• Aucun résultat trouvé

Lasry-Lions Regularization and a Lemma of Ilmanen

N/A
N/A
Protected

Academic year: 2021

Partager "Lasry-Lions Regularization and a Lemma of Ilmanen"

Copied!
10
0
0

Texte intégral

(1)

HAL Id: hal-00448255

https://hal.archives-ouvertes.fr/hal-00448255v2

Submitted on 4 Jan 2016

HAL

is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire

HAL, est

destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Lasry-Lions Regularization and a Lemma of Ilmanen

Patrick Bernard

To cite this version:

Patrick Bernard. Lasry-Lions Regularization and a Lemma of Ilmanen. Rendiconti del Seminario

Matematico della Università di Padova, University of Padua / European Mathematical Society, 2010,

124, pp.221-229. �10.4171/RSMUP/124-15�. �hal-00448255v2�

(2)

!"#$%&!'()# *+,-."$'/"0'() ")1 " !+22" (3 4.2")+)

-()!+89:"!#(!$5;7

<=>!?= @ AB1?=C> DE@F=% G= H=IBJ= >K= I0110LBJM BJI 5DNE7 F0JO01N>B0J 0E=C@>0CD @F>BJM 0J ?0NJH=H INJF>B0JD"!!!"!P

#$"5%7!#BJI

&

!"5&7$2

$%&!%%3"

@JH

"

#$"5%7!#DNE

&

!"5&7!2

$%&!%%3"

! G= K@O= >K= C=1@>B0J

#$5!"7# !#"$5"7!

!=F@11 >K@> >K=D= 0E=C@>0CD I0CQ D=QBRMC0NED/ BJ >K= D=JD= >K@>

#$&#' ##$$' @JH #"$&#"' ##"$$'

I0C @11$!6 @JH'!6/ @D F@J ?= FK=FS=H ?T HBC=F> F@1FN1@>B0J% #0>= @1D0 >K@>

BJI""#$"5%7""5%7"#"$"5%7"DNE"

I0C =@FK $!6 @JH =@FK %'!% ( INJF>B0J "!!!"! BD F@11=H (RD=QBR F0JF@O=/ ("6/ BI >K= INJF>B0J %!""5%7! %%%3#( BD F0JF@O=% G= LB11 0FF@DB0J@11T F0JDBH=C D=QBRF0JF@O= INJF>B0JD LKBFK >@S= O@1N=D BJ U! ($$ (7% )K= INJF>B0J"BD F@11=H(RD=QBRF0JO=V BI!"BD(RD=QBRF0JF@O=%

( INJF>B0J"BD$RD=QBRF0JF@O= @JH NEE=C D=QBRF0J>BJ0ND BI @JH 0J1T BI B>

?=10JMD >0 >K= BQ@M= 0I >K= 0E=C@>0C#$/ >KBD I0110LD IC0Q <=QQ@ 2 @JH

<=QQ@ W ?=10L% ( INJF>B0J BD F@11=H D=QBRF0JF@O= BI B> BD(RD=QBRF0JF@O= I0C D0Q=("6% ( INJF>B0J"BD D@BH)2$2BI B> BD XC=FK=> HBII=C=J>B@?1= @JH BI >K=

MC@HB=J> 0I"BD <BEDFKB>Y% #0>= >K@> @ F0J>BJN0ND INJF>B0J"!!!"!BD)2$2

5;7 +JHBCBYY0 H=11Z(%P 8"!"'($"/ *JBO=CDB>=[ -@CBDR$@NEKBJ=/ -1@F= HN '@C=[FK@1= H= <@>>C= H= )@DDBMJT/ \]\\]/ -@CBD 8=H=V 2^/ XC@JF=%

"RQ@B1P E@>CBFS%?=CJ@CH_F=C=Q@H=%H@NEKBJ=%IC

(3)

BI @JH 0J1T BI B> BD D=QBRF0JF@O= @JH D=QBRF0JO=V/ D== <=QQ@ ]% <=> ND C=F@11

>L0 BQE0C>@J> C=DN1>D BJ >K@> 1@JMN@M=P

)A".!"' 2 5<@DCTR<B0JD/ U^`7% *+$ " ,+ - ,."/0+0 1"/2$3./4 5.6 6!'!$7 $8+ 1"/2$3./#"'&#$" 3' )2$2-/07 31 " 3' "/31.69:& 2./$3/"."'7 $8+/

3$ 2./;+6<+' "/31.69:& $. " =8+/ $!"64

)A".!"'3 5+1Q@J=J/ U]`7% *+$ "!; ,+ $=. ,."/0+0 1"/2$3./' ./ ! '"28 $8-$ " -/0!; -6+ '+93>2./2-;+4 #8+/ $8+6+ +%3'$' - )2$21"/2$3./ = '"28 $8-$ "!=!;4

.NC M0@1 BJ >K= EC=D=J> E@E=C BD >0 aaM=J=C@1BY=ZZ DBQN1>@J=0ND1T ?0>K 0I

>K=D= C=DN1>D @D I0110LDP

)A".!"' W% #8+ .?+6-$.6 @$!##"$&#3$&#"$ 8-' $8+ 1.::.=3/< ?6.?>

+6$3+'A

) @+<":-63B-$3./ A 5.6 +-28 1"/2$3./ 1 !!!"!-/0 +-28 $"67 $8+

1"/2$3./ @$5173' )2$2?6.;30+0 3$ 3' :.2-::& ,."/0+04 #83' 8.:0' 1.6 -:: $"6 31 1 3' ,."/0+04

) C??6.%39-$3./ A D1 1 !!!"!3' "/31.69:& 2./$3/"."'7 $8+/ @$517 3' )2$2-/0 2./;+6<+' "/31.69:& $. 1 -' $!"64

) E3/283/<A D1 $8+6+ +%3'$' - (>'+93>2./2-;+ 2./$3/"."' 1"/2$3./ "

-/0 - (>'+93>2./;+% 2./$3/"."' 1"/2$3./ ; '"28 $8-$ ;"1""7 $8+/7 1.6 -:: $'`6$(`7 =+ 8-;+ "!@$517!;7 -/0 @$5173' )2$24

)K=0C=Q W H0=D J0>/ EC0E=C1T DE=@SBJM/ M=J=C@1BY= )K=0C=Q ]% A0LR

=O=C/ B> 0II=CD @ J=L 5@1>K0NMK DBQB1@C7 @JDL=C >0 >K= D@Q= EC0?1=QP @ER EC0VBQ@>BJM NJBI0CQ1T F0J>BJN0ND INJF>B0JD 0J AB1?=C> DE@F=D ?T )2$2 INJF>B0JD LB>K @ DBQE1= =VE1BFB> I0CQN1@%

:=F@ND= 0I B>D DTQQ=>CBF I0CQ/ >K= C=MN1@CBYBJM 0E=C@>0C @$ =Jb0TD D0Q= JBF=C EC0E=C>B=D >K@J >K= <@DCTR<B0JD 0E=C@>0CD% X0C =V@QE1=/ BI1 BD )2$2/ >K=J B> I0110LD IC0Q >K= EBJFKBJM EC0E=C>T >K@>@$1 #1 I0C $DQ@11

=J0NMK%

)K=0C=Q 3/ F@J ?= EC0O=H NDBJM )K=0C=Q W ?T >@SBJM =#@$"/ I0C$ DQ@11 =J0NMK% #0>=/ BJ OB=L 0I <=QQ@ W ?=110L/ >K@>@$"##"$&#$"LK=J$ BD DQ@11 =J0NMK%

)K=0C=Q W F@J ?= D0Q=K0L =V>=JH=H >0 >K= F@D= 0I IBJB>= HBQ=JDB0J@1 0E=J D=>D 0C Q@JBI01HD OB@ E@C>B>B0J 0I NJB>T/ @> >K= ECBF= 0I 100DBJM >K=

DBQE1BFB>T 0I =VE1BFB> =VEC=DDB0JD% <=> F ?= @ E@C@F0QE@F> Q@JBI01H 0I HBQ=JDB0J// =cNBEE=H 0JF= @JH I0C @11 LB>K @J @>1@D 5!3$3'!7 F0QE0D=H 0I

333 -@>CBFS :=CJ@CH

(4)

FK@C>D!3!G/!"F/ LK=C=G/BD >K= 0E=J NJB> ?@11 0I C@HBND 0J= F=J>=C=H

@> >K= 0CBMBJ BJ!/% G= @DDNQ= BJ @HHB>B0J >K@> >K= BQ@M=!35G/7 BD @ C=R 1@>BO=1T F0QE@F> 0E=J D=>/ @JH >K@> >K= D=>D !35G/7$3'!I0CQ @ 10F@11T IBJB>= 0E=J F0O=CBJM 0I >K= Q@JBI01H F% <=> ND IBV/ 0JF= @JH I0C @11/ @ E@C>B>B0J 0I >K= NJB>T<3DN?0CHBJ@>=H >0 >K= 0E=J F0O=CBJM 5!35G/7$3'!7%

+> Q=@JD >K@> >K= INJF>B0J <3 BD J0JRJ=M@>BO= LB>K DNEE0C> BJDBH=!35G/7/

@JH >K@>#

3

<3#2 5J0>= >K@> >KBD DNQ BD IBJB>= @> =@FK E0BJ>7% <=> ND H=IBJ=

>K= 0E=C@>0C

H$5"7!#$

3

%@$-3&

5<3"7&!3'(

&!!23 $

LK=C=-3$3'!@C= E0DB>BO= C=@1 JNQ?=CD% +J >KBD =VEC=DDB0J/ L= F0JDBH=C

=@FK 0I >K= >=CQD%

@$-3&

5<3"7&!3'(

&!!23 @D H=IBJ=H 0J >K= LK01= Q@JBI01H FLB>K >K= O@1N= 6 0N>DBH= 0I!35G/7% )K= DNQ BD >K=J 10F@11T IBJB>= K=JF=

L=11RH=IBJ=H% G= D@T >K@> @ INJF>B0J"!F!"!BD 10F@11T D=QBRF0JF@O=

BI/ I0C =@FK 3'!/ >K=C= =VBD>D @ F0JD>@J> ,3 DNFK >K@> >K= INJF>B0J

"&!3! %!%3#,3 BD F0JF@O= 0JG/%

)A".!"'4% *+$ "!; ,+ $=. 2./$3/"."' 1"/2$3./' ./ F '"28 $8-$ "

-/0!; -6+ :.2-::& '+93>2./2-;+4 #8+/7 $8+ 6+-: /"9,+6' -32-/ ,+ 28.'+/

'"28 $8-$7 1.6 +-28 $'`6$2`-/0 +-28 1"/2$3./ 1 '-$3'1&3/< "!1!;7 =+

8-;+A

) #8+ 1"/2$3./ H$5173' :.2-::& )2$24

) D1 1 3' 2./$3/"."'7 $8+/ H$517 2./;+6<+' :.2-::& "/31.69:& $. 1 -'

$!"64 ) "!H$517!;4

G= LB11 MBO= D0Q= EC0E=C>B=D/ Q0D> 0I LKBFK @C= L=11RSJ0LJ/ 0I >K=

0E=C@>0CD#$@JH#"$BJ &=F>B0J 2/ @JH H=CBO= >K= EC00I 0I >K= Q@BJ C=DN1>D BJ

&=F>B0J 3%

I.$+' -/0 C2(/.=:+0<+9+/$'4 )K=0C=Q 3 @EE=@CD BJ +1Q@J=JZD E@E=C U]` @D <=QQ@ 4d% &=O=C@1 EC00ID @C= DS=>FK >K=C= ?N> J0J= BD H=>@B1=H% )K=

EC00I L= H=>@B1 K=C= I0110LD 1BJ=D DBQB1@C >0 0J= 0I >K= DS=>FK=D 0I +1Q@J=J%

)KBD D>@>=Q=J> @1D0 K@D @ Q0C= M=0Q=>CBF F0NJ>=CE@C>/ <=QQ@ 4" BJ U]`% ( H=>@B1=H EC00I 0I >KBD M=0Q=>CBF O=CDB0J BD MBO=J BJ U3`/ (EE=JHBV% 'T @>R

>=J>B0J L@D @>>C@F>=H >0 >K=D= D>@>=Q=J>D @JH >K=BC C=1@>B0JD LB>K C=F=J>

EC0MC=DD=D 0J DN?RD01N>B0JD 0I >K= A@QB1>0JRe@F0?B =cN@>B0J 5D== U4/ 2/ \`7

?T -B=CC= 8@CHB@1@MN=>/ (1?=C> X@>KB @JH '@VBQ= f@OBH0OBcN=% (1?=C>

X@>KB @JH '@VBQ= f@OBH0OBcN= @1D0 C=F=J>1T LC0>= @ H=>@B1=H EC00I 0I

(5)

)K=0C=Q 2/ D== UW`% )KBD E@E=C @1D0 EC0O=D K0L >K= M=0Q=>CBF O=CDB0J I0110LD IC0Q )K=0C=Q 3% )K=C= @C= Q@JT DBQB1@CB>B=D ?=>L==J >K= >001D ND=H BJ >K= EC=D=J> E@E=C @JH >K0D= ND=H BJ U2`% '0C=0O=C/ '@VBQ= f@OBR H0OBcN= 0?D=CO=H BJ U\` >K@> >K= =VBD>=JF= 0I)2$2DN?D01N>B0JD 0I >K= A@R QB1>0JRe@F0?B =cN@>B0J BJ >K= HBDFC=>= F@D= F@J ?= H=HNF=H IC0Q )K=0C=Q 3% A0L=O=C/ BD D==QD >K@> >K= Q@BJ C=DN1> 0I U2` 5>K= =VBD>=JF= 0I )2$2 DN?D01N>B0JD BJ >K= F0J>BJN0ND F@D=7 F@JZ> ?= H=HNF=H =@DB1T IC0Q )K=0R C=Q 3% #=B>K=C F@J )K=0C=Q 3 ?= H=HNF=H IC0Q B>%

56 78+ (9+$"0($#!"")1!""() :'.;+$0 #9"<+#6

)K= EC00ID 0I >K= >K=0C=QD I0110L IC0Q D>@JH@CH EC0E=C>B=D 0I >K= 0ER

=C@>0CD#$@JH#"$>K@> L= J0L C=F@11 BJ H=>@B1D%

!"##$%& 5.6 +-28 1"/2$3./ "!!!"!7 $8+ 1"/2$3./ #$" J=8328 $-(+'

;-:"+' 3/U! ($$(7K7 3' $>'+93>2./2-;+ -/0 "??+6 '+93>2./$3/."'4 #8+

1"/2$3./ #"$" J=8328 $-(+' ;-:"+' 3/ 5! ($$(`K7 3' $>'+93>2./;+% -/0 :.=+6 '+93>2./$3/"."'4 F.6+.;+67 31 " 3' (>'+93>2./2-;+7 $8+/ 1.6 +-28 $!(

$8+ 1"/2$3./#"$" 3'5(!$7>'+93>2./2-;+4 L393:-6:&7 31 " 3' (>'+93>2./;+%7

$8+/ 1.6 +-28 $!( $8+ 1"/2$3./ #$" 3'5(!$7>'+93>2./;+%4

-!..X% G= DK@11 EC0O= >K= D>@>=Q=J>D F0JF=CJBJM#$% G= K@O=

#$"5%7! %%%3#$#BJI

&

&

"5&7$ %&!%%3#$! %%%3#$'

#

#BJI

&

&

"5&7$ %&%3#$!3%*&#$'

$

>KBD INJF>B0J BD F0JF@O= @JH NEE=C D=QBRF0J>BJN0ND @D @J BJIBQNQ 0I F0JR

>BJN0ND 1BJ=@C INJF>B0JD% .J >K= 0>K=C K@JH/ L= K@O=

#$"5%7$ %%%3#:#BJI

&

&

"5&7$ %&!%%3#$$ %%%3#:'

!

&=>>BJM 15%$&7!#"5&7$ %&!%%3#$$ %%%3#:/ >K= INJF>B0J BJI&15%$&7 BD @ F0JO=V INJF>B0J 0I%BI1 BD @ F0JO=V INJF>B0J 0I 5%$&7% )KBD BD >CN= BI"BD(R D=QBRF0JO=V/$!(/ @JH:#(!$?=F@ND= L= K@O= >K= =VEC=DDB0J

15%$&7#"5&7$ %&!%%3#$$ %%%3#:#5"5&7$ %&%3#(7$ )))))

: ($

*

&! ))))( :$

*

% ++

++ +

++ ++ +

3

!

#

334 -@>CBFS :=CJ@CH

(6)

dBO=J @ NJBI0CQ1T F0J>BJN0ND INJF>B0J "!!!"!/ L= H=IBJ= B>D Q0HN1ND 0I F0J>BJNB>T "567!U6$(7!"U6$(7 ?T >K= =VEC=DDB0J "567# DNE%$+"5%$6+7!"5%7$LK=C= >K= DNEC=QNQ BD >@S=J 0J @11%'!@JH @11+ BJ >K= NJB> ?@11 0I !% )K= INJF>B0J " BD J0JRH=FC=@DBJM/ B> D@>BDIB=D

"56$6+7""567$"56+7/ @JH B> F0JO=CM=D >0 Y=C0 BJ Y=C0 5>KBD 1@D> I@F> BD

=cNBO@1=J> >0 >K= NJBI0CQ F0J>BJNB>T 0I "7% G= D@T >K@> @ INJF>B0J

"!U6$(7!"U6$(7 BD @ Q0HN1ND 0I F0J>BJNB>T BI B> D@>BDIB=D >K=D= EC0E=CR

>B=D% dBO=J @ Q0HN1ND 0I F0J>BJNB>T "567/ L= D@T >K@> @ INJF>B0J " BD "R F0J>BJN0ND BI,"5&7!"5%7,""5%&!%%7 I0C @11%@JH&BJ!%

!"##$'& D1 "!!!"!3' "/31.69:& 2./$3/"."'7 $8+/ $8+ 1"/2$3./'

#$" -/0 #"$" 2./;+6<+ "/31.69:& $. " =8+/ $!"64 F.6+.;+67 <3;+/ - 9.0":"' .1 2./$3/"3$& "7 $8+6+ +%3'$' - /./>0+26+-'3/< 1"/2$3./

#5$7!U6$(7!"U6$(7 '-$3'1&3/< 1BQ

$!"6#5$7#6 -/0 '"28 $8-$7 1.6 +-28 ">

2./$3/"."' ,."/0+0 1"/2$3./ "7 =+ 8-;+A

) #$" -/0#"$" -6+">2./$3/"."' 1.6 +-28 $!64

) "!#5$7"#$"5%7"" -/0 ""#"$"""$#5$71.6 +-28 $!64

-!..X% <=> ND IBV &'!/ @JH D=> ;5%7#"5%$&7% G= K@O=

"5%7!"5%&%7";5%7""5%7$"5%&%7% (EE1TBJM >K= 0E=C@>0C #$ MBO=D

#$"5%7!"5&7"#$;5%7"#$"5%7$"5&7!.J >K= 0>K=C K@JH/ L= K@O=

#$;5%7#BJI

B

&

"5B$&7$ %B!%7%3#$'

#BJI

B

&

"5B7$ %B!5%$&7%3#$'

#

#$"5%$&7$

D0 >K@>

#$"5%7!"5%&%7"#$"5%$&7"#$"5%7$"5%&%7!

G= K@O= EC0O=H >K@>#$"BD"F0J>BJN0ND BI"BD/ >K= EC00I I0C#"$"BD >K= D@Q=%

+J 0CH=C >0 D>NHT >K= F0JO=CM=JF=/ 1=> ND D=>#5$7#DNE

6"6

5"567!63#$7% G=

K@O=

#5$7#DNE

6"6

&

"56 )) -$

7!63'

"DNE

6"6

&

56$27"5 )) -$

7!63'

#"5 ))

-$

7$"35 )) -$

7#4!

G= F0JF1NH= >K@> 1BQ

$!"6#5$7#6% G= J0L F0Q= ?@FS >0 >K= 0E=C@>0C#$/ @JH 0?D=CO= >K@>

"5&7! %&!%%3#$!"5%7!"5%&!%%7$ %&!%%3#$!"5%7!#5$7 I0C =@FK%@JH&/ D0 >K@>

"!#5$7"#$"""!

#

(7)

!"##$ (& 5.6 +-28 1"/2$3./ "!!!"5! ($$ (`7 =+ 8-;+

"

#$&#$5"7"" -/0 $8+ +M"-:3$& #"$&#$5"7#" 8.:0' 31 -/0 ./:& 31 " 3'

$>'+93>2./;+% -/0 :.=+6 '+93>2./$3/"."'4 L393:-6:&7 <3;+/ - 1"/2$3./

;!!!"U! ($$ (77 =+ 8-;+ #$&#"$5;7!;7 =3$8 +M"-:3$& 31 -/0 ./:&

31 ; 3' $>'+93>2./2-;+ -/0 "??+6 '+93>2./$3/"."'4

-!..X% <=> ND LCB>= =VE1BFB>1T

"

#$&#$"5%7#DNE

& BJI

B

&

"5B7$ %B!&%3#$! %&!%%3#$'

!

)@SBJMB#%/ L= 0?>@BJ >K= =D>BQ@>=#"$&#$"5%7"DNE&"5B7#"5B7% <=> ND J0L LCB>=

"

#$&#$"5%7$ %%%3#$#DNE

&

BJIB

&

"5B7$ %B%3#$$53&#$7*5%!B7' LKBFK ?T @J 0?OB0ND FK@JM= 0I O@CB@?1= 1=@HD >0

"

#$&#$"5%7$ %%%3#$#DNE

&

BJIB

&

"5B7$ %B%3#$$&*5%!B7'

!

G= C=F0MJBY= K=C= >K@> >K= INJF>B0J#"$&#$"5%7$ %%%3#$BD >K= <=M=JHC=

?BHN@1 0I >K= INJF>B0J "5%7$ %%%3#$% +> BD L=11RSJ0L >K@> @ INJF>B0J BD

=cN@1 >0 B>D <=M=JHC= ?BHN@1 BI @JH 0J1T BI B> BD F0JO=V @JH 10L=C D=QBR

F0J>BJN0ND% #

=6 >$((3 (3 08+ 2"') $+#-.0#6

-!..X .X)A".!"'W% X0C =@FK INJF>B0J1@JH =@FK$"6/ >K= INJF>B0J

"

#$&#3$&#"$ 1BD ?0>K$RD=QBRF0JF@O= @JH$RD=QBRF0JO=V% +> BD$RD=QBRF0JO=V

?T <=QQ@ 2/ @JH B> BD D=QBRF0JF@O= ?=F@ND=#3$5#"$ 17 BD 3$RD=QBRF0JF@O= ?T

<=QQ@ 2/ LKBFK BQE1B=D/ D>B11 ?T <=QQ@ 2/ >K@> #"$&#3$&#"$ 1 BD$RD=QBR F0JF@O=% (D @ F0JD=cN=JF=/ <=QQ@ ] ?=10L BQE1B=D >K@> >K= INJF>B0J@$1BD )2$2 EC0OBH=H B> BD 10F@11T ?0NJH=H% )K= INJF>B0J @$517 BD ?0NJH=H BI1 BD

?0NJH=H/ K=JF= B>D BD)2$2BJ >KBD F@D=%

+J >K= F@D= LK=C=1 BD NJBI0CQ1T F0J>BJN0ND/ <=QQ@ 3 BQE1B=D >K@>

1!#53$7"@$517"1$3#5$7!

(D @ F0JD=cN=JF=/ @$517 BD F0JO=CMBJM NJBI0CQ1T >0 1/ @JH B> BD 10F@11T

?0NJH=H K=JF=)2$2%

G= J0L F0JDBH=C >L0 F0J>BJN0ND INJF>B0JD"@JH;DNFK >K@>"@JH!;

@C=(D=QBRF0JF@O=/ @JH DNFK >K@>;""% G= F1@BQ >K@>

"!1!;#."!#$&#"$ 1!;@JH"!#"$&#$ 1!;

33^ -@>CBFS :=CJ@CH

(8)

I0C$"(% )KBD F1@BQ BQE1B=D >K@>"!#"$&#3$&#"$ 1!;LK=J"!1!;@JH

$"(% <=> ND J0L EC0O= >K= F1@BQ F0JF=CJBJM#"$&#$/ >K= 0>K=C E@C> ?=BJM DBQB1@C% &BJF=;BD(RD=QBRF0JO=V @JH F0J>BJN0ND/ L= K@O=#"$&#$;#;I0C

$"(/ ?T <=QQ@ W% )K=J/

"!1!#"$&#$ 1!#"$&#$;#;

LK=C= >K= D=F0JH BJ=cN@1B>T I0110LD IC0Q <=QQ@ W/ @JH >K= >KBCH IC0Q >K=

0?OB0ND I@F> >K@> >K= 0E=C@>0CD#$@JH#"$@C= 0CH=CREC=D=COBJM%

G= K@O= EC0O=H >K@>;"@$517""BI;"1""@JH$"(% X0C$'`6$(`/

>K= INJF>B0J@$517 BD >KND 10F@11T ?0NJH=H K=JF=)2$2% # -!..X .X )A".!"' 4% <=> -3 ?= FK0D=J DNFK >K@> >K= INJF>B0JD 5<3"7&!3 @JH!5<3;7&!3 @C=-3RD=QBRF0JF@O= 0J!/ 5LK=J =V>=JH=H ?T 6 0N>DBH= 0IG/7% )K= =VBD>=JF= 0I C=@1 JNQ?=CD-3LB>K >KBD EC0E=C>T I0110LD IC0Q <=QQ@ 4 ?=10L% dBO=J"!1!;/ L= F@J @EE1T )K=0C=Q W I0C =@FK3

>0 >K= INJF>B0JD

5<3"7&!3!5<317&!3!5<3;7&!3

=V>=JH=H ?T Y=C0 0N>DBH= 0IG/% G= F0JF1NH= >K@>/ I0C$'`6$2`/ >K= INJF>B0J

@$-355<317&!37 BD)2$2@JH D@>BDIB=D

5<3"7&!3!@$-355<317&!37!5<3;7&!3! (D @ F0JD=cN=JF=/ >K= INJF>B0J

%@$-3&

5<317&!3'(

&!!23 $

=V>=JH=H @D @ INJF>B0J 0JF=cN@1 >0 6 0N>DBH= 0I!35G/7/ BD)2$2% )K= INJF>B0J H$517 BD >KND 10F@11T @ IBJB>= DNQ 0I )2$2 INJF>B0JD K=JF= B> BD 10F@11T )2$2% '0C=0O=C/ L= K@O=

"#$

3

<3"!H$517!$

3

<3;#;!

# G= K@O= ND=HP

!"##$)& *+$ "!G/!"!,+ - ,."/0+0 1"/2$3./ '"28 $8-$ "! %!%3#- 3' 2./2-;+7 1.6 '.9+ -"64 5.6 +-28 2.9?-2$:& '"??.6$+0 /./>/+<-$3;+ )3 1"/2$3./ <!G/!"!7 $8+ ?6.0"2$ <" J+%$+/0+0 ,& B+6. ."$'30+ .1 G/K 3' '+93>2./2-;+ ./!/4

-!..X% &BJF= " BD ?0NJH=H/ L= F@J @DDNQ= >K@> "!6 0J G/% <=>

N/G/?= @ F0QE@F> DN?D=> 0I >K= 0E=J ?@11G/LKBFK F0J>@BJD >K= DNEE0C>

(9)

0I<BJ B>D BJ>=CB0C% &BJF= >K= INJF>B0J"! %!%3#-BD F0JF@O= 0JG2B> @HQB>D DNE=CRHBII=C=J>B@1D @> =@FK E0BJ>% (D @ F0JD=cN=JF=/ I0C =@FK%'G// >K=C=

=VBD>D @ 1BJ=@C I0CQ:%DNFK >K@>

6""5&7""5%7$:%*5&!%7$ %&!%%3#-

I0C =@FK&'G2% '0C=0O=C/ >K= 1BJ=@C I0CQ:%BD ?0NJH=H BJH=E=JH=J>1T 0I

%'N% G= @1D0 K@O=

6"<5&7"<5%7$0<%*5&!%7$)%&!%%3

I0C D0Q=)"6/ I0C @11%$&BJ!/% )@SBJM >K= EC0HNF>/ L= M=>/ I0C%'N@JH

&'G//

"5&7<5&7""5%7<5%7$5<5%7:%$"5%70<%7*

*5&!%7$)%&!%%3$)%&!%%W$)%&!%%4 LK=C= )"6 BD @ F0JD>@J> BJH=E=JH=J> 0I %'N @JH&'G// LKBFK Q@T FK@JM= IC0Q 1BJ= >0 1BJ=% (D @ F0JD=cN=JF=/ D=>>BJM*%#<5%7:%$"5%70<%/ L= 0?>@BJ >K= BJ=cN@1B>T

5<"75&7"5<"75%7$*%*5&!%7$)%&!%%3 5<7

I0C =@FK%'N@JH&'G/% +I L= D=>*%#6 I0C%'!/!N/ >K= C=1@>B0J 5<7 K01HD I0C =@FK%'!/@JH&'!/% X0C%'N@JH&'G// L= K@O= @1C=@HT EC0O=H B>% &BJF= >K= 1BJ=@C I0CQD*%/%'N@C= NJBI0CQ1T ?0NJH=H/ L= F@J

@DDNQ= >K@>*%*5&!%7$)%&!%%3!6 I0C @11%'N@JH&'!/!G/?T

>@SBJM ) 1@CM= =J0NMK% )K=J/ 5<7 K01HD I0C @11 %'N @JH &'!/% X0C

%'!/!N @JH& 0N>DBH= 0I >K= DNEE0C> </ >K= C=1@>B0J 5<7 K01HD BJ @J 0?OB0ND L@T/ ?=F@ND=<"5%7#<"5&7#6/ @JH*%#6% X0C%'!/!N@JH&

BJ >K= DNEE0C> 0I</ >K= C=1@>B0J K01HD EC0OBH=H >K@>)!Q@V 5<"7#03/ LK=C=

0BD >K= HBD>@JF= ?=>L==J >K= F0QE1=Q=J> 0IN@JH >K= DNEE0C> 0I<% )KBD BD @ E0DB>BO= JNQ?=C DBJF=NBD @ F0QE@F> D=> F0J>@BJBJM >K= DNEE0C> 0I<BJ B>D BJ>=CB0C% G= F0JF1NH= >K@> >K= INJF>B0J 5<"7 BD D=QBRF0JF@O= 0J!/% #

X0C F0QE1=>=J=DD/ L= @1D0 EC0O=/ I0110LBJM X@>KBP

!"##$*& *+$ "!!!"!,+ - :.2-::& ,."/0+0 1"/2$3./ =8328 3' ,.$8 (>'+93>2./2-;+ -/0 (>'+93>2./;+%4 #8+/ $8+ 1"/2$3./ " 3' )2$27 -/0 ^#(

3' - *3?'283$B 2./'$-/$ 1.6 $8+ <6-03+/$ .1 "4

-!..X% +> BD L=11 SJ0LJ >K@> @ 10F@11T ?0NJH=H F0JO=V INJF>B0J BD F0J>BJN0ND% G= F0JF1NH= >K@>"BD F0J>BJN0ND% <=>"?= @ F0J>BJN0ND INJFR

>B0J LKBFK BD ?0>K(RD=QBRF0JF@O= @JH(RD=QBRF0JO=V% )K=J/ I0C =@FK%'!/

33g -@>CBFS :=CJ@CH

(10)

>K=C= =VBD>D @ NJBcN=:%'!DNFK >K@>

,"5%$&7!"5%7!:%*&,"%&%3#(!

G= F0JF1NH= >K@>:%BD >K= MC@HB=J> 0I"@>%/ @JH L= K@O= >0 EC0O= >K@> >K=

Q@E%0!":%BD <BEDFKB>Y% G= K@O=/ I0C =@FK%/&@JHBBJ!P

:%*5&$B7! %&$B%3#(""5%$&$B7!"5%7":%*5&$B7$ %&$B%3#(

:5%$&7*5!&7! %&%3#(""5%7!"5%$&7":5%$&7*5!&7$ %&%3#(

:5%$&7*5!B7! %B%3#(""5%$&7!"5%$&$B7":5%$&7*5!B7$ %B%3#(!

)@SBJM >K= DNQ/ L= 0?>@BJ

,,5:%$&!:%7*5&$B7,,"%&$B%3#($ %&%3#($ %B%3#(!

:T @ FK@JM= 0I O@CB@?1=D/ L= M=>

,,5:%$&!:%7*5B7,,"%B%3#($ %&%3#($ %B!&%3#(!

)@SBJM%B% # %&%/ L= 0?>@BJ

,,5:%$&!:%7*5B7,,"^%B%%&%#(

I0C =@FKBDNFK >K@>%B% # %&%/ L= F0JF1NH= >K@>

%:%$&!:%%"^%&%#(!

#

!"X"!"#8"&

U2` -% :"!#(!$/ "VBD>=JF= 0I)2$2FCB>BF@1 DN?RD01N>B0JD 0I >K= A@QB1>0JRe@F0?B

=cN@>B0J 0J F0QE@F> Q@JBI01HD/ (JJ% &FB% "% #% &% 547/ ?@/ J0% W 5366\7/

EE% 44]R4]3%

U3` -% 8(!$(<+(d*")/ 56./$ E6.?-<-$3./ E6.,:+9' =3$8 I./:.2-: #+69' O/ e%

'@>K% (J% (EE1%/=A@536627/ EE% ]\3R^62%

UW` (% X()A+ R '% f(,+$.,+h*"/ D/'+6$3./ .1 )2$2 1"/2$3./' -/0 D:9-/+/P'

*+99-4

U4` (% X()A+ R (% &+8.#.<X+/ Q%3'$+/2+ .1 )2 263$32-: '",'.:"$3./' .1 $8+

!-93:$./>R-2.,3 +M"-$3./% +JO=J>% '@>K%/5BB/ J0% 3 536647/ EE% W^WRWgg%

U]` )% +<'(#"#/#8+ *+;+:>L+$ 5:.= ./ - F-/31.:0/ -C0F% &TQE0DB@ -NC= '@>K%/

B?/ J0% 2 52iiW7/ EE% 2iWR364%

U^` e% '% <(&!jR -% <% <+.#&/C @+9-6( ./ @+<":-63B-$3./ 3/ !3:,+6$ L?-2+'/

+DC@=1 '@>K% e%/BB/ J0% W 52ii^7/ EE% 3]\R3^^%

U\` '% f(,+$.,+h*"/Q%3'$+/2+ .1 )2$2263$32-: '",'.:"$3./' 3/ 03'26+$+ =+-( NCF

$8+.6&% )0 @EE=@C BJ e0NCJ@1 0I '0H=CJ $TJ@QBFD%

'@J0DFCB>>0 E=CO=JN>0 BJ C=H@YB0J= B1 ^ 1NM1B0 3626%

Références

Documents relatifs

To test whether the vesicular pool of Atat1 promotes the acetyl- ation of -tubulin in MTs, we isolated subcellular fractions from newborn mouse cortices and then assessed

Néanmoins, la dualité des acides (Lewis et Bronsted) est un système dispendieux, dont le recyclage est une opération complexe et par conséquent difficilement applicable à

Cette mutation familiale du gène MME est une substitution d’une base guanine par une base adenine sur le chromosome 3q25.2, ce qui induit un remplacement d’un acide aminé cystéine

En ouvrant cette page avec Netscape composer, vous verrez que le cadre prévu pour accueillir le panoramique a une taille déterminée, choisie par les concepteurs des hyperpaysages

Chaque séance durera deux heures, mais dans la seconde, seule la première heure sera consacrée à l'expérimentation décrite ici ; durant la seconde, les élèves travailleront sur

A time-varying respiratory elastance model is developed with a negative elastic component (E demand ), to describe the driving pressure generated during a patient initiated

The aim of this study was to assess, in three experimental fields representative of the various topoclimatological zones of Luxembourg, the impact of timing of fungicide

Attention to a relation ontology [...] refocuses security discourses to better reflect and appreciate three forms of interconnection that are not sufficiently attended to