Lasry-Lions regularization and a lemma of Ilmanen

!"#$%&!'()# *+,-."$'/"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

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

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

)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)<sup>2$2</sup>DN?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

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$7"#_$"""!

#

!"##$ (& 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

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>

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

>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%