenoteby
u

L
+
c
),ifforevery
continuouspiecewiseC
1
curve
γ
:[
a,b
]

M,a<b
,wehave
bZu
(
γ
(
b
))

u
(
γ
(
a
))

L
(
γ
(
s
)

˙(
s
))
ds
+
c
(
b

a
)
.
(DOM)
aInfactthisissimplyadifferentwaytodefinethenotionofviscositysolution
for
H
.Morepreciselywehave,see[11]or[15,Proposition5.1,page12]:
Theorem2.4.
A
u
:
M

R
isdominatedby
L
+
c
ifandonlyifitisaviscosity
subsolutionoftheHamilton-Jacobiequation
H
(
x,d
x
u
)=
c
.Moreover,wehave
u

L
+
c
ifandonlyif
u
isLipschitzand
H
(
x,d
x
u
)

c
almosteverywhere.
NotethatRademacher’sTheoremstatesthateveryLipschitzfunctionisdiffer-
entiablealmosteverywhere.FortheproofthatdominatedfunctionsareLipschitz
seeB.2.Itisnotdifficulttoseethatafunction
u
:
M

R
isdominatedby
L
+
c
ifandonlyif

t>
0
,

x,y

M,u
(
y
)

u
(
x
)

h
t
(
x,y
)+
ct.
Withthesenotations,weobservethatafunction
u
isacriticalsubsolutionifand
only
u

L
+
c
(
H
).
Wenowgivethedefinitionofcalibratedcurves.If
u
:
M

R
and
c

R
,we
saythatthecurve
γ
:[
a,b
]

M
is(
u,L,c
)-calibratedifwehavetheequality
bZu
(
γ
(
b
))

u
(
γ
(
a
))=
L
(
γ
(
s
)

˙(
s
))
ds
+
c
(
b

a
)
.
aIf
γ
isacurvedefinedonthenotnecessarilycompactinterval
I
,wewillsay
that
γ
is(
u,L,c
)-calibratedifitsrestrictiontoanycompactsubintervalof
I
is
(
u,L,c
)-calibrated.
Infact,thisconditionofcalibrationisusefulonlywhen
u

L
+
c
.Inthis
case
γ
isan
L
-minimizer.Moreover,if[
a
0
,b
0
]isasubintervalof[
a,b
],thenthe
restriction
γ
|
[
a
0
,b
0
]isalso(
u,L,c
)-calibrated.
Likein[11],if
u
:
M

R
isacriticalsubsolution,wedenoteby
I
˜(
u
)the
subsetof
TM
definedas
I
˜(
u
)=
{
(
x,v
)

TM
|
γ
(
x,v
)
is(
u,L,c
(
H
))-calibrated
}
,
where
γ
(
x,v
)
isthecurve(alreadyintroducedinTheorem2.1)definedon
R
by
γ
(
x,v
)
(
t
)=
πφ
tL
(
x,v
)
.
Thefollowingpropertiesof
I
˜(
u
)areshownin[11]:

8

Theorem2.5.
Theset
I
˜(
u
)
isinvariantundertheEuler-Lagrangeflow
φ
tL
.If
(
x,v
)
∈I
˜(
u
)
,then
d
x
u
exists,andwehave
L∂d
x
u
=(
x,v
)
and
H
(
x,d
x
u
)=
c
(
H
)
.
v∂Itfollowsthattherestriction
π
|
I
˜(
u
)
oftheprojectionisinjective;therefore,ifwe
set
I
(
u
)=
π
(
I
˜(
u
))
,then
I
˜(
u
)
isacontinuousgraphover
I
(
u
)
.
Moreover,themap
x
7→
d
x
u
islocallyLipschitzon
I
(
u
)
.
Sincetheinverseoftherestriction
π
|I
˜(
u
)
isgivenby
x
7→L

1
(
x,d
x
u
)
,and
theLegendretransform
L
is
C
1
,itfollowsthattheinverseof
π
|I
˜(
u
)
isalsolocally
Lipschitzon
I
.
Usingthesets
I
˜(
u
),onecangivethefollowingcharacterizationoftheAubry
setanditsprojection:
Theorem2.6.
TheAubryset
A
˜
isgivenby
\A
˜=
I
˜(
u
)
,
SS∈uwhere
SS
isthesetofcriticalviscositysubsolutions.TheprojectedAubryset
A
,
whichissimplytheimage
π
(
A
˜)
,isalso
\A
=
I
(
u
)
.
SS∈uNotethatthefactthattheAubrysetisalocallyLipschitzgraph(i.e.part(2)
ofTheorem2.1)followsfromtheaboveresults,since
A
˜
⊂I
˜(
u
),foranycritical
subsolution
u
.Moreover,Theorem2.2alsofollowsfromtheresultsabove.

2.3Mathersemi-distanceandcriticalsubsolutions
AsitwasobservedbythefirstauthortogeneralizeMather’sexamples[29],see
theannouncement[13],arepresentationformulafor
δ
M
intermofC
1
criticalsub-
solutionsisextremelyuseful.ThishasalsobeenusedmorerecentlybySorrentino
.]43[Toexplainthisrepresentationformula,likeinTheorem2.6,wecall
SS
the
setofcriticalviscositysubsolutionsandby
S

thesetofcriticalviscosity(or
weakKAM)solutions.Hence
S

⊂SS
.If
u
:
M

R
isacriticalviscosity
subsolution,werecallthat

x,y

M,u
(
y
)

u
(
x
)

h
(
x,y
)
.
In[16],FathiandSiconolfiprovedthatforeverycriticalviscositysubsolution
u
:
M

R
,thereexistsaC
1
criticalsubsolutionwhoserestrictiontotheprojected

9

Aubrysetisequalto
u
.RecentlyPatrickBernard[4]hasevenshownthat
u
can
beassumedC
1
,
1
,i.e.differentiableeverywherewith(locally)Lipschitzderivative,
seealsoAppendixBbelow.Inthesequel,wedenoteby
SS
1
(resp.
SS
1
,
1
)the
setofC
1
(resp.C
1
,
1
)criticalsubsolutions.Therepresentationformulaisgiven
bythefollowinglemma:
Lemma2.7.
Forevery
x,y
∈A
,
δ
M
(
x,y
)=max
{
(
u
1

u
2
)(
y
)

(
u
1

u
2
)(
x
)
}
u
1
,u
2
∈S

=
u,
m
u
a

x
SS
{
(
u
1

u
2
)(
y
)

(
u
1

u
2
)(
x
)
}
21=max
1
{
(
u
1

u
2
)(
y
)

(
u
1

u
2
)(
x
)
}
u
1
,u
2
∈SS
=max
1
,
1
{
(
u
1

u
2
)(
y
)

(
u
1

u
2
)(
x
)
}
.
u
1
,u
2
∈SS
Proof.
Let
x,y
∈A
befixed.First,wenoticethatif
u
1
,u
2
aretwocritical
viscositysubsolutions,thenwehave
(
u
1

u
2
)(
y
)

(
u
1

u
2
)(
x
)=(
u
1
(
y
)

u
1
(
x
))+(
u
2
(
x
)

u
2
(
y
))

h
(
x,y
)+
h
(
y,x
)=
δ
M
(
x,y
)
.
Ontheotherhand,ifwedefine
u
1
,u
2
:
M

R
by
u
1
(
z
)=
h
(
x,z
)and
u
2
(
z
)=
h
(
y,z
)forany
z

M
,bythepropertiesof
h
thefunctions
u
1
,u
2
arebothcritical
viscositysolutions.Moreover
(
u
1

u
2
)(
y
)

(
u
1

u
2
)(
x
)=(
h
(
x,y
)

h
(
y,y
))

(
h
(
x,x
)

h
(
y,x
))
=
h
(
x,y
)+
h
(
y,x
)=
δ
M
(
x,y
)
,
since
h
(
x,x
)=
h
(
y,y
)=0.Thusweobtaineasilythefirstandthesecond
equality.ThelastinequalitiesisanimmediateconsequenceoftheworkofFathi
andSiconolfiandthatofBernardrecalledabove.

2.4Norton’sgeneralizationofMorseVanishingLemma
WewillneedinacrucialwayNorton’selegantgeneralizationofMorseVanishing
Lemma,see[30,31].Thisresult,likeFerry’sLemma(seeLemmaA.3)arethe
twobasicpiecesthatallowtoprovegeneralizationsoftheMorse-SardTheorem
(seeforexampletheworkofBates).
Lemma2.8
(TheGeneralizedMorseVanishingLemma)
.
Suppose
M
isan
n
-
dimensional(separable)manifoldendowedwithadistance
d
comingfromaRie-
mannianmetric.Let
k

N
and
α

[0
,
1]
.Thenforanysubset
A

M
,wecan
findacountablefamily
B
i
,i

N
of
C
1
-embeddedcompactdisksin
M
ofdimen-
sion

n
andacountabledecompositionof
A
=

i

N
A
i
,with
A
i

B
i
,forevery
i

N
,suchthatevery
f

C
k,α
(
M,
R
)
vanishingon
A
satisfies,foreach
i

N
,

y

A
i
,x

B
i
,
|
f
(
x
)

f
(
y
)
|≤
M
i
d
(
x,y
)
k
+
α
(1)
01