On the rectifiability of defect measures arising in a micromagnetics model

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On the re tiability of defe t measures arising

in a mi romagneti s model  Luigi Ambrosio y , Bernd Kir hheim z , Myriam Le umberry x , Tristan Rivière { May 30, 2002 1 Introdu tion

Given a bounded domain of R

2

, we onsider the spa e of maps u : ! C

satisfying juj=1 a.e. in divu=0 in D 0 (): (1.1) Equivalentely, takingu=r ? g :=(  x2 g; x1

g),this spa e oin ides withthe spa e

of all fun tionsg :!R solving

jrgj 2

=1 a.e. in:

Inside this large spa e we will restri tour attention to the following lass of ve tor

elds: M div ():= 8 < : u:!C s.t. divu=0and 92L 1 () satisfying u=e i and U  :=div(e i^a

) is anite Radon measure in R

9 =

;

(1.2)



Tobepublishedinavolumededi atedto O.Ladyzhenskaya

y

S uolaNormaleSuperiore,PiazzadeiCavalieri,56100Pisa,Italy,luigiambrosio.sns.it

z

MaxPlan kInstitut,Leipzig,Germany,Bernd.Kir hheimmis.mpg.de

x

Laboratoire de Mathématiques, Université de Nantes, 44322 Nantes Cedex 03, Fran e,

Myriam.Le umberrymath.univ-nantes.fr {

on the liftingin (1.2) isnonlinear, unlike the divergen e-free onstraint.

Thespa eM

div

() wasintrodu edin[RS2℄andisthe naturallimitspa eof the

two dimensional variational problem modelising mi romagnetism without vorti es

(see [RS1℄ and [ARS℄ for a detailed presentation of this problem). In brief, we

onsider the energy

E  (u):= Z jruj 2 + 1  Z R 2 jH u j 2 dx; where H u

(the so- alleddemagnetizingeld) isthe url-free ve toreld relatedtou

by the PDE div(~u+H

u

)=0, u~ being the extension of u toR

2

n with the value

0. Assuming that E  (u  )  C and u  = e i  with   2 H 1 uniformly bounded in L 1

,inTheorem1 of[RS2℄ itisshown that thefamily 



has limitpoints(inthe L

1

topology) as  ! 0

+

and that any limit point fulls (1.2). Moreover, we have the

liminf inequality

liminf k!1 E  k (e i" k )2jU  1 j(R) whenever  " k ! 1

. In[Le℄ this ompa tnessresult hasbeen extended totheM

div

spa e, see Theorem 3.6.

Theproofofthesefa tsisbased,amongotherthings,onsomemethodsdeveloped

in[ADM℄and in[DKMO1 ℄ inthe very lose ontextoftheAviles-Gigaproblem(see

[AG1℄, [AG2℄). In this settingone onsiders the energy fun tionals

F  (v):= Z jr 2 vj 2 + (1 jrvj 2 ) 2  dx;

so that the ve tor elds rv, up to a rotation, are exa tly divergen e-free but take

their values onS

1

only asymptoti ally.

At this stage a full - onvergen e theorem in the mi romagneti s ase (and in

the Aviles-Giga problem as well) is still missing, although as we said the liminf

inequality isknown to hold ingeneraland the limsup inequalityhas been proved

in some parti ular situations. Besides, the results in [RS2℄ and [ARS℄ lead to a

hara terization of energy minimizing ongurations.

The ompleteness of the -limit analysis of this variational problem requires a

deeperunderstandingofthespa eM

div

(). Inparti ular,amorepre isedes ription

of the singular sets of arbitrary maps inM

div

() is avery naturalquestion.

As explained in [RS2℄, the measure div(e

i^a

) dete ts the singular set of :

for instan e, it is proved in [LR℄ that  is lo ally Lips hitz in if and only if

div(e

i^a

)=0inD

0

(R). In theparti ular ase wherethelifting isafun tion

in BV) that the measure dive is arried by S 

, where S

is the ountably H

-re tiable set where  has a dis ontinuity of jump type, in an approximate sense

(see Se tion2). Pre isely, forany  2BV() su h that dive

i =0one has div(e i^a )= f <a< + g (e ia e i )
  H 1 J  ; (1.3) where  

are the approximate limits of  on both sides of S

and 

is hosen

in su h a way that  < 

+

, and 

f <a<

+ g

is the hara teristi fun tion of the

interval( ; + )inR. FinallyH 1 J 

denotesthe1-dimensionalHausdormeasure

restri ted to J

 .

Our main motivation in this work is to extend su h a des ription of the jump

set to liftings of ve torelds inM

div

(). In[ADM ℄ anexample ofa ve toreld in

M div

() whi h isnot inBV(;S

1

)isgiven. Pre isely, the authorsgiveanexample

of a map in the so- alled Aviles-Giga spa e AG

e

(see [AG1℄, [AG2℄, we follow the

terminology of [ADM ℄) whi h is not in BV(). We re all that AG

e

() is made by

all solutionsu of the eikonal equationsu h that

div  u   3 ;  u   3 !

isa nite Radon measure in

for any orthonomal basis (;) of R

2

. Be ause of the similaritiesbetween the two

spa es it happens that this map an be made also in M

div

(the te hni al reasons

is that smalljumps are penalized with a power faster than 1, see (3.5) and [RS1℄).

Therefore the BV spa eistoosmallfor ouranalysisand thereisnohopetoa hieve

our goal by using the lassi al resultsof the BV theory.

Itis proved in[RS2℄ thata lifting of ave toreld inM

div

solves the following

kineti equation : ie ia
r x [((x) a)℄= a dive i^a
in D 0 (R); (1.4)

wheredenotesthe hara teristi fun tionofR

+

. Byapplyingnow lassi alresults

of regularity of velo ity averaging of solutions to kineti equations (see [DLM ℄),

one gets that solutions to (1.4) for whi h the jump distribution div(e

i^a

) is a

nite Radon measure are in W

;p () for any  < 1 5 and p < 5 3 . Taking advantage

of the spe i ity of the solution f = [((x) a)℄ solving the general equation

ie ia
r x f = a g,whereg =div(e i^a

),P.E.JabinandB.Perthamein[JP ℄improved

the Sobolev exponents and showed that

2W ;p () 8 < 1 3 and p< 3 2 : (1.5)

Still being a ni e improvement, this is far from being enough to tellus something

on the stru ture of the singularset of  (one would like for instan e to get as lose

not the most appropriate one to explore our non linear spa e M div

(), we adopt

hereamoredire tapproa hworkingdire tlyonthesingularsetthroughablow-up

analysis of the measure 

(B):=jU

j(BR).

Our main resultis the followingstru ture theorem.

Theorem 1.1. Let  be a lifting of u2M

div

() as in (1.2). Then

(i) The jump set J

is ountably H

1

-re tiable and oin ides, up to H

1 -negligible sets, with :=  x2: limsup r!0 +   (B r (x)) r >0  : In addition div(e i^a ) J  = f <a< + g (e ia e i )
  H 1 J  8a2R: (1.6) (ii) For H 1 -a.e. x2nJ 

we have the following VMO property:

lim r!0 + 1 r 2 Z B r (x) j j=0;

where  is the average of  on B

r (x).

(iii) The measure Æ:=

 (nJ  ) is orthogonal to H 1 , i.e. B Borelwith H 1 (B)<+1 =) Æ(B)=0:

ComparingthisresultwiththeBV theory,weexpe tthat(ii) ouldbeimproved,

showing also onvergen e of the mean values as r ! 0

+

(and thus existen e of an

approximatelimitatH

1

-a.e. x2nJ

). Moreover,by(1.3)andtheVMO ondition

out of J

we expe t also that the measures divT

a

u are on entrated onJ

. If this

is the ase, by the formula(see Theorem 3.2(ii))

  = Z R jdive i^a jda (1.7)

one would get that the measure Æ in(iii)is identi ally0 andfull re tiabilityof the

measure 

. All theseproblems arebasi ally open, anditwould beinteresting even

toshowthatÆ issingularwithrespe ttothe 2-dimensionalLebesguemeasure, thus

showing that Æ is a Cantor-type measure (a ording tothe terminologyintrodu ed

in [DeGA ℄, [A℄ for BV fun tions). We prove that Æ is identi ally 0 by making

an additional mild regularity assumption on , namely H

1

(\ n) = 0, see

dilating at a point where the 1-upper density of the jump measure is nonzerois

strongly related tothe uniqueness result established in[ALR℄.

It is likey that this analysis an be extended to s alar rst order onservation

lawswithstri tly onvexnon-linearities,wherethe lassi alOleinikuniquenessresult

plays the role of our uniqueness result in [ALR℄. Pre isely, given a solution  on

RR + of  t + (AÆ) x =0 for A 00

>0and assuming that,for any S 2Lip(R),one has that

m= (SÆ) t + (QÆ) x 2 M lo (RR + ); where S 0 A 0 =Q 0 and whereM l o (RR +

) denotes the the distributionswhi h are

Radon measures in RR

+

, then we expe t a similarstru ture theorem tobe true

for the measure m.

Now we briey des ribe the ontents and the te hniques used in this paper.

Se tion2 ontainssomebasi materialaboutBV fun tions,approximate ontinuity,

approximate jumps. The main result is Proposition 2.3, wherewe nd a ne essary

and su ient for a lifting to bea fun tion of bounded variation.

Se tion3 ontains the mainbasi properties ofthe spa eM

div

. In parti ularwe

show the identity (1.7) and, as a onsequen e, the absolute ontinuity of 

 with

respe t to H

1 .

InSe tion4westudysome propertiesof on avefun tionswhosegradient

satis-es the eikonal equations. Thesepropertiesare used inthe lastse tionof the paper

for the lassi ation of blow-ups.

Se tion 5 is devoted to some abstra t riteria for the re tiability of sets and

measures in the plane. We use a lassi al blow-up te hnique (see [Pr℄ for mu h

more onthe subje t), studying the asymptoti behaviourofthe res aled and

renor-malized measures arounda point. The renormalizationfa tor we use is simply the

radius of the ball (see Denition 5.1). The new observation here is that very weak

informations about the stru ture of blow-ups allow to show that points where the

upper 1-dimensionalspheri aldensity is positive are indeed pointswhere the lower

1-dimensional spheri al density is positive, see Theorem 5.2. In our problem, this

information is used to show that 

(nS

) has zero 1-dimensional density, and

therefore is orthogonalwith respe t to H

1 .

Se tion 6 is devoted to the lassi ation of blow-ups. Here we use the idea

that any ve tor-valued measure be omes, after blow-up, a onstant multiple of a

positive measure at a.e. blow-up point. This idea was rst used by E. De Giorgi

nite perimeter. Here this idea is pushed further, onsidering the measures Z R e ia dive i^a da; Z R g(a)dive i^a da;

all absolutely ontinuous with respe t to

, and blowing up at Lebesgue points of

alltherespe tivedensities. Weshowinthiswaythatanyblow-upiseither onstant,

or jumps ona line, or jumps on a haline, with a uniform (i.e. independent of the

hosen subsequen e) lowerbound on the width of the jump. This su es to apply

the results of the previousse tions, and toinfer re tiability.

While ompleting this work we learned that C. De Lellis and F. Otto

indepen-dentlyestablishedin[DO℄ astru ture theorem similarto Theorem1.1 forthe

Aviles-Giga spa e. Their proof, stillbased on a blow-up argument, is more elaborate, sin e

in the aseof theAviles-Gigaspa e the lassof blow-ups isaprioriri her. Itisalso

interesting to noti e thatno onne tion with thetheory of vis osity solutions isused

in their paper.

We lose this introdu tion with the following table, summarizing the notation

used without further explainationinthe paper.

A bounded open set inR

2

a^b The minimum of a and b

a_b The maximum of a and b

v
w The s alar produ t of v and w

\

(v;w) The angle  2[0;℄ su h that v
w=jvjjwj os

v ?

The anti- lo kwise =2 rotationof v, ( v

2 ;v 1 ) e ia

The ve tor ( osa;sina)

B r

(x) The ballwith entre xand radius r (x=0 an be omitted)

H 1

Hausdor 1-dimensional measure inR

2 S

1

Unit spherein R

2

M(X) Finite Radonmeasures inX

M +

(X) Positiveand niteRadon measures in X

 B Restri tion of  toB, dened by 

B .

2 Continuity points, jump points, BV fun tions

Let us introdu e some weak notions of ontinuity and jump, well studied in the

ontext of BV fun tions. All of them have a lo alnature and, to x the ideas, we

givethe denitionsfor some fun tion f 2L

1 lo (R 2 ;R m ).

exists a2R m su h that lim r!0 + 1 r 2 Z Br(x) jf(y) ajdy=0:

The ve tor a whenever exists isunique and is alled the approximate limitof

f atx. We denoteby S

f

the set of pointswhere f has noapproximate limit.

 (Approximate jump points) We say that x is a jump point of f if there

exist a + ;a 2R m and  x 2S 1 su hthat a + 6=a and lim r!0 + 1 r 2 Z B  r (x) jf(y) a  jdy=0; where B  r (x) = fy 2 B r (x) : (y x)
 x

> 0g are the two half balls

determinedby x . Thetriple(a + ;a ; x

)isuniquelydetermineduptoa hange

of orientationof  x andapermutationof (a + ;a ). Wedenoteby J f the set of jump pointsof f.

It isnot hard toshow (see [AFP℄) that S

f

; J

f

are Borelsets, that J

f S f , and that S f is Lebesgue negligible.

The following Lemma has been proved in [A1℄ in a more general ontext. For

the sake of ompletenesswe in lude the proof.

Lemma 2.1. Let (

l

) be a family of ontinuous fun tions dened on R whi h

sep-arates points. Let 2L

1 (R 2 )and set  l := l

Æ. Thenthe followingimpli ations

hold:

(i)  has an approximatelimit atx ifand only ifall fun tions

l

have an

approx-imate limit at x;

(ii) If xiseither anapproximate ontinuitypoint or ajump pointfor allfun tions

 l

, with the same normal to the jump, then the same is true for .

Proof. (i) We prove only the nontrivial impli ation, the "if" one. Let us set X :=

[ kk

1

;kk

1

℄. By the Stone-Weierstrass theorem the algebra A generated by the

family (

l )

l 2N

is dense inthe set of ontinuous fun tion of X,C(X), endowed with

the sup norm. If 

l

Æ has an approximate limitat x for any l we inferthat f Æ

has anapproximatelimitatxforany f 2A. Sin eAisdenseinC(X),theidentity

fun tion is the uniform limit of a sequen e of fun tions of A, so that  has an

approximate limitatx.

(ii)Theproofissimilar,workinginthetwohalfspa esdeterminedbythe ommon

x is a jumppointfor at least one of the fun tions l

, then x must be a jumppoint

of , by (i).

We are going to apply this result with 

l (x) = (x_b l )^ l , where (b l ; l ) is a

family of open intervals. It is easy to he k that the family (

l

) separates points if

and only if the losed set Rn[

l (b

l

;

l

) has anempty interior.

Were allalsosomebasi fa tsaboutBV fun tionswhi hwillbeusedthroughout

the paper. We say that u 2L

1

(;R

m

) is a BV (bounded variation) fun tion, and

we write u2BV(;R

m

)(R

1

an beomitted), if its distributionalderivatives D

i u, i.e. hD i u; i:= Z  x i udx 2C 1 (); i=1; 2

are representable bynite R

m

-valuedRadon measures in. We denoteby jDuj()

the total variation of the R

2m -valued measure Du = (D 1 u;D 2 u). When u 2 W 1;1 (;R m )we have Du=ruL 2 and therefore jDuj() = Z jrujdx:

Were allthat the jumpset of aBV fun tion uis ountablyH

1

-re tiableand that

Z J u ju + u jdH 1 jDuj(): (2.1) Moreover, H 1

-a.e. any approximate dis ontinuity point isa jumppoint, i.e.

H 1 (S u nJ u )=0: (2.2)

Nowweinvestigateunderwhi h onditions aliftingofafun tionu2BV(;S

1 ) is itselfa BV fun tion. Proposition 2.3. Let 2L 1 () be su h that (i) u:=e i 2BV(;S 1 ); (ii) U  :=dive i^a 2M(R). Then 2BV() and jDj() C[jU  j(R)+jDuj()℄

Proof. Let  0

2BV() be given by Lemma 2.4 below, satisfyinge

0

=e . Then

there exists a unique k 2 L

1

(;Z) su h that  = 

0

+2k. The goal is to show

that k 2BV(;Z). Itis lear, sin e 0 2BV()\L 1 ,that div(e ia^ 0 )2M(R). Therefore we an dedu e that     Z Z R osa e ia^ e ia^ 0

r dadx     Ck k 1 (2.3) forany 2C 1 (),with C=jU  0 j(R)+jU  j(R). Noti ethat jU  0 j(R)

an be estimated(see (3.5)) withjD

0

j() andthis, inturn, an beestimated with

jDuj(). We observe that e ia^ =e ia^( 0 +2k) =e i(a 2k)^ 0

. Fixing x 2and assuming

k(x)>0 tox the ideas, we dedu e fromthe remark above that

Z R os a e ia^(x) e ia^0(x)
da= Z  0 (x)+2k(x)  0 (x) osa(e ia e i0(x) )da =e i0(x) Z 2k(x) 0 os(b+ 0 (x)) e ib 1
db =e i 0 (x) (k(x) os 0 (x) ik(x) sin 0 (x)) =k(x)  1 0  :

Combining this fa twith (2.3) we have proved that

    Z k  x 1      C k k 1 8 2C 1 ():

This shows thatD

1

k isa niteRadonmeasure in. A similarargument(repla ing

osa by sina in(2.3)) works for D

2 k.

In the proof above we used the followinglemma, whi h ensures the existen e of

a BV lifting.

Lemma 2.4 (BV lifting). Letu2BV(;R

2

)su hthatjuj=1almosteverywhere

in . Then there exists 

0 2BV(;[ 2;2℄) verifying (i) u=e i 0 a.e. in ; (ii) jD 0 j()C 0 jDuj(), where C 0 is an absolute onstant.

Proof. Let  0

be a smooth fun tion from R into [

2

;+

2

℄ su h that for any z =

(x 1 ;x 2 ) in S 1 verifying x 1  0,  0 (z) is the angle in [  2 ;  2 ℄ su h that e i0(z) = z. Similarlyweintrodu e 

tobeasmooth mapfromR

2

into[0;2℄su hthat forany

z =(x 1 ;x 2 )2S 1 verifying x 1  3 4 , 

(z) isthe anglein [0;2℄su h that e

i  (z) =z. Sin e u =(u 1 ;u 2 ) is in BV(;R 2

), by the mean value theorem and the oarea

formulainBV we may nd  2[ 1 4 ; 1 2 ℄ su h that jD fx: u1(x)g j() 4jDu 1 j(); thus E =fx2 : u 1

(x)  g is a nite perimeterset. By virtue of the Volpert's

hain rule (see for instan e [AFP℄, Theorem 3.96), we have that both 

0 Æu and   Æuare inL 1

\BV() and theirtotal variations an be estimatedwith jDuj().

Using nowthe de omposability theorem ([AFP℄, Theorem 3.84), we have that

 0 := E  0 Æu+ nE   Æu is inBV() and jD 0 j()[k 0 k 1 +k 1 k 1 ℄jD E j()+jD( 0 Æu)j()+jD( 1 Æu)j(): By onstru tionwe have e i

=u a.e. in and  isa solutionof our problem.

3 The spa e M

div ()

In this se tion we introdu e the main obje tof study of the present paper.

Denition 3.1. We denote by M

div

() the spa e of two-dimensional ve tor elds

u in L 1 (;S 1 ) satisfying (P1) divu=0 inD 0 ();

(P2) thereexists alifting2L

1

(),i.e. amap satisfyingu=e

i

,su hthat the

distribution U  inD 0 (R) dened by hU  ; (x;a)i:= Z R Z e i(x)^a
r x (x;a)dxda

is anite Radon measure inR.

For a 2 R we set T a u := e i^a 2 L 1 (;S 1

) (this is a slight abuse of notation,

sin e T

a

u depends on the lifting and not only on u, but it is justied by the fa t

that in the followingthe liftingof u will bekept xed), sothat

hU  ; i:= Z R hdivT a u; (
;a)ida: 8 2C 1 (R):

Noti e that, sin e  2 L (), then (P1) implies that divT u = 0 for all a 2 R

su h that jaj >kk

1

. Finally, we denote by 

the proje tion of jU

 j on the rst variable,i.e.   (B):=jU 

j(BR) for any B Borel.

In the following theorem we state some basi properties of the trun ated ve tor

elds T

a

u and a useful representation formula forU

 .

Theorem 3.2. Let u2M

div

(). Then, the followingproperties hold:

(i) The map a7!divT

a

u satises the Lips hitz ondition

  hdivT a u; i hdivT b u; i   L n ()kr k 1 jb aj 8 2C 1 (): (3.1) (ii)   (B)= R R jdivT a

uj(B)da for any Borel set B . In parti ular divT

a

u is

a nite Radon measure in for a.e. a2R.

(iii) For a.e. a2R we have

1 2Æ Z a+Æ a Æ divT b udb ! Æ!0 + divT a u in M 0 ():

Proof. (i) Follows by the elementaryinequality jT

a

u T

b

ujjb aj.

(ii)Forany (x;a)=f(x)g(a), with f 2C

1 () and g 2C 1 (R) wehave hU  ; (x;a)i= Z R g(a) Z T a u
rf(x)dxda:

By approximation, the same identity holds if g is a bounded Borel fun tion with

ompa tsupport. Now, hoosinganopenset Aandf 2C

1 (A)withkfk 1 1 and g = (a;a+Æ) weget jU  j(A(a;a+Æ℄) Z a+Æ a Z T b u
rf(x)dxdb; so that d da jU  j(A( 1;a℄) Z T a u
rf(x)dx 8a2R:

Being f arbitrary,this givesthat divT

a

u isa nite Radon measure in Aand

d da jU  j(A( 1;a℄)jdivT a uj(A)

jU  j(AR) Z R jdivT a uj(A)da (3.2)

for any open set A. On the other hand, the inequality

jU  j(R) Z R jdivT a uj()da: (3.3)

is easy to prove, using the denition of U

. From (3.2) and (3.3) we obtain the

oin iden e of the measures 

 and R jdivT a ujda.

The property (iii) is an easy onsequen es of (ii) and of the Lips hitz

prop-erty (3.1): it su es to hoose Lebesgue points of the integrable fun tion a 7!

jdivT

a uj().

The following overing te hni allemma willbeused toshow theabsolute

onti-nuity of 

with respe t to H

1 .

Lemma 3.3. Let K be a ompa t set of . Then, there exists a sequen e (

n )  C 1 (;[0;1℄)su h that: (i) n =1 on K and spt n !K as n !1;

(ii) limsup

n!+1 Z jr n jdxH 1 (K). Proof. Let L = H 1

(K). By the denition of Hausdor measure, for any n  1 we

an nd anitenumberofballs B

i

=B(x

i

;r

i

)whoseunion overs K and su h that

r i < 1=n and P i 2r i

< L+1=n. By the subadditivity of perimeter, the open set

A n :=[ i B i

has perimeter less than L+=n. Then, we set

n = An  n , where  n

<1=nis hosensosmallthatstill

n

=1onK(itsu es that

n

<dist(K;A

n ))

and the supportof

n

is ompa t. Sin ethe total variationdoesnot in rease under

onvolution(see forinstan e Proposition3.2( ) of [AFP℄) we have

Z jr n jdx=jD n j()jD An j()L+  n and therefore n

has allthe statedproperties.

Theorem 3.4 (Absolute ontinuity). The measure 

is absolutely ontinuous

with respe t to H

1

, i.e. (B)=0 whenever B is a Borel H

1

that, for all ompa t sets K  , 

(K)  CH

1

(K). We will prove that, for all

a 2 R su h that divT

a

u is a Radon measure on , the inequality jdivT

a

uj(K) 

2H 1

(K) holds for any ompa t set K  . Then, sin e divT

a

u = 0 as soon as

jaj>kk

1

, by Theorem 3.2(iii) we obtain

  (K)= Z R jdivT a uj(K)da2kk 1 H 1 (K):

Let a 2 R be su h that  := divT

a

u is a nite Radon measure on . By

the Hahn de omposition theorem, there exists two disjoint Borel sets A

+

, A su h

that, if 

+

and  denote respe tively the positive and negative parts of , then

  = A  . Sin e jj = +

+ , it su es to prove that 

+ (K) H 1 (K) for any K A + ompa t and  (K)H 1

(K) forany K A ompa t.

Let K  A

+

be ompa t and let (

n )  C 1 (;[0;1℄) be given by Lemma 3.3. Wehave  + (K) = (K) lim n!1 Z n d = lim n!1 Z T a u
r n dx  limsup n!1 Z jr n jdxH 1 (K):

A similar argumentworks for ompa t sets K A .

Inthe ase when 2BV

lo

() one an use Volpert's hainrule inBV toobtain

an expli it formulafor divT

a

u, see [RS2℄: it turns out that

divT a u=(a; + ; )(e ia e i ^ + )
  H 1 J  ; (3.4) where (a; + ; ):= 8 < : 1 if  <a< + 1 if  + <a< 0 else.

Moreover, thedivergen e free onditiongivese

i +
  =e i
  atanypointinJ  . In parti ular, hoosing  

in su h a way that 

+

>  , Fubini theorem and (2.1)

give jU  j(R) = Z R jdivT a uj()da = Z R Z J   f a + g je ia e i jdH 1 da  Z J  j +  j(2^ 1 2 j +  j)dH 1 2jDj(): (3.5)

The following lemma provides an integral representation of the divergen e,

Lemma 3.5. Let u 2 L (R ;R ) and let K  R be ountably H -re tiable. If

divu is a Radon measure in R

2 and H 1 (K\S u nJ u )=0, then divu K =(u + u )
H 1 K\J u :

Proof. Arguing as in Theorem 3.4 and using Lemma 3.3 one an easily show that

divu<< H

1

,hen e divu K isrepresentablebyH

1

K for somedensityfun tion

. The fun tion  an be hara terized by a blow-up argument, using the fa t that

K be omes a line (here the re tiability of K plays a role) after blow-up and u

be omesajumpfun tionora onstant fun tionatH

1

-a.e. blow-up pointof K.

Thefollowing ompa tnessresulthasbeenprovedin[Le℄adaptingthetrun ation

argument of [RS2℄.

Theorem 3.6 (Compa tness). For any onstant M 0 the set

f2L 1 (): kk 1 +jU  j(R)Mg is ompa t in L 1

() with respe t to the strong topology.

4 Some properties of on ave fun tions

In this se tion we study some properties of on ave fun tions g whose gradient

satises the eikonal equation. Were all that the superdierentialg(x) of g at xis

the losed onvex set dened by

g(x):=  p2R 2 : g(y)g(x)+p
(y x) 8y2R 2 :

Itfollowsimmediatelyfromthedenitionthatthegraphofg,i.e. f(x;p): p2g(x)g

is a losed subset of R

2

R

2

. Moreover, the Lips hitz assumption on g gives

g(x)B

1

for any x. Finally,g(x)=frg(x)gatany dierentiability pointof g.

Forany! 2S

1

and any x2R

2

,the leftand rightdire tionalderivativealong!

of g at xare dened by r  ! g(x):= lim r!0  g(x+r!) g(x) r : For any x 2J rg

we denote in the following by (rg

+

;rg ;

x

)the triple dened in

Se tion 2.

Proposition 4.1. Let g :R

2

!R be on ave and satisfying jrgj=1 a.e. in R

2 .

setting D x :=fx+trg(x): t <0g, for H 1 -a.e. y2D x , rg has an

approxi-mate limit at y equal to rg(x).

(ii) Let J be the set of approximate jump points of rg and let x 2 J. For any

! 2S

1

su h that !


x

>0, the partial derivativesr

 ! g(x) exist and r ! g(x)=!
rg (x)!
rg + (x)=r + ! g(x): (4.1) Moreover, setting D  x :=fx+trg  (x): t<0g, for H 1 -a.e. y2D  x , rg has

an approximate limit equal to rg

 (x).

(iii) For all >0, we dene the following sets

J  :=fx2J : jrg + (x) rg (x)j g   :=fx2R 2 : diam(g(x)) g: Then, J    and   is losed.

Proof. The rst two statements an be proved in the same way and we prove only

the se ond. By the denition of J there exist 

x 2 S 1 and rg + (x); rg (x) 2 S 1 su h that lim r!0 + 1 L 2 (B  r (x)) Z B  r (x) jrg(y) rg  (x)jdy =0: (4.2) For r>0,let usdene g r (y):= g(x+ry) g(x) r ; y 2B 1 : Then, rg r (y)=rg(x+ry). By(4.2), rg r onverge inL 1 (B 1 ) whenr !0 + to the fun tion G 0 (y):=  rg + (x) if y
 x >0 rg (x) if y
 x <0:

By Sobolev embedding, this implies that (g

r ) uniformly onverges in B 1 to a 1-Lips hitz fun tion g 0 satisfyingrg 0 =G 0 . Sin e g r (0)=0 we have that g 0 (0)=0 and therefore g 0 is uniquely determined: g 0 (y):=  y
rg + (x) if y
 x 0 y
rg (x) if y
 x 0

But, for any ! 2S

1 ,we have r + ! g(x)= lim r!0 + g(x+r!) g(x) r = lim r!0 + g r (!)=g 0 (!)

r ! g(x)= lim r!0 + g(x r!) g(x) r = lim r!0 + g r ( !)= g 0 ( !):

Therefore, if we assumethat !


x >0,then r ! g(x)=!
rg (x) and r + ! g(x)=!
rg + (x): Moreover, we have r ! g(x)r + !

g(x) sin e the restri tion of g toR! is on ave.

Letusnowprovethese ondpart. Letx2Jandlety 2D

x

. Sin etherestri tion

of g to D x is on ave, wehave r rg (x) g(y)r + rg (x) g(y)r rg (x) g(x)=rg (x)
rg (x)=1:

Sin e g is 1-Lips hitz we obtain that

r rg (x) g(y)=r + rg (x) g(y)=1: (4.3) By (2.2), for H 1 -a.e. y 2 R 2

either rg has an approximate limit at y, or y is an

approximate jumppointof rg. If y2D

x

, y an't be a jumppoint of rg. Indeed,

assuming that y2J and applying (4.1) with !=rg (x), we have

r rg (x) g(y)=rg (y)
rg (x)rg + (y)
rg (x)=r + rg (x) g(y): By (4.3), rg (y)
rg (x) = r +

g(y)
rg (x) = 1. Thus, rg (y) = rg

+

(y) =

rg (x),whi h ontradi tstheassumptiony2J. Therefore,rghasanapproximate

limitequal torg (x) atH

1

-a.e. y 2D

x

. The same argument an be used for D

+ x

and (ii) isproved.

(iii) First, let us show that J

 

. Indeed, sin e g is dierentiable a.e., for any

x 2 J

we an nd dierentiability points x

 h

onverging to x su h that rg(x

 h

)

onverge to rg



, hen e the losednedd os the graph of g gives that rg

+

(x) and

rg (x)are ing(x). Thus, diamg(x) and x2

 .

The losedness of 

is an immediate onsequen e of a ompa tness argument

based onthe losedness of the graphof g and onthe fa t thatg(x)B

1

for any

x.

5 Re tiability of 1-dimensional measures in the

plane

Inthisse tionwe onsiderameasure2M

+

()absolutely ontinuouswithrespe t

to H

1

, i.e. vanishing onany H

1

-negligibleset. We dene

  (;x):=liminf r!0 + (B r (x)) r ;   (;x):=limsup r!0 + (B r (x)) r : (5.1)

A general property is that  (;x) is nite for H -a.e. x (see for instan e [AFP℄)

hen e the absolute ontinuity assumption gives that 



(;x) is nite for -a.e. x.

Wedene also  +  :=fx2:   (;x)>0g;   :=fx2:   (;x)>0g (5.2)

and noti e that 

 

are Borel sets and 



 

+ 

. Noti e also that 

+ 

is -nite

with respe t to H

1

, asall the sets

  :=fx2:   (;x) g (5.3) satisfy H 1 ( 

)  2()= (see [AFP℄, Theorem 2.56). Therefore, by the Radon

Nikodým theorem, we an represent

=  +  + (n +  )=fH 1  +  + (n +  ) (5.4) for some f 2 L 1 (H 1  + 

). Noti e that the residual part 

r :=  ( n  +  ) is "orthogonal" toH 1

in the followingsense:

H 1

(B)<+1 =) 

r

(B)=0:

This is a onsequen e of the fa t that 

 ( r ;x) is 0for  r -a.e. x.

The following denition is a parti ular ase of the general one given in the

fun-damental paper[Pr℄.

Denition 5.1 (Tangent spa e to ). Given x 2 and r > 0, we dene the

res aled measures  x;r 2M(( x)=r)by  x;r (B):= (x+rB) r

for any Borel set B ( x)=r, sothat

Z (y)d x;r (y)= 1 r Z ( y x r )d(y) 82C (( x)=r):

We denote by Tan(;x) the olle tion of all limit points as r ! 0

+ of  x;r , in the duality with C (R 2 ).

Noti e that the denition above makes sense be ause the sets ( x)=r invade

R 2 as r!0 + . Moreover, as  x;r (B R )= (B Rr (x)) r 1+R   (;x) 8R >0

for r su iently small (depending on R ), a simple diagonal argument shows that

Tan(;x) is not empty whenever



that for some x2 + 

the following properties hold

(i) The density fun tion f(r):=(B

r

(x))=r is ontinuous in (0;Æ) for some Æ 2

(0;dist(x;)) ;

(ii) 



(;x) is nite;

(iii) There exists

x

> 0 su h that any nonzero measure  2 Tan (;x) is

repre-sentable by H

1

L, where 

x

and L is either a line or a haline (not

ne essarily passing through the origin).

Then x2

 .

Proof. We introdu e rst some notation. Given a line or a half lineL interse ting

the open ball B

1

, we denote by

^

L the line ontaining it and by  its dire tion (if

L=

^

Lthe orientationdoes not matter). We denoteby h

L

2[0;1)the distan e of

^ L

from the origin. Finallywe dene d

L 2[ 1;1℄so that y2L\B 1 () y2 ^ L\B 1 and y
 > d L :

An elementary geometri argument shows that, if d

L  0 and H 1 (L\B 1 )  1=2, then h L  p 3=2.

Weassumeby ontradi tion thatx2= 



,i.e. 



(;x)=0. Hen eforth,wexa

positive number q < minf

x

=2;



(;x)g and nd a de reasing sequen e (R

i ) with f(R i )<q=4and then r i <R i su h that f(r i )=q and f(t)<q for t2(r i ;R i ℄ (r i is the rst r below R i

atwhi h f hits q). Noti e that ne essarily R

i =r

i

4.

Possibly extra tinga subsequen e, by assumptions(ii),(iii)we anassume that

the res aled measures 

i

= 

x;r i

weakly onverge, in the duality with C

(R 2 ), to a Radon measure  = H 1

L, where L iseither a lineora halineand 

x . As  i (B 1 ) = q we obtain that (B 1 ) = (B 1

)  q. On the other hand, as

 i

(B r

)qr for any r 2(1;4) weobtain

(B 1

)=q and (B

r

)qr 8r2(1;4):

In parti ular the right derivativeof g(r):=(B

r

)=r atr=1 isnonpositive.

On the other hand, we have

(B r )=  d L + q r 2 h 2 L  8r1; so that d dr + g(r)     r=1 = d dr + d L + p r 2 h 2 L r     r=1 = h 2 L d L p 1 h 2 L p 1 h 2 L :

L L H 1 (L\B 1 )= q  q x < 1 2 ; hen e h L  p 3=2 and h 2 L > p 1 h 2 L

. Therefore the derivative above is stri tly

positivein any ase. This ontradi tion proves the theorem.

The following re tiability result is part of the folklore on the subje t, but we

in lude aproof for onvenien e of the reader.

Theorem 5.3 (Re tiability riterion). Assume that for -a.e. x 2 

 there

exists a unit ve tor  = (x) su h that any measure  2 Tan(;x) is on entrated

on a line parallel to . Then 

 is ountably H 1 -re tiable. Proof. For n1, letS n be dened by S n :=  x2:   (;x) 1 n  As H 1 S n  2n it follows that H 1 (S n

) < +1, therefore by the de omposition

theorem (see Corollary 2.10 in[F℄)we an write S

n =S r n [S u n , where S r n \S u n =;, S r n is ountably H 1 -re tiable and S u n

is purely unre tiable, i.e. its interse tion

with any re tiable urveis H

1

-negligible. LetusshowthatH

1 (S u n )=0. Then,  

willbe ontained ina ountable unionof re tiable urves andTheorem 5.3willbe

proved.

Let us dene, for any dire tion ! 2 S

1

, for any angle  2 (0;

 2 ), x 2 R 2 and r >0, S r (x;!;) asthe interse tion of B r

(x)with the one

n y2R 2 nfxg: j os \ (y x;!)j>j osj o :

having x+R as axis. Sin e S

u n

is purely unre tiable,by Theorem 3.29 in[F℄, for

H 1 -a.e. x2S u n we have limsup r!0 + H 1 (S u n \S r (x;!;)) r  1 6 sin 8!2 S 1 ; 82(0;  2 ):

In parti ular, xing , wehave

limsup r!0 +  S r (x; ? (x);)
r  1 12n sin for H 1 -a.e. x2S u n .

Assumingby ontradi tionthatH (S n

)>0, hoosex2S

n

wherethe abovedensity

property holds and a sequen e r

i #0 su hthat  x;r i ! lo allyweakly in R 2 and lim i!1  S r i (x; ? (x);)
r i  1 12n sin: (5.5)

By assumption we know that  is on entrated on a line L parallel to , and (5.5)

gives  S 1 (0; ? (x);)
 1 12n sin >0:

Wewillobtaina ontradi tionby showing thatthelineLpassesthrough theorigin.

If not, there is > 0 su h that (B

) =0, so that (B r i (x))=r i is innitesimal as

i!1. This isnot possible be ause x2

 .

6 Classi ation of blow-ups and re tiability

Inthisse tionweanalyzetheasymptoti behaviourofgoodliftingsofve torelds

u2M

div

(). InProposition6.1and Theorem6.2we showthat generi allya

blow-up produ es alifting

1

with spe ialfeatures, i.e. either approximately ontinuous

orjumping onalineoronahaline. Moreover, thereisari hfamily oftrun ations

whi hturns 

1

into aBV

lo

ve tor eld.

Then, inTheorem 6.3we prove re tiability of the 1-dimensionalpart of 

 by

showing thatthe normaltothe jumpis independent ofthe sequen e of radii hosen

for theblow-up,and alowerbound onthe widthof thejumpof

1

. The rst

infor-mation omes hoosing a Lebesgue point for the density fun tion

~ H hara terized by Z R e ia divT a uda= ~ H  :

These ondinformation omes hoosingLebesguepointforthe densityfun tions

~ H k hara terized by Z R e ika divT a uda= ~ H k   ; k 2(1;2)\Q:

This aspe t of the proof isquite deli ate,sin e a priori the jump an bearbitrarily

small and no universal onstant in the lower bound an be expe ted, unlike in the

theory of minimal surfa es. A linearization around k = 1 shows that small jumps

are uniquely determinedby allve tors

~ H k . Proposition 6.1. Let u2 M div () and let  2L 1 () be a lifting satisfying (P2) in Denition3.1. For   -almost every x 0

2, from any sequen e r

n

!0

+

i r i 0 i  1 in L 1 lo (R 2 ). Moreover setting u 1 :=e i 1

, the following properties hold:

(i) There exist a nonnegative Radon measure  on R

2

and a Lips hitz map h :

R!R su h that divT a u 1 =h(a) 8a2R:

(ii) Thereexistsa niteor ountablefamilyof opensegments(possiblyunbounded)

I l =(b l ; l ) su h that (a) Rn[ l I l

has an empty interior;

(b) for all l, divT

l b l u 1 =0;

( ) for all l, either divT

a b

l u

1

is a nonnegative measure for all a 2 I

l or divT a b l u 1

is a non-positive measure for all a2I

l .

Proof. ByTheorem3.4weknowthat

isabsolutely ontinuouswithrespe ttoH

1 ,

hen e(see Se tion5)theupperdensity

 (  ;x) isnitefor  -a.e. x. Hen eforth, we hoose x 0

with this property. Sin e 

r (B R ) =   (B Rr (x 0 ))=r is equibounded

with respe t to r for any xed R , the ompa tness Theorem 3.6 and a diagonal

argument ensure the rst part of the statement. We an also assume that the

res aled measures (

 )

x0;ri

as inDenition5.1 weakly onverge, in the duality with

C

(R 2

), tosome Radon measure .

In order toobtain the property stated in(i) weimpose additional(but generi )

onditions on x

0

. By Theorem 3.2(ii) we have that, for all g 2 C

(R ), the Radon measure R R g(a)divT a

uda is absolutely ontinuous with respe t to

. Let D be a

ountable set dense in C

(R) and set  g := Z R g(a)divT a uda 8g 2D:

Then, by the Radon-Nikodým theorem there exist fun tions h

g 2 L 1 (;  ) su h that  g =h g  

. ByProposition 3.2(ii)again we obtain

Z (h g h g 0 ) dx= Z R (g(a) g 0 (a))hdivT a u; idasupjg g 0 j Z j jd  for any 2 C 1 () and any g; g 0 2 C (R), hen e kh g h g 0 k 1  supjg g 0 j (the L 1

normis omputed using 

asreferen e measure).

Letus onsider theBorel set

0

=n[

g2D S

hg

of approximate ontinuitypoints

of all maps h g , for g 2 D. Let B 1 ( 0

on , endowed with the sup norm. By the previous estimate, the map R whi h

asso iates tog 2D the fun tion

R g (x):=ap lim y!x h g (y); x2 0

is 1-Lips hitz between D and B

1 (

0

). By a density argument R extends to a

1-Lips hitzmapdenedonthewholeofC

(R)andea hpointxof

0

isanapproximate

ontinuity pointof allfun tions h

g

, g 2C

(R),with approximate limitR

g (x). We x x 0 2 0 . Res aling  g as inDenition 5.1we obtain ( g ) x 0 ;r =h g (x 0 +r
)(  ) x 0 ;r

and the approximate ontinuity of h

g

at x

0

, together with the fa t that the upper

density is nite, ensures that (

g )

x0;ri

weakly onverge, inthe duality with C

(R 2 ), to R g (x 0

). On the other hand, the identity

( g ) x 0 ;r i = Z R g(a)divT a u r i da

and the onvergen e inthe sense of distributions of divT

a u r i to divT a u 1 give Z R g(a)hdivT a u 1 ;ida=R g (x 0 ) Z R 2 d 8g 2C (R 2 ); 2C 1 (R 2 ): Nowwex 0 2C 1 (R 2 )su hthat R R 2  0

d =1(assumingwithnolossofgenerality

that (R

2

)>0)and noti e that onsequently

jR g (x 0 )jkr 0 k 1 Z R jg(a)jda: Ifparti ular,if g k

weakly onverge totheDira mass ata, thenR

g k

(x 0

)isbounded,

and any limitpointh satises

hdivT a u 1 ;i=h Z R 2 d 8 2C 1 (R 2 ):

This impliesthat hdoesnot depend ontheapproximatingsequen e, but onlyona.

The Lips hitz property of h follows dire tly by Proposition 3.2(i), using 

0

as test

fun tion.

Letusnowprovethat(i)implies(ii),assumingwithnolossofgeneralitythatis

anonzeromeasure. Thenitsu estotakeasintervalsthe onne ted omponentsof

fh6=0g and the onne ted omponents of the interior of fh=0g. By onstru tion

Theorem 6.2. Let u 2 M div

() and let  2 L () be a lifting satisfying (P2) in

Denition 3.1. For 

-almost every x

0

2 , from any sequen e r

n ! 0 + one an extra t a subsequen e r i

su h that the fun tions 

ri (x) := (x 0 +r i x) onverge to in L 1 lo (R 2 ) to  1

. Moreoverthe jump set J

 1 of  1 oin ides, up to H 1 -negligible

sets, either with the empty set, or with a line or with a haline K, not ne essarily

passing through the origin.

If K isa line and! K 2S 1 , A2R 2

aresu h thatK =A+R!

? K

(see Figure 1),

then 

1

is onstant in the halfspa es

 denedby + :=  y2R 2 : (y A)
! K >0 ; :=  y2R 2 : (y A)
! K <0; : (6.1) If K is a haline and ! K 2S 1 , A 2 R 2

are su h that K = fA+t!

? K

: t >0g

(see Figure 1), thenthe approximatelimits 

+ 1 and  1 are onstantH 1 a.e. onK. Moreover,  1 is equal to   1 a.e. in  A , where  A :=  \  y2R 2 : y A jy Aj
! ? K  u  l
! K ) :

Proof. KeepingthenotationofProposition6.1,inthe followingwedenotebyL

0 the

setofalllsu hthatI

l

isnota onne ted omponentoftheinterioroffh=0g. Then,

if l 2= L 0 , divT a u 1 =0for anya2I l . Ifl 2L 0 ,eitherdivT a u 1 isnonnegativeand

nonzero forany a2I

l

ordivT

a u

1

is nonpositive and nonzero forany a 2I

l . Let us set u l := e i( 1 _b l )^ l . Then u l

is divergen e free, be ause divT

b l u 1 = divT l u 1 =0and e i(1_b l )^ l +e i1^b l =e ib l +e i1^ l : Sin e e i(1_b l )^a +e i1^b l =e ib l +e i1^a ; (6.2)

we obtain that divT

a u l =divT a u 1 for a2I l , therefore divT a u l =  h(a) if a 2I l 0 else. (6.3) In parti ular u l 2M div

(). Moreover, either divT

a u

l

is nonnegativefor any a2R

or divT

a u

l

is non-positive for any a 2 R. If we are in the rst situation, by

Theorem I.1 of [ALR℄, any fun tion g

l 2 W 1;1 (R 2 ) su h that u l = r ? g l is a

vis osity solution of the eikonal equation jrgj

2 1 = 0 on R 2 . Therefore, g l is on ave and u l 2 BV lo (R 2

) (see [AD ℄). If we are in the se ond situation, for any

fun tion g l 2W 1;1 (R 2 ) su h that u l =r ? g l

we have the same statement. In both

ases, by applyingProposition 2.3, we obtainthat 

l 2BV lo (R 2 ).

1 

l

. If l 2= L

0

, by the previous dis ussion we obtain that any fun tion g

l satisfying u l = r ? g l

is ane (being on ave and onvex) and therefore u

l

is onstant. As

U 

l

=0, fromProposition 2.3we obtain that 

l

is onstant as well.

In the following we onsider l 2 L

0

and, to x the ideas (sin e the argument

is similar for both ases), we assume that divT

a u

l

is a nonzero and nonnegative

measure forany a2I

l .

We denote by J

l

the set of approximate jump points of 

l , by ! l a unit normal of J l and by  + l ;  l

the orresponding approximate limits of 

l on ea h side of J l . Sin e divu l = 0, then ! l
e i + l = ! l
e i l on J l . Thus, ! l = e i 2 ( + l + l ) and we hoose ! l =e i 2 ( + l + l )

. Then, the expli it formula(3.4) given in Se tion3 gives

divT a u l =(a; + l ; l )(e ia e i l )
! l H 1 J l ; where (a; + l ; l ):= 8 < : 1 if  l <a< + l 1 if  + l <a< l 0 else. But, divT a u l

is nonnegativefor all a2R. Then, j

+ l

 l

j<2, sin e, otherwise,

therewouldexista2Rsu hthat(a;

+ l ; l )(e ia e i l )
! l <0. Inparti ularJ l is

alsothe set of approximate jumppointsof u

l . If l > + l ,then (e ia e i l )
! l 0 for any a2[ + l ; l

℄. Therefore, we must have 

+ l > l and j + l  l j<2 H 1 -a.e. on J l and divT a u l = ( l ; + l ) (a)(e ia e i l )
! l H 1 J l : (6.4) Claim 1.  + l = l and  l =b l H 1 -a.e. onJ l .

First of all, we noti e that

l   + l >  l  b l H 1 -a.e. on J l . Assuming by ontradi tion that f + l < l g has positive H 1

-measure, we an nd  > 0 su h

that f + l < l g\ f +   l > g has positive H 1

-measure, and then an interval

(; 0 )(b l ; l

) withlength less than=2 su h that

E :=f + l 2(; 0 )g\f + l  l >g has positive H 1

-measure. From (6.4) we inferthat divT

a u l E =0for a 2( 0 ; l ), while divT a u l

(E) >0 for a2 ( =2;). Sin e h >0 on (b

l

;

l

), this ontradi ts

(6.3). The argument for 

l

is similar.

Claim 2. Forany hoi e of l; m2L

0 we have H 1 (J l nJ m )=0.

Suppose that there exist l; m 2 L

0 and A  J l nJ m su h that H 1 (A) > 0. Sin e

A\J m

=;,(6.4) yieldsdivT u

m

A=0foranya2Rand (6.3)yieldsh(a)(A)=

0, so that (A)= 0. On the other hand, the fun tion 

( l ; + l ) (a)(e ia e i l )
! l is onstant H 1 -a.e. on J l

by Claim1. Moreover, this onstant is not 0 for any a 2I

l .

Sin e (A)=0,then jdivT

a u

l

j(A)=0 and thereforeH

1 (J l \A)=0. Sin e AJ l , then H 1

(A)=0 whi h ontradi tsthe hypothesis and provesthe laim.

Claim 3. Forany l 2L

0

, J

l

is ontained in one line.

Letusre allthatthenormalunitve tor!

l toJ l isgivenbye i 2 ( + l + l ) andis onstant H 1 -a.e. onJ l

. Letusassumethat thereexistx

1 ; x 2 2J l su h that(x 2 x 1 )
! l 6=0

and assume (up toa permutationof x

1

and x

2

) that the s alar produ t is positive.

We set ! := x 2 x 1 jx2 x1j , sothat !
! l

>0. Sin e the restri tion of g

l

to the lineR! is

on ave, we must have

r + ! g l (x 1 )r ! g l (x 2 ): By Proposition4.1 weget r + ! g l (x 1 )=!
rg + l (x 1 )=!
(e i + l ) ? and r ! g l (x 2 )=!
rg l (x 2 )=!
(e i l ) ? ; so that !
(e i + l ) ? !
(e i + l ) ?

. On the other hand, sin e !
!

l >0 and  + l > l , then !
(e i + l ) ? <!
(e i l ) ?

, a ontradi tion (this inequality an beeasily he ked

in a frame where  + l + l = 0, so that ! 1 > 0). Therefore J l must be ontained

in one line. By Claim 2, all sets J

l

with stri tly positive H

1

-measure (i.e. those

orresponding to l 2L

0

) are ontained in the same line. Letus denote this lineby

R .

Claim 4. There exists a losedset K

l R su hthat H 1 (K l
J l )=0.

Let us re all that J

l

oin ides with the set J

rg l

of approximate jump points of

rg l = (e i l ) ? , where g l

is on ave and satises jrg

l j = 1. Sin e  + l = l and  l = b l H 1 -a.e. on J l , taking  = je i l e ib l

j, it is lear that the losure K

l of J  := fx 2 J g l : jrg + l (x) rg l (x)j   g ontains H 1 -almost all of J l . By Proposition 4.1, J     , where   := fx 2 R 2 : diam(g l (x))   g is a losed set. Therefore K l    . But,    S rg l , where S rg l

is the set of points where

rg l

doesn't have anapproximate limit. Indeed, by Proposition4.1 ,atany pointx

where rg

l

has anapproximate limitthefun tion g

l isdierentiable,hen e g l (x)is a singleton. By (2.2)weinfer H 1 (K l nJ l )H 1 (  nJ l )H 1 (S g l nJ g l )=0: For any l 2 L 0 , for H 1 -almost every x 2 K l , rg l

has an approximate limit at

H 1

-almosteveryy 2D

x

andtheapproximatelimitata.e. pointinthestrip

S

x2K l

D x

+

Γ

Γ

−

+

Κ

A

K

Γ

_Α

−

Γ

_Α

when K isa lineor ahaline

is equal torg l (x)=(e ib l ) ? , sin e  l is onstant equal tob l H 1 -a.e. onK. In the

same way, one an showthat rg

l

has anapproximate limitat a.e. pointin

S x2K l D + x equal to (e i l ) ? . If K l

is the whole line, then

S x2K l D  x =  , where 

are the sets

dened in(6.1). Therefore u l is onstant a.e. in  and equal toe i  l . By

Proposi-tion 2.3 weobtain that 

l

is onstant inthe two halfspa es aswell.

Now, letusassumethatK

l

isnotthe wholelineandletusshowthatK

l

mustbe

ahaline. AssumethatK

l

isnot onne ted. Thereexists abounded openintervalS

ontainedinRnK l ,whoseendpointss 1 ; s 2 belongtoK l . WewilldenotebyK 1 ; K 2 the omponentsofK l ontainings 1 ands 2 respe tively. SetR  i := S x2K i D  x ,i=1;2. TheregionR 2 n(R 1 [R + 1 [R 2 [R + 2

) anbedividedintothreepartsA

+

;A ; C (see

Figure 2). If y2A

+

is a point of approximate ontinuity of rg

l , then rg l (y)must be equalto(e i l ) ?

, otherwise the halineD

+ y would rossR 1 orR 2

and this would

ontradi t the result of Proposition 4.1 (ii). If y 2 A is a point of approximate

ontinuity of rg

l

, by the same argument, rg

l (y) = (e ib l ) ? . If y 2 C is a point of approximate ontinuity of rg l , then rg l

(y) an only be equal to (e

i l ) ? or (e ib l ) ?

(see Figure 2). Then, C ontains a set of approximate jumppointsof rg

l . But,by hypothesis, K l \C =;,hen e H 1 (J l \C)=0. Therefore, K l must be onne ted. If K l

is not the whole line, then K

l

has one or two endpoints. Let A be one

endpointofK

l

andlet!

K

betheunitnormaltoK

l su hthatK l fA + t! ? K : t 0g.

K1

K

2

R

2

-R

+

2

+

C

A

-

A

+

R

1

-R

1

Figure 2: K must be onne ted

Let usdene the one C by

C :=  y2R 2 nfAg: y A jy Aj
! ? K  e i l
! K : LetC 0

beany open set ontainingC su hthat C

0 \K =fAg and C 0 \K =;. Then, divT a u l =0inD 0 (C 0

)forany a2R. Usingtheresultof [LR℄,

l

islo allyLips hitz

inC

0

. Therefore,fora.e. xinC

0

,r

l

(x)existsanddive

i l (x)=(e i l (x) ) ?
r l (x)= 0. Then, r l (x) is parallel to e i l (x) for a.e. x 2 C 0

. Therefore for any a 2 R the

tangent to the level set f

l = ag at x is orthogonal to e i l (x) whi h is equal to e ia onf l

=ag. Hen e,the level sets f

l

=ag are straightlines oriented by (e

ia

) ?

and

the only possible ongurationin C is the one des ribed inFigure 1.

Finally, we an ex lude the ase of K

l

is asegmentor a singlepoint(Figure 3).

Indeed, hoose R>0su h thatK

l B R . Sin e H 1 (J l nB R

)=0,the sli ing theory

of BV fun tions (see [AFP ℄, Theorem 3.108)shows that fora.e. r 2(R ;R+1) the

restri tionof

l

toB

r

is(equivalentto)a ontinuousBV fun tion. Therefore u

l has

a ontinuous liftingin B

r

and its topologi al degree is 0. This is in ontradi tion

with the fa tthat there are vorti eswhi h have the same orientation 

l atthe two endpointsof K l . Therefore, K l isa haline.

K

Figure3: K an't be a segment

By Claims 2 and 4 we obtain that all lines (or halines) K

l , l 2 L 0 , oin ide. Hen eforth we set K =K l

. By Lemma 2.1and Remark 2.2we obtain that 

1 has an approximate limit H 1 -a.e. inR 2 nK and H 1

-a.e. point of K is ajump point of

 1

. Moreover, asall limits

 l

are onstanton J

 1

, the same is true for 

 1

.

Theorem 6.3 (Main re tiability theorem). Let u 2 M

div

() and let  2

L 1

() be a lifting satisfying (P2) in Denition3.1. Then the set

:=fx2:   (  ;x)>0g (6.5) is ountablyH 1

-re tiableand oin ides,uptoH

1

-negligiblesets,withJ

 . Moreover, for H 1 -a.e. x2nJ  we have lim r!0 + 1 r 2 min 2R Z Br(x) j(y) jdy=0: (6.6)

Proof. Step 1. We show that 

0

:= f

 (

;
)>0g is ountably re tiable, using

Theorem5.3. Tothisaimweshowthatfor-a.e. xany=lim

i (  ) x;r i 2Tan (  ;x)

is supported on aline whosedire tion depends on x only.

We proved in Theorem 6.2 that (possibly passing to a subsequen e) we an

assume that  r i =(x+r i y)! 1 in L 1 lo (R 2

K, the empty set, aline ora haline,su h that H (K
J 

1

)=0. Denotingby !

K

the orientation of K su hthat e

i( + 1 + 1 )=2 =! K

,now we showthat

divT a u 1 =(T a u + 1 T a u 1 )
! K H 1 K 8a 2R (6.7)

using Lemma 3.5. To this aim, we need only to he k that T

a u

1

is

divergen e-free in nK. If a belongs to some interval (b

l

;

l

), this follows by the identity

divT a u 1 = divT a u l

(see (6.2) and by (3.4), be ause J

 l

 K up to H

1

-negligible

sets. In the general ase one an argue by approximation, using the fa t that the

omplementof [ l (b l ; l

) has anempty interior.

One an show, by a dire t omputation based on (6.7), that the ve tor-valued

measure R R e ia divT a u 1 da is oriented by ! K , and pre isely Z R e ia divT a u 1 da= 1 2  + 1  1 sin( + 1  1 )
! K H 1 K (6.8)

(this omputationiseasilydone inaframewhere !

K =(1;0),sothat + 1 =  1 +

4k for some k 2 Z and the periodi ity and the odness of the integrandshow that

the integral of the se ond omponent is 0). Moreover, the ve tor-valued measure

 1 := R R e ia divT a

uda satises,by Theorem 3.2(ii),the inequality j

1

j

. Thus,

thereexistsave tor-valuedfun tion

~ H 2L 1 (;  )su hthat 1 = ~ H  andj ~ Hj1.

In addition to the previous generi onditions imposed on x

0

, assume also that x

0

is aLebesgue pointof

~

H, relativeto the measure 

 . Then ( 1 ) x0;r i ! ~ H(x 0 ) inM 0 (R 2 ):

On the other hand,the onvergen e of 

r i to  1 implies ( 1 ) x 0 ;r i = Z R e ia divT a u r i da! Z R e ia divT a u 1 da in D 0 (R 2 ): Therefore Z R e ia divT a u 1 da= ~ H(x 0 ):

Comparing this expression with (6.8) we obtain

1 2  + 1  1 sin( + 1  1 )
! K H 1 K = ~ H(x 0 ): (6.9) Therefore ! K

doesnot depend onthe sequen e hosen, but only onx

0 .

Step 2. We show that 

(n

0

) = 0 using Theorem 5.2. Sin e H

1 (S\S 0 ) = 0 wheneverS 6=S 0

are ir les,thefamilyofall ir lesS su hthat

0

of this set, so that the density fun tion f(r) :=

 (B r (x 0 )) is ontinuous. In order

to he k ondition (iii) of Theorem 5.2, for k 2 Q\(1;2) we dene the measures

 k := R R e ika divT a

uda, allabsolutely ontinuous with respe t to

 , we denoteby ~ H k 2 L 1 (; 

) their densities with respe t to 

and we hoose a Lebesgue point

x 0

for allfun tions

~ H k (relativeto   ).

Assuming that  is not identi ally 0, we have to show that  = H

1 K with  (x 0 )>0. By(6.9)andsin e  1

are onstantonK,weknowthat = H

1 K,

where is onstant onK. Moreover,

j ~ H(x 0 )j= 1 2   ( + 1  1 ) sin( + 1  1 )   : (6.10) Therefore, ifj + 1  1 j=2,wehave (=2 1)=2be ausej ~ H(x 0 )j1. Setting d :=j + 1  1

j=2>0, in the followingwe show that d (and therefore , by (6.10))

is uniquely determined by ~ H k (x 0

) whenever d =4. We an assume with no loss

of generality (possibly making a rotation and adding to 

1 an integer multiple of 2)that ! K =(1;0), + 1 =d and  1

=d. Then, arguingas inStep 1 weget

Z R e ika divT a u 1 da= ~ H k (x 0 ) 8k2 (1;2)\Q:

Ontheotherhand, omputingtheleftsidewendthatitsrealpartequals

2 k(k 2 1) F d (k)H 1 K, where F d

(k):=(sinkd osd k oskdsind):

Then F d (k)6=0if and onlyif ~ H k (x 0 )
! K 6=0 and = 2 k(k 2 1) F d (k) ~ H k (x 0 )
! K : (6.11)

It turns out that the ratios

 k;m (d):= F d (k) F d (m) = k(k 2 1) m(m 2 1) ~ H k (x 0 )
! K ~ H m (x 0 )
! K (6.12)

(when dened) depend onx

0

, k and m but not on d, so that the fun tions F

d and F

d 0

areproportionalwheneverd; d

0

satisfy(6.12). ATaylorexpansionatk =1gives

F t (k)=(k 1)(t sint ost)+(k 1) 2 tsin 2 t: Therefore F t

(k)6= 0 for k 1 su iently small and the onstant ratio between F

d and F d 0 must beequal to d sind osd d 0 sind 0 osd 0 and dsin 2 d d 0 sin 2 d 0 :

Therefore g(d)=g(d), where g(t):= t sint ost tsin 2 t :

A dire t omputation shows that g is stri tly de reasing in (0;=4). Therefore

d=d

0 .

Step 3. Now weshow the lastpart of the statement. Sin e we knowthat 

is

a re tiable measure, by Theorem 2.83 of [AFP℄ we know that Tan(

;x), is a

singleton for H

1

-a.e. x 2 , therefore Tan(

;x) is a singleton for H

1

-a.e. x 2 .

Coming ba k to (6.9) we obtain that the jump 

+ 1  1 is uniquely determined H 1

-a.e.,andthesameistruefor

+ 1 + 1 modulo2. Hen e, + 1 isonlydetermined modulo 2, H 1 -a.e. on  and  1 is given by  1 =  + 1 ( + 1  1 ) when  + 1 is known.

Let usdene the followingmeasures, allabsolutely ontinuous with respe t to 

 :  k := Z 2(k+1) 2k div T a uda; 8k 2Z: Let usdenoteby t k 2L 1 (; 

)their densitieswith respe t to

and letus hoose

a Lebesgue point x 0 of allfun tions of t k . As in Step 1,we have Z 2(k+1) 2k divT a u 1 da=t k (x 0 ) 8k 2Z: By (6.7), divT a u 1 = 0 as soon as a 2= [ 1 ; + 1 ℄. Let us dene X 0 := fk 2 Z : t k (x 0 )=0g. Then,k 2X 0 if andonlyif(2k;2(k+1))\[ 1 ; + 1 ℄=;. Letk 0 2Z be su h that  + 1 2 [2k 0 ;2(k 0 +1)). Then,  1 2 [2(k 0 l 0 );2(k 0 l 0 +1)), wherel 0 2Ndepends onlyon + 1  1 and k 0 depends onX 0

inthe followingway:

ZnX 0 =fk 0 j : 0j l 0 g. Sin e X 0 onlydepends onx 0 , thenk 0 onlydepends on x 0 and  + 1

is uniquely determined. Thus 

+ 1

and 

1

are uniquely determined

-a.e. on. Hen eforth H

1

-a.e. x

0

2is ajump point of .

Finally,(6.6) and thein lusionJ

followbythe fa tthat any blow-uplimit

 1

at points x 2=  is onstant. Indeed, e

i 1 = r ? g 1 is onstant, (being g 1

on ave and ane, see [ALR℄) and U

 1 = 0, so that  1 is onstant by Proposi-tion 2.3.

In on lusion, the statements made in Theorem 1.1 of the introdu tion follow

by Theorem 6.3 with the only ex eption of (1.6). The latter follows by applying

Lemma 3.5to the ve toreld T

a

u, with K =J

 .

Theorem 6.4. Let u;  as in Theorem 6.3 and assume that

H 1

dimensional re tiable measure.

Proof. Let g be a 1-Lips hitz fun tion su h that u = r

?

g and re all that 

oin ides, up toH

1

-negligiblesets, with J

. The blow-up argumentin[ALR℄ shows

that g is a vis osity solution of the eikonal equation jrgj

2

1 = 0 in set n ,

sin e U

1

= 0 for any blow-up fun tion 

1

at any point x 2 n. Therefore, g

is lo ally semi on ave in the open set A := n and its gradient (and u as well)

is aBV

lo

fun tion inA. ByProposition 2.3we obtainthat 2BV

lo (A) and (3.4) gives   A = 0 be ause A\J  is H 1 -negligible. Therefore   is supported on 

and the absolute ontinuity of

withrespe t toH

1

leads ustothe on lusion.

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