Méthode de la frontière élargie pour la résolution de problèmes elliptiques dans des domaines perforés. Application aux écoulements fluides tridimensionnels
187
0
0
Texte intégral
(2) ´ Ecole doctorale de Sciences Math´ ematiques de Paris Centre UFR 921 – Science du calcul et ing´ enierie Math´ ematique. M´ ethode de la fronti` ere ´ elargie pour la r´ esolution de probl` emes elliptiques dans des domaines perfor´ es. Application aux ´ ecoulements fluides tridimensionnels. ` THESE pr´esent´ee et soutenue publiquement le 26 mai 2004 pour l’obtention des grades :. Docteur de l’Universit´ e Pierre et Marie Curie – Paris VI et. ´ Docteur de l’Ecole Nationale d’Ing´ enieurs de Tunis (sp´ ecialit´ e Math´ ematiques Appliqu´ ees) par . Mourad Composition du jury Rapporteurs :.
(3) Michel Ad´elia
(4) . Examinateurs :. Christine
(5) !!"
(6) !#$ Ta¨ıeb %&"#$'$
(7) ! Yvon ()"#*,+ Bertrand (!
(8) -+. Invit´es :. (Responsable de la cotutelle) (Responsable de la cotutelle) (Directeur de th`ese). Silvia Bertoluzza Lassaad .0/214356. Laboratoire Jacques-Louis Lions (UMR 7598). Laboratoire d’Ing´ enierie Math´ ematique ´ Ecole Polytechnique de Tunisie.
(9)
(10)
(11)
(12)
(13)
(14) ! "
(15) #$&%' "
(16) ( ) *+ ,-+. 0/1, !2 3 "
(17) 3+
(18) 5463+/1, !7,.
(19) 3+
(20) 7",! /1, 1
(21)
(22) (*+.8 , 8#9)
(23) !:';<>=$ .
(24) 3+
(25) ..?A@B ?2+#C7DFE7G+'.",$=: H,
(26) .3+1
(27) II ,
(28) .I..?J
(29) KG+ ,H463+
(30) ,.
(31) -L*
(32) 2$MN
(33) O P(MQ0R7STMNUV *M PBWI.4 M4 6M4$?XMZYF @[P(M \V
(34)
(35) M P]#463 6^.
(36)
(37)
(38)
(39) !#"%$'&. *( ),+-/.102/0435!-765!8:965!8:9*;=<>+8:9@? A 5!-B65!-/<C5 A BD?FE.G5!-7.F<C0HI5!8 A BD? 65!8:9*;=<>+8:9@0=HI0 A .J<C0DRTBKS UW5VL=B'XZ;=Y!-/[]9=8:.G\ZMD^`BO_NQX!P[ a.
(40) aCa.
(41)
(42) A X A
(43) 3
(44) +3?+ ":
(45) (/" , + !,,T 9 !^,H" 1
(46) ;A !. #
(47)
(48) 0
(49) #$36,ZM+ 46, /1 ,X+$;# ,3 * +:1 !, * MJ1 +
(50) :, , !+0 1?:
(51) * M! "
(52) J , ##(. 46 3+! .# 1A$
(53) *.^Z.+S)4
(54) 0! . #:
(55) 46,3
(56) 3*6+ M ?+,$= . #
(57) ]
(58) ]3#
(59) 3+1
(60) H$T .?...
(61) : "
(62) B+,
(63) 3+"
(64) 7/1,Z+..
(65) ]3+ +../1, ; + 3 .,[
(66) #/",A ;< "
(67) #?*+, , + ,(! ?
(68) 36+. ^ QA! ,
(69) :, A2+.( ; * .. #!($.
(70) ,7H
(71) 3H
(72) B 6;<
(73) ], : 3Z+ ( ; * .8 #!(+# ^" ;< ( #+ $ ( 1
(74) 3
(75) *^ % 3+ B ,?$. #
(76) FH',+ , ''-& 7 , [3/":, 4 /1",
(77) ..[.3.46 [
(78) , $ , .
(79) [ ,(! , . Z + -, !,$=' ^ ! "
(80) ! Q-+
(81) . # "+
(82) F ;</1
(83) EF,X ; ..:8 +, .3;<
(84) F1
(85) \
(86) > =$! . .46!
(87) . : + ]Q0 46 ,3 +
(88) ].
(89)
(90) 3, +
(91) ")
(92) (;0/ ! Q0$F.DB.4+
(93)
(94) .: ! Z A+
(95)
(96) F#Z'
(97) +;0 1 $4-!
(98) H+ OI, H$A* !? *B+ OX $, $! '^1 ; OI,$ * , ?2 B+2
(99) ,
(100) $ .- , ; ! , ...I + [72+? 3+
(101) ./1,X; .X$..43: ;0/ H\ .9)
(102) /1, HOX,6^ S)F/",Z+..
(103) 4, "
(104) 4#Z/1, -
(105) 01,+. - "
(106) AT ,0?T*+, , +, ?T 5
(107) ! 1
(108) $,2+? 3+
(109) .6^6" ! $ ! "
(110) 8 ,7
(111) , 77 .*M'+. /1, :T ,73 $ .?...
(112) (+.46#, +4 , 1
(113)
(114) 3Z+46^ QA ,
(115) -,463+ , /", Q0 ,7* &=
(116) [ 8 .2S)/1, .3#+ "
(117)
(118) 8]3+
(119) ,8
(120)
(121) (
(122) 6^9"(B ! !$,
(123) # 8/1,X; .7 "
(124) 3 2
(125) ,#$, +1, .
(126)
(127) 7 , ; ."
(128) ; :
(129) ]/",X; . "
(130) F <: !
(131) 36+. ^ ]
(132) [ 4"+ ! 1
(133) = >=$.! -
(134) ,
(135) FH Z+K+ : S1..)2>
(136) .,7? : ,[0 3 .*M +#.
(137) 8
(138) F3: , 8 .2+? 3+
(139) +, 3 . 72 @BADCFEHGJI(# K^ WZ+.AZ+
(140) -$,(+?T 3+
(141) .1 /", L, ,V' *M+ : !0 (
(142) 3+1
(143) J/1,
(144) K+'M+
(145) I3+ I ,
(146) A, 6+"
(147) 3+4N1M 3 , ! "
(148) AZ+ $.
(149) -+%'
(150) +-+, 46 O P8 Q ,
(151) , ! ?
(152) , "
(153) F! . ! ! "
(154) ^ 27 ! ") .) 2+Z.+.
(155) F
(156) /1 ,,. 9( ! 46 "6
(157) ^ R].
(158)
(159) ]C].3+,.
(160) *MSR ! 9!
(161) AYF..) 0\ . *+, , 0 ,$= .A/",X; . ;< "
(162) -
(163) B ; .1
(164) ; :
(165) F/1,X; . "
(166)
(167) >: !
(168) 3* +. ^ A$ #
(169) -+, $ J/1, /", I..46 IT ,J $ I 463+
(170) .
(171) , [+,$=H 4"+K+
(172) ,V$,H46 , :
(173) 36+.=T5
(174) )]DF,! /1, >5 U LVR W[8 ,XMQ0
(175) . XA Z+$V
(176) R Y *9^9 : ! AT , 2 -# + -/1,X; . ;< 1
(177) A :! -# +1
(178) V2[ 3+?...
(179) A 2F ) $.Z+
(180) $, 46 ,77
(181) 36+.X +"
(182) F, +
(183) # ^ X:?
(184) ! "
(185) $, 2+? 3+
(186) .: .
(187) ]?*+, , T V $#.
(188) 3+
(189) -
(190) F
(191) 3 /1, 6^ , +! @B+ A ,. 6M6X..2+ , 3"
(192) *M +
(193) . X 463
(194) X, X ..
(195) , 1
(196) !-. - ! ! "
(197) *^EF ]T 3 Z+
(198) ,. B6 , Q0
(199) 2+@7) ,- * .
(200) K+ +..6^ "7B ! 8). ! "
(201) FT ,[ ; .# 8 7) ,! "
(202) *^ UV!O,6M ; +$2B Z+ [ +.0Z+
(203) $,( +?T 3+
(204) . '; 4 +
(205) +
(206) /", 0 ; \I .91
(207) 3/", #OX,6^X ; +?2+ 11.)2+!/1,J9 46 #
(208) 7 ,.
(209) 8$, , ! "
(210) :# ! ? ^ B[ ! 7
(211) , ^E ] BZ+
(212) ,. ]
(213) , 7 36 UV ]# , ? ,$= 3 .*M ,3+4 ! 1
(214) ]
(215) 3 +#.
(216) 6^ X Z+ ! 1
(217)
(218) #
(219) ( 3,H + ]: * $(] ) 3+
(220) 3+
(221) , ^ ! T 3+? 1
(222) Z/", 1 3+
(223) T , , + 6^ " .
(224) Z+
(225) ,. FQ0
(226) . - Z+$ M$O ? B $ M$W3+ W ,.. ? *M -
(227) -\I
(228) Z+!6M 336 UV ]# M *+ W #CB ?*+,
(229) -
(230) 3+ +,9^. _^. Q Z `. 3# 8 7 -,. D 7 . W. =. ]Y3 <. \[. . 4 # \$ * Z+ d. 4. 01 3 \. `b. a8. -,. 0/. - # $, - $c. a aCa.
(231) $ . 4#. @
(232) ,
(233)
(234) J "
(235) Z/1, 6MN
(236) 3+#! "
(237) + * $A'; , 0 ) T 3+
(238) XMN."98 , , B
(239) 3 $#.
(240) 3+
(241) .6^ ! !$&%' "
(242) 3+? $#.
(243) 3+
(244) /1, M , ! ! "
(245) , , +,
(246) 6M ;< 1
(247) +
(248) , + , 6^ 463+
(249) .
(250) , 6 Z+
(251) ,. ! 1
(252) +! +
(253) - .Z EF. .
(254) $, * ? Q-+
(255) Z+4 >M +! WZ+3+Z C]3+4 2+
(256) "
(257) Z+
(258) Z+ ; D >M 3 +! @B *M + $.
(259) ,....)
(260) .
(261) ,
(262) W3+ *+ QA) 3+
(263) >M +! A.46.
(264)
(265) [+...9 U0C @]U >M +! S$+ $. X ! .46 @].
(266) 2+
(267) ."
(268) Z+
(269) Z+ A ;<EF. .
(270) [\V [
(271) +[Q0, >M +! F! M [ ,Z+
(272)
(273) VX 2. \I .91
(274) 3/", :7OX,
(275) ,]S$+. HD]+
(276) Z+8'; 4 , OX, >^ J W QA1/1, J
(277) 8Q0
(278) . + +:T ,B ,8+#.
(279) (
(280) B ,]4 1
(281) ..(3+ ..#.
(282) 6^ F, F!
(283) , ,$= /",B "
(284) Z+
(285) 3+4 ,
(286) ! /1,
(287) $
(288) 3+ LA!. #Z+46 H$, ?,*+, @ .?Z+ M + 2 3M ,.*M'X ,*M DF 2 *M)\V... Z
(289) [S) . >M1
(290) ? ) #! "
(291) -A+,
(292) *9 ,$=
(293) 3+A$, +?T 3+
(294) . " /1, X ,-' *^)UV Z+
(295) ,. BL F"
(296) . 6M F,46,
(297) MTQA .6M@B .. 6MTW ...Z$M B+!$ M B+# $ M 8+. MTX Z+M +
(298) .XMS)
(299) RF. 1
(300) *^ -*+, , ';< "
(301) F ,$= 1
(302) , "
(303) +.
(304) IZ+
(305) FV! ?A[ ; , ,![ ; +,
(306) [!0 ,$= 2+?6 '; , . M+ ; +X XB- ! Z+ ! "
(307) FZ+ 7/",X;< HZ+
(308) 3+48'; +,
(309) ?T ! ! 1
(310) >=)
(311) 3 ? ,
(312) ; + >=6M+,5Z+ XM ^^^ (LS1
(313) >M! ,7
(314) +#.
(315) . 6M
(316) 3 12+?...
(317) #
(318) B ,7B?T B! ! 1
(319) 8/1, ( ;< Z+
(320) 3+46^ ;<T /1,X; -
(321) 3+ , 3 * 2
(322) 3 $.
(323) 2 $, $$ MJ 3 , ! "
(324) ]T , 82+. 7+4 A+]+, ' , . .F Z+T 791H?T +
(325) +
(326) /", ^ YF +,3# 8?T*+, , +. >^ \I 3 E]+ >M\V. M M
(327) BQA . >MW >MTQ0
(328) >MT+, 1
(329) >MYF..) *M F>= ^YF 2 3Z+
(330) HZ 8! 3 &= 8!
(331) *; .: + ?,*+, L >M F9"+ >M " !
(332) F! . >M'\J+,
(333) ] >M *+ ',. [
(334) O +Z+ >^ Z+#!
(335) -+, 8K #!FW3+ 36M VT , 3 +#.
(336) 8. 6M K( 12+?...
(337)
(338) 3($ .?...
(339) 3+ [..#.
(340) 6^ ]+
(341) M + *M Z+! - .Z >M 8+! )
(342) BZ+'M 03+,3+I #! "
(343) V , ! , F
(344) 8 ( + B.B! .
(345) ,Z+
(346) *^ 7 , ,
(347) Z+3+1
(348) *^ :X;< ,?. 3+
(349) 3+. ! "
(350) Z B!8+# 8: ; +,
(351) #
(352) #H2 $.
(353) 3+ L 6*M Z+!'M)WZ+ $M$Q0 1 M "+?Z+.2+
(354) -S) ,Z+. ^ T+, B
(355) ,
(356) 3+A$, ^1E .. ';<1 . Z+
(357) ,. F ,Z , +#87 46, 8 +
(358) 6MZOX,*M Z
(359) .*M DB+? , M.^^^
(360) ;< Z >^ Q0Z , 1 M ? "
(361)
(362) #+ '; +#.
(363) 3+
(364) .
(365) 6^ !8Z+ ,*M #. F, , $&% 1
(366) + !H
(367) #.Z+
(368) , , !7?,
(369) *^ , $.8B., E !T !Z+
(370) ,. , 8+#..2$M F 'M -$M "+! M [ ,Z+
(371)
(372) B+# $^ . [9 ,$= + $, 1 ,*^ +# 46 MT
(373) ,
(374) , ,7
(375) H! ,
(376) ! 3+V + ]3 2+.
(377) 6^'Q];<
(378) F46 #
(379) . /1, 7 #! : FB! F+. 6^ 7 ! 8
(380) ,
(381) 7(? . +#..8
(382)
(383) ,
(384) 7 +#..BT , ,F ,
(385) X^
(386) F$ ^ +,$= ,. X 2H
(387) -CB 3+'M1! Z ,
(388) ,
(389) - /1, , - +.
(390) 0T ,0! ^)S$+ A , 07) Z+ X; +,3+.
(391) 0Z
(392) [Z+ ..6^ Z; + ?T*+, , + ]
(393) 1
(394) 3
(395) *M
(396) 3+#! "
(397) A/",Z+ !- ,3+.
(398). O 4 % % 4 $ ' & [ % 1 % `b Q 1 & 01 Pd[P 1 8W& D % 1 N & \ '& N01 8 (8 _ &* %/ %/ '& \ 8 -# > B % '& % 8
(399) " "!$#&%')(+*, !$-$-$ 9 /.0 +12(3-$5 4-$-$!$5# , 6# 4!$7(!$#8 49 :(247-$<;&4>= ?@ A-CB 4.+;24D4((<-E44! F & L , 8 8 "+ + 5G % & )T , <U % & _ D% H &-$'!$IK'J5F?& "L8 ! NMC4O4, 8@6(!$30 -E4P ' RQW ? % S 8G TVUW I XZYN\I [$]L^VW"G G_E Y+UE`a^\[[ I'& 3 NM , & % '& % %#,>B (!$ 64;!$6!$, -$'% d%-E 4)bc#&+.g#4B !$-$-$ d3.0R & !\bI!R4!e*,4*a 4%Dk -$-$!?f F? !R& (K3*-$K(g%Dl& 6m&= ?h&@ & d-$"V8*4'( % 4ijnii4poI!$* i 4 iVq"i$r B & 3&0bc0# gst !$4u-E4N; % -CB 4-$-$!$% ("-E4g#w8st 4!$x & ,y % 8 '& % + _v '& & 9 -, % ># !$ , & + bI:6'!$ ii&i & .0&#!V4!zL '< 5# &!$-$ & % " > `b D4a :4I D4, : ;-$!$a-$< ! <{5 w.g4R & % $& VG )G + $ v V % '& V 2y 2 R $ , X8 % ` Na1 0v & : 7 ,y | :|G _, Q [ ": 5}a 8 _ >+ "{d~('!$4! bI '5 '5 j, < &(5(6OI. _, DN ( Z -# 2ZT. < P+ 7 -# * ] . P. a. U. `. O X8=*.
(400) . . O % D 6,. B8 & (. D T. U. -$.+ [ 3 A , +,
(401) ,VA2F
(402) 3+? +#..2+ [U
(403) .-
(404) [ >^ BCB CB , . ., +
(405)
(406) 1
(407) (
(408) 6^TC] (
(409) '7+1, .
(410) ]:4"+46 8 2+
(411) 6^ H
(412) #. : Z+B ,$= /", 8 T ,3+ ++ ! BZ+FF!
(413) *M
(414) F/1,X;< ,$= , +, "
(415) ,3+4
(416) (2 Z+
(417) .. (Z+4 , /",X; +, ? ,
(418) L , *MJ!! 3 Z+ "
(419) M$H31 ,*M1! B
(420) A
(421) [ , ]463+ ! 6^$R 6[ ,3+4 ! 1
(422)
(423) -
(424) B ,
(425) X;< "
(426) ++ +.
(427) +,
(428) *^ !463+ 6M $.3+.
(429) /", + #
(430) 3 3 ,$%T3+
(431) , +
(432)
(433) $, .2 3+? ! "
(434) !
(435) ,! ,!!$. @] , ^^^
(436) !
(437) #3+ "
(438) + #
(439)
(440)
(441) #
(442) .+47
(443) 3 ++ 4638 + :! ! .6^ 67 Z+&='M 7
(444) *;< ,?. 3 ++*^ 8
(445) .
(446) +#..]
(447) -3+ - ,
(448) B .]/1,'8Z*9FF&=]., - *M) #!BH, 3,! ]?*+, , '; +! ,[
(449) A, ]Z+
(450) B>= #2+.6^ ] F3+,3+- #! "
(451) [2H ! [T., $.]T ]/1, ; + +.
(452) 06+2+.
(453) AF , ] 12 ^^^ ? /", F + A ] "
(454) >=$
(455) 6M) ! . -/1, B,7$.8, 8
(456) !8
(457) /1,X; .X
(458) ,1, 7
(459) ,
(460) F \I ,[28
(461) .
(462) *M H/", ; +T$]
(463) 3* +. M ;<6 8 . .,$.(L >= , ! T463+ 6^ \V #*M , , B
(464) *;<>=$./", 3+ ^. . -#. 2 _6, T U 5P V[ P 7 N V vaTW aU [ 6!$n 6 X B U. ". Z. >. v. W ;:. T. .
(465) 7a.
(466)
(467) . "!$#%'& ()*! - ,.!0/1 "!23%546(7,8/)9;:<*! > 43/?"/),@46A;! C A'/?()4D;&EF/),46AG%5H*A;HI()3. +, += B J. MK LON"P"QSRUT VGWDX"Y[ZS\]P"QSY[^`_aRUbaNYOcdZSe fgRDXhRDZSZiQjb]PkQSlm\aRDX_nL[^aXo_aRDXo_aYpfgL[QS^nRDX bqRNrsYONWDX t u ;9 v,8/?()wB x /?k/y Sz{(?/}|~H}/)9 4D;*!2}+ ,.!/?3A'/?*! B6B [ "00oqqqqqqqqq) [ }m a ¡ Fk¢£¡¤ ¥¤ a ¤F¦a¤F3¤F§k¤¨ a ¢Oqqq?© ª© «0£¬"k¡Iy0 a ® k]3¤F£0]£¡0¡¥¤ a ¤F¦p 1¤Fi0 a 0qq1¯ ª©" «¬*°¤±¢} a 0£ a y¦±²´³ H HI0=9@B'022µ 5!- A 2Z+9·¶±¸ 0=.8¹5O2?º 1¯ ª©" «¬*0aF¥¤ a p3 0 a ¢ qqqqq)» ª©"ª© «¼¤ a ¤F¦a* 1¤F }¤F¤F§k qqqqqq?½ ª©"¾¯ ¿ ¹*¶'BO+< 8CB qqqqqqqqqq?À ª©"» Ák§°¤F¦FÂÃÄ«MÅSÆmÃÄ« qqqqqqqÇ" ª©"ªÈ Ák§°¤F¦FÉ a Ê 0¡3 a 0ËÌÆÃp« qqqqÇ ª©"Í Î" s¬ky¤y¤F*ÏM a Ê 03 a ¡ F£ a ¦ qqqqÇF© 7a a.
(468) I'[ GSGK I EDU ;G. 7a aCa. u 9 v,/?()
(469) ~H}/)9 4D ;¼@ ()46A'/),I(7H*@()%6,!#"%$&('*),+-./.&10 5J " 2j}"k 0 a M0]F a ¤¨ a qqqqqqqq "ª O a a §' k¤±y¬*2qqqqqqqqq "¾© Â3±°¤7 a F¥± a Åjy¤ a qqqqqqqq "¾©k Ám¤F¡] F a a k¤F"ÏM a a £¬kyF¦0`qqq "¾©kª Ám¤F¡] F a a k¤F"ÏM a a £k"Åj¬kyF¦0qq " ¯ Â3±°¤7 a F¥§6 a }kÏ" qqqqqqqq u 9 v,/?()
(470) 3 ~H}/)9 4D ;¼@ ()46A'/), I(7H*@()%6,5 4 ,@!1E()H}/),.!0k/),4'A }/y,.:<vH": "A6/?k/),46A 6;B ©k 7d0 ® ]k a m 0¤ a £2qqqqqqqq ©k 8¬F£: 9;=<?>A@BDCGF EG@BIHJ<?K qqqq ©k ª La¤¨§§'0D ® ® k]Fs¤7 a F qqqqqq ©k ¾© 2j¦¤F a a £§'k] m¢k 0 a ¬3¤¨£ a® 0wqq ©k ¯ O£ M0 0 a O NPHIQCSRTD;VU qqqqqq ©kª O£F0y0 a £ qqqqqqqqqq ©kª" Î" s¬0¥¤2£0 a Åj a £ 0 qqqqqqqq ©kª"ª ÃO£§3¤F 0aS Wª¤¨§§£1Ï a y¤¨ a Y Xn¦F a ¬ qqqq ©k¾© «¬*0a£Fk a Ì a y§k}¤¨ a qqqqq ©k¾©k O£ ¦F3¤F qqqqqqqqq ©k¾©kª O£ * 1¤F qqqqqqqqq ©k ¯ 8;i]*ky0 a® 0qqqqqqqqqq ©k ¯ ZD[ Z ¤F§k°¤F a ¤F y¤ a <§'0¢F] \a £_ ^ ¤F£a ` ¦¤¨
(471) b §3¤¨ a 0Çqqqqqqq u 9 v,/?()6 c !0/),@:<k/),46AG Sz I()()*&;(y;,@!1E()H}/),.!0k/),4'A v;(Ìe dgf=hifkjDf l B ¯ L]¤F§§6D ® 0 ® ]F¤¨ a qqqqqqq ¯ª ÃO a y¤¨ a S W £02qqqqqqqqqq ¯¾© mĤF a 3¤7 a FI a® qqqqqqqqq ¯¾©k 80aIy0 a ® k¤F¡ ¤F£S W £Fk a Ì " ¤Fd·Ïk¤F 0£
(472) ¯¾©kª 80aIy0 a ® k¤F¡ ¤F£S W ¤F§k§£1Ï a y¤¨ a ∂v
(473)
(474)
(475) ∂n. γ. FÈ F È F È F È ©F¯ }© » ¯I I¯ I¯ ¯}© ¯}© ¯¯ ¯¯ » »¨¯ »¨¯ È} Ƚ Ƚ Í ÍF© ÍFÈ Í Í Í Í.
(476) u ;9 v,8/?() J u 3!y *!yE46A;,8/),46A!$ > *&;:<3A;A 3&O45(7 & /1(746& l »" Xn¦F a ¬ qqqqqqqqqq »" 2jy§ }s¤7 a F qqqqqqqqqq »"ª© 80]I a ® oqqqqqqqqqq »"¾¯ mĤF a 3¤¨ a ¥I a® qqqqqqqqq KMLON"P"QSRT"T ¼ RNX ZSRDX W'Y[\nZSR fgRD^pP"Xd\aQi_aR )bdLON*PkQ '\nZSRDX. ½ ½} ½F¯ ½½ t
(477). u 9;v,8/?() ,.!1E()H}/),@!k/),46AG *!H& k/),46A!2 > }= ,I('/?45"! 3 Èk Z 0¡ ® 3¤¨ a ¡ a¤ /a iÅSÎ*F£ a a } a 00 a Fy§ a k ÀF¯ Èk É a £ a ¤¨ a Ì0£§¡¡ ® 3¤7 a F¡ n¤ /a SÅiÎ*£F0 qqq ÀF¯ Èk* Z ¤$y¬* 0a ¤F¤¨ 00 a a ® 2qqqqq À}» Èk* ZD¡0£¬"k¡§£ M 0 a qqqqqq ÀÈ Èkª© É a £ a ¤¨ a Ì00£§3¤F 0qk] ® 3¤¨ a k a¤ 7a SÅiÎ*£ F0Ìqqq À½ Èk¾¯ \aMy*0 §6= W a }£¤¨ 0 a 3 a Å §3¤Fi a 0qqqq ÀÀ Èk» 0 a ! ±§6£§'0 0 a 0±qqqqqqqq]} u 9;v,8/?()el D,@: &k/),46A! A&: H*(7", & "! $B # 3 Í" Á a "Åj a Ê a M°¤± s¬3¤F ]¤F"J W¾¤Fy¤F§3¤Fi a mkÏ" F¯ Í" Á a "Åj a Ê a M°¤± s¬3¤F ]¤F"kk"Ï6 0y0} È Í"ª© Ám¤ /a }£&¤ %°k §£0 qS W F¤F §¬0 a® qqq] À Í"¾¯ [ 0 }]¤Fk£¡S W £§¬k£$qqqqqq]? Í"» [ 0 }¡3¤F¡ 1¤¨3¤FD FI£3¤¨Ia0"ÏMis¤F 00]§¬0 a® 0 q]?FÀ Í"ªÈ [ 0 }]¤Fk£¡S W k± a §'is¤F}k6 mkÏ"0 qq]1©È u 46AE}@&!,@46AG}/vOI(7!vO*EF/),=6*! B 3' aÏ.
(478) I'[ GSGK I EDU ;G ^n^aR¡RDX ;A A I+ &;*"&;*!2()vvO*@!y&D/),@@*! BA6 3 X Â3±n ® ¤F¤7k¤FiÅVZ 0¦k£qqqqq¡1¯¯ X ª`Â3±°¤ a £nqqqqqqqqqqqq¡1¯I» X ¾© 2j0¦}¤F a nqÁm¤F s ¬ qqqqqqqqqq¡1¯I» A;A I+. - z{H*@H*:<*A'/2# A, Q B 6Sl A Æn 8;¤Fk¢¥¤¨ a km *£k0$qqqqqqq¡1¯}À Ænª 2j}0¦¤7 a FI a® £k]k W¾00y0Im£0¢0£0 qqqq¡1¯}À Æn¾© «¼¤7 a 0]k a ¦ a a £naky¤F£ qqqqqqq¡1¯}À A;A I+ u vvO*@!y %5HI46:<H}/?(7,@ B*JDB Án Xa¦0¡S WªÃO qqqqqqqqqq¡)» Ánª ;
(479) *¦k¡£§¬k a ® qqqqqqqqq¡)»F© . 1. Ï. C A;;}+ $,@.,45%5(7kv 9;,@. B*J5J B*J.
(480) . . ª ¾© ¯ ª» ¾È ªÍ ¾½ ¾À " "ª ©k ©kª ©k¾© ©k ¯ ©kª» ©k¾È ©kªÍ ©k¾½ ©k¾À ©k ©k ©k ) ¯ ¯ª »" »"ª »"¾© »" ¯ »"ª». 00 a ¨qqqqqqqqqqq à Ïk0y§k £0§£0£0I¤¨ a ¢¤2¦F0£ a nSW¾kF¥¤ a d§6i¢£¨Gqq) «¼¤ a °¤¨¦aki 0££¨aqqqqqqqqq1¯ à Ïk0y§k 1D ÅÎk¤¨k]°¤±0 a ¥¤F d°¤±" a F`qqq?È Ã Ïk0y§k²@ a a y0 a 0 º k]¥¤ a ¤F¦Ä Ω ¡ γ ±qqqq)Í Ã Ïk0y§k²@ a a y0 a 0 º k]¥¤ a ¤F¦Ä Ω ¡ B 5qqqq?½ «¼¤ a °¤¨¦a " ¤F0y0}¡¤F3¤F§"Fhqqqqqqqq?À ¿ 3à Ï"§ ^ 1¤¨¦ bÌÏ*0 a 0£0 a } a 0£¨6qqqÇ Á¹*¶ BO§+¤F< 8C¦B É a Ê 0£0 0]3 a pËÌÆmÃÄ« qqqqqqÇ Î" s¬0¥¤¤F"Ï a Ê 0¡3 a 0¡ a ¦FFqqqqqqÇF© à Ïk0y§k²@ a a y0 a 0 º k]¥¤ a ¤F¦Ä* 1¤¨ 0]¦ 3¤F.qqqÇÍ Ã Ïk0y§k y¤ a kd3¤F¡ 1¤F¡ a a a k.qqqqÇÍ 00 a ¨qqqqqqqqqqq ¯}© É a ¤¨¦¤Fn3 a ¤F}]0¡§'0¤¨£ 3 0 qqq ¯I» «¼¤ a °¤¨¦a¦F3¤¨´qqqqqqqqqqÇ»¨¯ à Ïk0y§k °¤$0 §6 a a F §3¤F¡ " 0§'k k = 4 qqqq Èk É a §'F a a km¡iÅ * ¡§¤F (x, y) qqqqqq È} O£ M0 0 a M£¡¤2¢@¤¨ γ qqqqqqqqq È© «¼¤ a °¤¨¦ a a a ¤F "¢@¤F 0S W¾k 0'¨ qqqqqq È© Xn§§k£* s¬ a § a 30F qqqqqqqqqq È}» Z ¤2" a FÌ 1¤F F qqqqqqqqqq Ƚ 2j£¢@¤¨ qqqqqqqqqqq Ƚ u=0 Z ¤2" a FÌ 1¤F 3¤Fk] 1¤F¡ §3¤Fi a 00hqqqqq ÈÀ 343 2j£¢@¤¨ q q q q qqqqqqq ÈÀ u=0 Ák3¦F¤¨ a ̦F0£ a® ¨'qqqqqqqqÇÍ ÃO££0£¡ " ¤F0m0]¦ 3¤F " a a
(481)
(482) ¬¦0¡£k γ ÇÍF½ Énk¦Fy a ® F qqqqqqqqqq ½ La¤ yk}]* 1¤F.;qqqqqqqqqq ½© 2j£ ¤F0£¡¤±£ k a k 0 a a F !#"%$$ ¬¦0¡ ½}» £¤F a }°¤$k a qqqqqqqqγq ½È 2j£ ¤F0£¡¡¦¤F a }¡¤2" a F k 1¤¨]k 5 ££0 qqqq ½}Í Ïa . ?. h.
Documents relatifs