INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE
The Beltrami Flow over Manifolds Nir Sochen — Rachid Deriche — Lucero Lopez
N° ???? Juillet 2003 THÈME 3
ISSN 0249-6399
apport de recherche
!" $#&% '(*),+ - ."/1032547689."/1(;:=(;."/70?4A@B7.30C)D-E@-GFH0CI JLKNMPORQ,STVUXWZY$QP[\^]PY_a`^W?KN`^ObORQDcdOe\^]Kf_gWNQih _gOe\3jkQ"lmhonf`^WfWNp"Q"lmhN]"`^WfWo\3_al$l\3WN]"Q"l q[$`3rQPYtstnGuGl$l$p"Q vt\3wfw1`^[YnfQx[$Q"]$KNQP[$]$KNQyW z {3{C{3{|TV}k~f_ggaQPYtiiiSTS3Rwo\3jkQ"l |
"mCfkk UXW|Oe\3WuRORQ"nG_a]m\3]"`^Obwf~fY$QP[_al$_a`^W|Y\^lGlYKNQ [$QPaQPi\3WZYnN\3Y\,_alL\3YY\^]KNQ"nY$` \tlw7Q"]P_o]Y_al$l~NQLl~N]Ke\^lYKNQ]"`^a`^WR`^[YKNQL]"`^[Y$QDJLKf_all_gY~o\3Y_a`^WR]m\3gal`^[[$QPji~f*\3[_g\3Y_a`^W Y$Q"]KfWf_aZ~NQ"lKf_a]$K?\3[$QxnfQDNWNQ"n`mkQP[l~f[ d\^]"Q"lm¡¢Qt_gWZY[$`9nG~N]"Qx_gWYKf_alLwo\3w1QP[LYKNQx£LQPgY[\3Ob_ ¤ `¥`kQP[AOe\3Wf_`^anflHJLKf_al¦WNQP¢[$QPji~f*\3[_g\3Y_a`^WY$Q"]KfWf_aZ~NQ`mkQP[$]"`^ORQ"lYKNQ `kQP[ c§lOR``^YKf_gWfj `3YKNQ¨© ¤ `ª\3WNneYKNQlY\3_g[$]m\^l$_gWfjQD«Q"]PY$l`3YKNQ ¨¬ ¤ `mh9YKo\3YQP[$Q[$Q"]"QPWZYguel~fjijkQ"l$Y$Q"n 9_*\bYKNQKo\3[OR`^Wf_a]&Oe\3wEORQPYKN`9nflmJLKNQ,kQPu®`3¯`^~f[°\3wfwf[$`k\^]$K±_al N[$lYY$`|]P*\3[_²u?YKNQ,g_gWf ³ QPY§Q"QPW?YKNQ°_gWY[_gWNl_a]&´µ^¶¸·k¹3º^µ^»e¹k¼"½¾¿µ^À\3WNn®YKNQ°_gObwfg_a]P_gYxÁ¹^ÂDÃRµ^ÀN¾¿¼ÅÄPÀ1ÄD¿Æ^·ÇDÈ9À ¼"½¾¿µ^À ¹^¶ \3WNnYKNQPWÉ~Nl$QtYKNQtjkQ"`^ORQPY[_a]m\3A~fWNnfQP[$lY\3WNnG_gWfj|`3HYKNQx£LQPgY[\3Ob_AÊHa`ËY$`ÅjkQPWNQP[\3g_gmQx_gYY$` _gOe\3jkQ"l°`^WÌQDGwfg_a]P_gYgu±\3WNn_gObwfg_a]P_gYgu±nfQDNWNQ"n±WN`^W ¤ \3Y°l~f[ ¿\^]"Q"lU§Yx_al°lKN`WÌYKo\3Y°`^WN]"Q \3jZ\3_gWYKNQ£LQPgY[\3Ob_ ¤ `m=_gWZY$QP[w7`^*\3Y$Q"l ³ QPY§Q"QPWÅYKNQ¯¨©¯\3WNn,¨¬ ¤ `ml`^W,WN`^WGc ¤ \3Yl~f[ d\^]"Q"lm JLKNQ_gObwfaQPORQPWZY\3Y_a`^Wl$]$KNQPORQL`3oYKf_al ¤ `=_alwf[$Q"l$QPWY$Q"nb\3WNni\3[_a`^~NlQD9w7QP[_gORQPWZY\3f[$Q"l$~fgY$l ` ³ Y\3_gWNQ"n&`^Wb\l$QPY`3N^\3[_a`^~NlH[$Qm\39_gOe\3jkQ"lH_ggg~Nl$Y[\3Y$Q¯YKNQw1QP[ ¿`^[Oe\3WN]"Q"l`3NYKNQ\3wfwf[$`k\^]KR\^l QPg¦\^lLYKNQ°nG_«7QP[$QPWN]"Q"l_gYKÉYKNQxKo\3[OR`^Wf_a]°Oe\3w ¤ `mlmJLKf_al QD9Y$QPWNl_a`^W®`3YKNQ°£LQPgY[\3Ob_ ¤ `ÍY$`eYKNQ]m\^lQ,`3 WN`^W ¤ \3Ytl~f[ d\^]"Q"lt`^w1QPWNlWNQPÎw7QP[$lw7Q"]PY_gkQ"l_gWEYKNQ,[$QPji~f*\3[_g\3Y_a`^WÌ`3 WN`^_al$unN\3Y\bnfQDNWNQ"n?`^W®Oe\3Wf_`^anfl Ï®ÐkÑ7Ò$ÓÔ CÕk vQPji~f*\3[_g\3Y_a`^W¥`^W¢Oe\3Wf_¿`^anflmhH£QPgY[\3Ob_Êa`mh¦q`^guZ\3k`m=\^]PY_a`^WAhHÖx\3[ c OR`^Wf_a]Oe\3wNlmh ×9]m\3*\3[\3WNnEØ`^a`^[ _gOe\3jkQ°[$QPji~f*\3[_g\3Y_a`^WAhoÙt`^W ¤ \3Y l~f[ d\^]"Q"l
Unité de recherche INRIA Sophia Antipolis 2004, route des Lucioles, BP 93, 06902 Sophia Antipolis Cedex (France) Téléphone : +33 4 92 38 77 77 — Télécopie : +33 4 92 38 77 65
L Ë#Ú
1 % Û Î#&%ËÜÝ 7$Þ
Þ% ßà CáHâ à äã \3WNlwfg~Nl_aQP~f[$lY\^]$KNQ"lLnfQ9_al_a`^W|wo\3[`^[$nG_gWo\3Y$QP~f[mhGaQ"lnf`^WfWNp"Q"l¯_gObw7`^[Y\3WZY$Q"l l`^WZY¯nfpDNWf_aQ"ll~f[¯nfQ"lY_al$l$~Nl ]"`^ObORQaQ]"`^a`^W|`^~RaQ]"`^[Y$QDØQPYY$Q l_gY~o\3Y_a`^W|nfQPOe\3WNnfQnfQ"l Y$Q"]KfWf_aZ~NQ"lnfQ"l°[$pPji~f*\3[_al\3Y_a`^W=Z~f_ Y[\m^\3_ggaQPWYnG_g[$Q"]PY$QPORQPWY&l~f[°aQ"l,l~f[ ¿\^]"Q"l ã \3WNl,]"QPY \3[Y_a]PaQihWN`^~Nl wf[$p"lQPWZY$`^WNlaQ ¤ `^Y¯nfQ£LQPgY[\3Ob_ l$~f[¯nfQ"li\3[_apPY$p"lm ØQPYY$QWN`^~fkQPgaQY$Q"]KfWf_aZ~NQ nfQÅ[$pPji~f*\3[_al>\3Y_a`^Wål~f[wo\^l$l$Q&aQÅwf[$` ³ aMPORQRnfQÅ*\l$~f[ cL[$pPji~f*\3[_al>\3Y_a`^W¥nG~ ¤ `^Y¨©&QPYxdæçQD«QPY nfQè§Oe\3[$]$KNQ"l°næçQ"l]m\3g_aQP[ é&nG~ ¤ `^Yt¨¬"h7~f_HpPY\3_gYx[$p"]"QPObORQPWZY°9_*\eaQ"l ORpPYKN`9nfQ"l°nfQ"l]m\3[Y$Q"l Ko\3[OR`^Wf_a~NQ"lmê¦\]PaQDnfQWN`^Y[$Q \3wfwf[$`]KNQQ"l$Y næë\ ³ `^[$n|]P*\3[_oQP[¯*\x[$QP*\3Y_a`^WQPWZY[$Qdæa¹Z¼"½¾µiÀ ì Äe´µ^¶í·k¹3º3µ^»Ch_gWY[_gWNl$M"~NQihQPYbdæë\3wfwf[$`9]$KNQÉ_gObwfg_a]P_gY$Q?nfQ"l?¼$¹^ÂP½§Ä>îï9¹^ÂDÃRµ^ÀN¾¿ðPÈfÄDîDhQPYb\3wf[$M"l ~fY_gg_alQP[R*\±[$QPwf[$p"l$QPWY\3Y_a`^Wjkp"`^ORpPY[_a~NQ?nG~ ¤ `^YenfQ?£QPgY[\3Ob_w1`^~f[b*\ÌjkpPWNpP[\3g_al$QP[\3~G _gOe\3jkQ"lxnfpDNWf_aQ"lxnfQOe\3Wf_aMP[$Q_gWY[_gWNl$M"~NQ&QPYt_gObwfg_a]P_gY$QÅl~f[xnfQ"ll~f[ d\^]"Q"ltWN`^WGcdwf*\3WNQ"lm°sW OR`^WY[$QxZ~NQ *ñ,\3~Nl$l_dhaQ ¤ `^YLnfQ £LQPgY[\3Ob_l~f[nfQ"ll~f[ d\^]"Q"lWN`^W|wf*\3WNQ"l_gWZY$QP[w7`^aQxQPWZY[$Qt*\ WN`^[ORQ¨ © QPY*\°WN`^[ORQ¨ ¬ êAQ l]$KNpPOe\°W~fORpP[_aZ~NQxnfQdæ _gObwfapPORQPWY\3Y_a`^W®Q"lY¯wf[$p"lQPWZY$pihGQPY wfg~Nl$_aQP~f[$l°[$p"l~fgY\3Y$l,QDGw7pP[_gORQPWZY\3aQ"l&` ³ Y$QPWZ~NQ"l\Éwo\3[Y_g[,nfQ"l°_gOe\3jkQ"l[$pm\3aQ"l_ggg~NlY[$QPWZYaQ"l w7QP[ ¿`^[Oe\3WN]"Q"lnfQbdæë\3wfwf[$`9]$KNQRQPY°aQ"lnG_«pP[$QPWN]"Q"l\mkQ"]baQ"l ¤ `^Y$lnfQ"l]m\3[Y$Q"l,Ko\3[OR`^Wf_aZ~NQ"l ØQPYY$QQD9Y$QPWNl$_a`^WnG~ ¤ `^YnfQ¯£LQPgY[\3Ob_f\3~,]m\^lHnfQ"lHl~f[ d\^]"Q"l¦WN`^WGcdwf*\3WNQ"l`^~f[$QnfQ"l¦WN`^~fkQPgaQ"l w7QP[$lw7Q"]PY_gkQ"lnN\3WNlaQLY[\3_gY$QPORQPWZYnfQ"l nf`^WfWNp"Q"lO,~fgY_gi\3g~Np"Q"l¯l$~f[nfQ"l¯l~f[ ¿\^]"Q"lWN`^Wbwf*\3WNQ"l òÔ m Ò Zó à ® vpPji~f*\3[_al\3Y_a`^WEnfQ°nf`^WfWNp"Q"ll~f[ nfQ"li\3[_apPY$p"lmhNÊa`^YnfQ£LQPgY[\3Ob_dgh ô]PY_a`^W nfQ°q`^guZ\3k`m7hØL\3[Y$Q"lKo\3[OR`^Wf_a~NQ"lmhovpPji~f*\3[_al\3Y_a`^W?nfQnf`^WfWNp"Q"ll$]m\3*\3_g[$Q"l QPY nfQ,]"`^~faQP~f[ l$~f[nfQ"l_gOe\3jkQ"lWN`^W®wf*\3WNQ"l
õZö3÷ø÷úùçûüý>þÿ1ù>÷Xü ý Cÿ Dù
S
1°#ÅÛ¦ 1$ ¡¢QRKo\"kQl$Q"QPW¢_gW¢[$Q"]"QPWYukQm\3[$lÅ\3WåQDGwo\3WNl_a`^Wå_gW¢YKNQe~Nl$QR`3nG_«QP[$QPWZY_*\3jkQ"`^ORQPY[uå\3WNn ]m\3a]P~fg~Nlt`3^\3[_*\3Y_a`^WE¿`^[^\3[_a`^~Nlwf[$` ³ aQPORl_gW?_gOe\3jkQwf[$`]"Q"l$l$_gWfjNho]"`^Obwf~fY$QP[t_al_a`^W\3WNn ]"`^Obwf~fY$QP[&ji[\3wfKf_a]"lmU§W¥wo\3[Y_a]P~f*\3[mh`^WNQ|]m\3W=WN`^Y_a]"Q|YKo\3YYKNQewf[$` ³ aQPO `3[$QPji~f*\3[_g"_gWfj WN`^_al$uxnN\3Y\ nfQDNWNQ"n,`^WWN`^W ¤ \3Ymh3_gObwfg_a]P_gYH`^[H_gWZY[_gWNl_a]ihil~f[ d\^]"Q"lAKo\^l ³ Q"QPWY\^]$9aQ"n_gYKY§` ~fWN]PY_a`^Wo\3alKf_a]$K`^w1QP[\3Y$Q`^WeYKNQ l$wo\^]"Q `3AQPO ³ Q"nfnG_gWfjOe\3wNl`3Av_aQPOe\3WfWf_*\3WOe\3Wf_`^anfl lQ" Q A`^[WN`^WGcdi\3[_*\3Y_a`^Wo\3\3wfwf[$`k\^]K Ð tÔ ó Ñ # " Ô $ fi % Ô & U§Y L\^l_gWZY[$`9nG~N]"Q"n|_gW|YKNQ £LQPgY[\3Ob_ ²[\3ORQP`^[' )(+*Gh (+,fh ! (i.h *G/h ,0§tU§Y~Nl$Q"l°\Ra`9]m\3\3WNn?_gWY[_gWNl_a]Dcdwo\3[\3ORQPY[_a]RnfQ"l$]P[_gwfY_a`^WÌ`3¯YKNQOe\3Wf_`^anfl \3WNnÉY[$Qm\3YYKNQ°ORQPY[_a]\^l \ÅnGuWo\3Ob_a]^\3[_*\ ³ aQi Ð 1 f3â Ô & %;®âå# 3H ¯v Q"]"QPWYgu~Nl$Q"nÉY$`RnfQPWN`^_al$Q,_gOe\3jkQ"l `^W®Oe\3Wf_¿`^anfl )(Z4h (mfh 2 Nhf h (+56§UXYx~Nl$Q"l°\3WÌ_gObwfg_a]P_gY,nfQ"l$]P[_gwfY_a`^W±¿`^[tYKNQbl$~f[ ¿\^]"Q&KNQP[$QÅYKNQÅWN`^_alunN\3Y\ \3[$Q]"`^WNlY[\3_gWNQ"nÉY$`Rg_gkQi JLKNQ°ORQPY[_a]°_al \&wo\3[\3ORQPY$QP[t`3YKNQwf[$`]"Q"l$l UXYL_alYKNQ\3_gO `3HYKf_alLwo\3w7QP[Y$`&N[$l$YL]P*\3[_ueYKNQx[$QP*\3Y_a`^W ³ QPY§Q"QPW®YKNQx_gWZY[_gWNl$_a]qH`^guZ\3k` \^]PY_a`^WÉ`3YKNQt£LQPgY[\3Ob_1[\3ORQP`^[7 )(+*G.h (+,Ghf- (i#h *G#h ,6\3WNnYKNQt_gObwfg_a]P_gYKo\3[OR`^Wf_a]°QPWNQP[jiu ~fWN]PY_a`^Wo\38 )(i9h (mGh7NhAf h (+50 ÅUXYx_alx`^~fWNn±YKo\3Y&\3gYKN`^~fjiK¢YKNQ&²~fWN]PY_a`^Wo\3al&\3[$Q ³ \^l_a]m\3ggu YKNQÌl>\3ORQihxYKNQP[$Qå\3[$Q¢nG_«QP[$QPWN]"Q"l®_gWËYKNQÌL\"uª^\3[_a`^~Nl®wf[$` ³ aQPORl\3[$Q¿`^[O,~f*\3Y$Q"n\3WNn ]"`^WNlQ"Z~NQPWYgu_gW5YKNQEL\"uÚYKNQ®~fWN]PY_a`^Wo\3al®\3[$QÌ\3wfwfg_aQ"n ×w1Q"]P_o]m\3ggu ¿`^[YKNQ]m\^lQE`3 nfQPWN`^_al$_gWfj_gOe\3jkQ"lb`^WÝWN`^WGc ¤ \3Ybl~f[ d\^]"Q"lÅYKNQP[$QÉ\3[$QnG_«QP[$QPWN]"Q"lÅ_gW=YKNQnfQDNWf_gY_a`^W`3YKNQ Oe\3Wf_¿`^anfl\3WNnÉ`3YKNQQPO ³ Q"nfnG_gWfje²~fWN]PY_a`^WNlm :tl~o\3ggu&Q Ko\"kQihk_gWe^\3[_a`^~Nl¯wf[$` ³ aQPORl¯_gWb9_al_a`^WAh\3We~fWNnfQP[g_gWNQOe\3Wf_¿`^anh ¤ \3Y¯`^[]P~f[kQ"nh \3WNnYKNQÉ¿Qm\3Y~f[$Q"l\3[$QEnfQDNWNQ"nÚ`kQP[|YKf_aleOe\3Wf_`^anÎô YXuwf_a]m\3xl_gY~o\3Y_a`^W5_alRKNQPWÚYKNQ _gOe\3jkQe[$Q"]PY\3WfjiaQe_al,YKNQR~fWNnfQP[gu9_gWfj ¤ \3YOe\3Wf_`^an=\3WNn¥\3Y&Qm\^]$Kåwf_fQP¯Q|\^l$l$_gjiWåi\3g~NQ"l l$~N]$K¢\^lx_gWZY$QPWNl_gYXukh¦]"`^a`^[mh7ji[\^nG_aQPWY^\3g~NQih7ji[\^nG_aQPWY,nG_g[$Q"]PY_a`^WAhOR`^Y_a`^WÌkQ"]PY$`^[mhAnG_al$wo\3[_gY§u kQ"]PY$`^[mhZY$QD9Y~f[$Q]Ko\3[\^]PY$QP[_alY_a]"lQPY$]iJLKf_al_al¯Qm\^l_ggunfQ"l$]P[_ ³ Q"nbOe\3YKNQPOe\3Y_a]m\3gguR\^l¯§ ¦
(
0¨z©«ª¬
¡¢QQDGY$QPWNnZÚ Y$`¡F~ ®~ l$~N]$KbYKo\3Y¯ _al YKNQL[$Q"lY[_a]PY_a`^We`3i5Y$`xJLKNQ Á¹^ÂDÃRµ^ÀN¾¿¼ ÃR¹ [ îL\3wfwf[$`k\^]$° K )(i±h (mGhGohN6A_alY$`Ra``^|`^[YKNQxOb_gWf_gOb_gmQP[t`3YKNQ°`^ga`_gWfj|QPWNQP[jiu ©Á  g ².³µ´B¶=· ¾
©
Á
½)½
Á
¦
¬Á¦ ©Á¦
¬
1µ¼DïGÄDÀ#V9WÄDÂD¾¿¼DïGÄXZY¦µ[fÄ]\ 5
U
ô gYKNQbQD9wf[$Q"ll_a`^WNl\ ³ `kQe\3[$QÅ]"`^WNl_anfQP[$Q"nÌ_gWÌYKNQbl$QPWNlQ&`3LnG_alY[_ ³ ~fY_a`^WNlmÅÙt`^Y_gWfjÉYKo\3Y ji_gkQPW¢\|l~f[ d\^]"Q,_gYKÌ\|~fWf_gYWN`^[Oe\3kQ"]PY$`^[ÕÓ h1YKNQÅ`^[YKN`^jk`^Wo\3wf[$`3rQ"]PY_a`^W±`^w7QP[\3Y$`^[ Ê ÍÌ `^WÉYKo\3Yl~f[ ¿\^]"Q_alLji_gkQPW ³ uÖn § gS ÊBÌm©×ÆØ ÓÓ KNQP[$Q × _alYKN¥Q ÙÀÚ2ÙRUúnfQPWZY_gY§uEOe\3Y[_E\3WNÛn Ó nfQPÒWN`^Y$Q"lYKNQY[\3WNlw1`ilQ"n?`3 YKNQkQ"]PY$`^Ç[ Ó1 JLKNQ°²~fWN]PY_a`^Wo\3¦]m\3W?[$QP[_gYY$QPWE\^lL¿`^ga`mlÔn Ò ¤ Á Ü ² ³µ´B¶ · Ü
à>7}il
.F
i
× ~fwfw7`il$QLQLKo\"kQ \xCc§nG_gORQPWNl_a`^Wo\3NOe\3Wf_¿`^an!¥_gYKba`9¦ ]m\3¬ N§ ¦ ]"`© `^§ [$¦ nG_gWo\3Y$Q"lá ¬ § á © QPO ³ Q"nfnfQ"n _g¦ WÌ\3WâÙCc§nG_gORQPWNl_a`^Wo\3Oe\3Wf_¿`^an7ã _gYK]"`9`^[$nG_gWo\3Y$Q"l h1YKNQ&QPO ³ Q"nfnG_gWfjÉOe\3w ³ ³ l äã _alji_gkQPW?QDGwfg_a]P_gYgu uYKNQS&~fWN]PY_a`^WNl`3 Å^\3[_*\ aQ"l ¦ ¤ ¤¦ ¤ ¬§ © ¬ á ¬ § á © ¨ § ¦ © ¤ á ¬ § á © ¨ § ¦ ¤ á ¬ § á © ¨J¨8¬ á á ¨Ø wyxzAiw|{
õZö3÷ø÷úùçûüý>þÿ1ù>÷Xü ý Cÿ Dù
¦
tÙ `^Y$QtYKo\3YYKNQ ³ l\3[$QxYKNQ°l$`^g~fY_a`^W®`3¤¦ å ¬ (¬D§¯¦ ô © ji[\"ucd© i§ \3¦ g~NQx_gOe\3jkQ°¤ _alL¬ [$§ QPwf© [$Q"l$QPWZY$Q"nho_gW © á © á YKf_alt[\3ORQP`^[7hH\^lxYKNQbQPO ³ Q"nfnG_gWfj ©Ôæ á á ¨J¨ l$Q"QÅÊ_gjN (D ds 2= gij d σ id σ j = dx 2 +dy2 +dI 2 σ2 d σ2
dI
ds
dσ
1
σ1
I y x
dy
dx
Ê_gji~f[$QÀ(FnJLKNQxji[\3wfKE`3YKNQx_gWZY$QPWNl$_gY§uÉOe\3wE\^l \Ål$~f[ ¿\^]"Q°QPO ³ Q"nfnfQ"n?_gWf~ u ¤
¤
ã QPWN`^Y$Q ³ u § ç ¨ YKNQx_gOe\3jkQxOe\3Wf_¿`^an®\3WN¦ n_gY$lORQPY[_a],\3WNn ³ u ã §è ¨ YKNQ°lwo\^]"QDc¿¿Qm\3Y~f[$Q Oe\3Wf_¿`^an?\3WNnÉ_gY$lORQPY[_a]ihfYKNQPW®YKNQ°Oe\3w l äã Ko\^lYKNQx¿`^ga`m_gWfj|QP_gjiKYn ¦ ¦!í ¤¦ ³ § ç Á ¬ Á ©+ê ççë0ì è í ,g ¨ ².³µé · ¸ ©Þ¹ ³ á á ë ì Ð
Ð
KNQP[$Q ç _alYKNQEnfQPY$QP[ObÒ _gWo\3WY®`3Y§KNï Q?_gOe\3jk§ Q§ORð QPY[_a]ih §óç ò ë6ì _al§ YKNQE_gWZkQP[$è l$Qí `3,YKNQE_gOe\3jkQ ORQPY[_a]ihYKNQ[\3WfjkQ`3 _gWNnG_a]"Q"l&_alÆî © Ý ¬)¬ \3WNnhñ © Ý ¬)¬ Ù\3WNn ³ _al&YKNQ|ORQPY[_a] `3tYKNQÉQPO ³ Q"nfnG_gWfj¥lwo\^]"Qi¥¡¢Q~Nl$QYKNQÉl~fObOe\3Y_a`^WÚ]"`^WZkQPWZY_a`^W±n?_gWNnG_a]"Q"lbYKo\3Ye\3wfw7Qm\3[ YX_a]"Q°\3[$Q ³ QP_gWfjbl~fObORQ"n`mkQP[m JLKf_al~fWN]PY_a`^Wo\3\^lN[$lYLwf[$`^w7`il$Q"n ³ ueq`^guZ\3k`m_gWYKNQ ]"`^WY$QD9Y`3Kf_gjiK?QPWNQP[jiuwfKu9l_a]"Ô l )(PF ¯ ¤ ê¦QPY~Nl `^[O~f*\3Y$Q&YKNQq`^guZ\3k`mÌ\^]PY_a`^W±¤ _gW?§]Oeõ \3Y[_®¿`^[fO n §ô ¨ _alYKNQ,_gOe\3jkQ&Oe\3Wf_¿`^an \3WNn¥_gY$lORQPY[_a]\^l ³ QD¿`^[$QiE×_gOb_g*\3[gukh ã ¨ _alYKNQelwo\3Y_*\3c¿¿Qm\3Y~f[$QOe\3Wf_`^anÝ\3WNn¥_gY$l ORQPY[_a] _gYe_al|l_gObwfgu ~ _gYK5YKNQ®~Nl~o\3ù
³§
¦
Ðø0ú
³¨ Ò
©
Ð
¿
¨
Ò
Ð
¿
1µ¼DïGÄDÀ#V9WÄDÂD¾¿¼DïGÄXZY¦µ[fÄ]\ *
JLKNQOe\3w
¦
U û äã
Ko\^l\ÅQP_gjiKY ²u·
¦
³}§ô§]õ
Á
¸ ©Þ¹
á
¬Á á +© ê ç ¦J [ ¤ö
¤
õ ¨
§
KNQP[$Q ç ©ýüeþ ÿ ô ¨ JLKNQji[\^nG_aQPWZYnfQ"l]"QPWZY Q"~o\3Y_a`^WNlmhf_gYK®[$Q"l$w1Q"]PYY$`bYKNQ°QPO ³ Q"nfnG_gWfje²~fWN]PY_a`^WNlho\3[$Q ¦ ¤ ¦ ¤ ¦ í ¤ ¦ ê çÜç ë6ì ³ ³ ç ë6ì í³ (mg © ê Ýç ë ¨ ë ¨ ì ¨ ¬ ì Ð Ð Ð Ð ¦ ¦ ¦ UXW¥`^~f[Å]m\^l$Q ¬ \3WNnÒ © \3[$QRffQ"nÝ\3WNnåQ|`^Wfgu¢]Koí \3WfjkQ ©kæ &EJLKNQ|QPO ³ Q"nfnG_gWfj l$wo\^]"Qx_aul å~N]Pg_anfQm\3W\3WNn®YKNQP[$QD`^[YKNQ,êAQP_cúØ_g9_gY\ ]"`Qe]P_aQPWY$lt\3[$QxmQP[$`f JLKNQ ¤ `mhf_gW®Oe\3Y[_¿`^[O®hf_alYKNQP[$QD¿`^[ à ² ê çeô÷¬ (F( © ê Ý ç nG_g ¾ ¬ ©ÄØ ð ê Ý ç Ã Ò JLKNQ|QD9Y$QPWNl$_a`^W¢¿`^[WN`^WG
c å~N]Pg_anfQm\3W QPO ³ Q"nfnG_gWfjl$wo\^]"QR_alY[$Qm\3Y$Q"nå_gW )(+*fh (+,GBh *Gh¦ifhH- ( § Ùt`^Y$Q±YKo\3Y®Q¢nG_anGWAæ Y?lw7Q"]P_²u5YKNQ±ORQPY[_a]ÌukQPYm U§ÅQ¢]KN``il$Q¿`^[?\ÝORQPY[_a]ÌYKNQÌN[$lY ~fWNnN\3ORQPWZY\3H¿`^[Oä`3YKNQ ³ \^l$QihfWN`^WGc ¤ \3Ymh Oe\3Wf_`^an®YKNQPW?YKf_al ¤ `Î_al YKNQ,\3Wo\3a`^j|`3 å 1 `gDÌ£L`^YK Q"~o\3Y_a`^WNlbnfQ"l$]P[_ ³ Q\ |ó % &HÐ f Ô 3 Ð CG Ô YKo\3YR\^]PY$lÅ`^W=YKNQ_gOe\3jkQ|~fWN]PY_a`^W Å?sWNQeOe\Cr`^[nG_«QP[$QPWN]"Q ³ QPYXQ"QPWÝYKNQe£LQPgY[\3Ob_²[\3ORQP`^[¥\3WNn¢YKNQ|Ko\3[OR`^Wf_a]|Oe\3w ¿`^[O,~f*\3Y_a`^Wå_alYKNQÅL\"u±`^WNQÅ]$KN`9`il$Q&YKNQÅORQPY[_a]i,UXWYKNQ&Ko\3[OR`^Wf_a]bOe\3w¢\3wfwf[$`k\^]K¢_gYx_al Y[$Qm\3Y$Q"n¢\^l°nN\3Y\Kf_a]KÌ_alxji_gkQPW ³ QD¿`^[$QPKo\3WNn&JLKNQÅORQPY[_a]ÅYKNQPWÌ_al Ô & "m & ¯ % & %â Ð UXWYKNQ&£LQPgY[\3Ob_[\3ORQP`^[ Ð â Ð m> %²7 %²=Õ iÑ & Naâ %² $ f0 %²f
ó Ð Ô
ÉÐ ?§Ô á & ÕË
Ñ â % & %aâ % % &Bc l h Ð ? á & i % Ô & Nó ¢ØL\3[[u_gWfj¢`^~fY&YKNQ]m\3a]P~f*\3Y_a`^W QNWNn¥YKNQe¿`^ga`m_gWfj Q"~o\3Y_a`^W l$Q"Q )(+ *>`^[YKNQnfQP[_gi\3Y_a`^W Rn ¦ ¦@í ¦ ¦@í à ² ³ ³ ç ç è í (mg Ð ³ Ð ë Ð ì è ³ í ©«ª § ê Ý ç à ç ë6ì ©ÄØ ðÝ ë6ì Ð ô ³ QP_gYYKNQ]"`^Obwfg_a]m\3Y$Q"n±\3wfw1Qm\3[\3WN]"Q&`3YKNQ,Ob_gWf_gOb_g\3Y_a`^WQ"~o\3Y_a`^W %;eZ &
Ð Ô ó $1Ð Õ & Nó Ñ %²kNóó Ñ ¯JLKNQ°[$Q"l~fgY_all_gObwfgu ç
ë0ì ©
Ð
ë
¦
³
Ð
ì
¦ í
è í ³ ¬
JLKf_alQ"~o\3Y_a`^WeKo\^lL\]PaQm\3[¯jkQ"`^ORQPY[_a]tORQm\3Wf_gWfjenUXY _al YKNQ_gWNnG~N]"Q"n|ORQPY[_a] `3AYKNQ lQ"]PY_a`^W UXWeYKNQtl_gObwfaQ"lYLl_gY~o\3Y_a`^W`3 ¤ \3YLnf`^Oe\3_gW|_gYK®\l]m\3*\3[¯oQPan|_dçQiYKNQ _gWY$QPWNl_gY§ukhG_gY_al¯NWNn wyxzAiw|{
õZö3÷ø÷úùçûüý>þÿ1ù>÷Xü ý Cÿ Dù Y$` ³ Q)(+,0ûn
,
© © ¬ (mSg Ý Ñ KNQP[$QÀ nfQPWN`^Y$Q"lYKNQ_gWY$QPWNl_gY§u?aQPkQPdUXW®YKNQ£QPgY[\3Ob_H[\3ORQP`^[7hoYKNQP[$QD¿`^[mhNYKNQ,Oe\3WGc _¿`^an=`3 _gWZY$QP[$Q"lYmh\3WNn=YKNQ|ORQPY[_a]ih _alY\3kQPWÝY$` ³ Q Ð ÕGmËáH ? f б_dçQi±YKNQji[\3wfK `3YKNQ|_gWZY$QPWNl_gYXuÌ~fWN]PY_a`^WA®U§W=`^YKNQP[,`^[$nflQ\3[$Qe_gWZY$QP[$Q"lY$Q"n¥_gW=YKNQelY[~N]PY~f[$Q`3YKNQ lQ"]PY_a`^W±`3YKNQ, ³ QP[ ³ ~fWNnGaQR\3WNnWN`^Yt_gWYKNQ ³ \^l$Q&Oe\3Wf_`^an,U§YxORQm\3WNlxYKo\3YxYKNQ&ORQPY[_a] _gY$lQPnfQPw7QPWNnfl¯`^WeYKNQ nN\3Y\GJLKf_al _al¯YKNQ\mubWN`^WGcdg_gWNQm\3[_gY_aQ"l\3[$Q_gWZY[$`9nG~N]"Q"n|_gWRYKNQ ¤ ` å¯Z~o\3Y_a`^W (F(DhN_gYK®YKNQ°_gWNnG~N]"Q"n®ORQPY[_a ] å (mgS DhfKo\^l¦ \Å]PaQm\3[jkQ"`^ORQPY[_a],ORQm\3Wf_gWfjN¯UXY _alYKNQxwf[$`3r Q"]PY_a`^W`3YKNQxORQm\3W]P~f[^\3Y~f[$Q°kQ"]PY$`^[ _gWÉYKNQ © nG_g[$Q"]PY_a`^±W n (PÜ © "!$ # KNQP[$Q _alYKNQxORQm\3W®¦ ]P~f[^\3Y~f[$QxOe\3jiWf_gYÒ ~NnfQi ! _alYKNQxWN`^[Oe\3Y$`ÅYKNQxl~f[ d\^]"Qx\3WNn&Í% _al \°~fWf_gY¯kQ"]PY$`^[¯_gWeYKNQ © ªnG_g[$Q"]PY_a`^W l$Q"Q Ê_gjNg DJLKf_al¯]$KN`^_a]"Qt`3YKNQ_gWNnG~N]"Q"n|ORQPY[_a] ji_gkQ"l\l$_gObwfaQÅjkQ"`^ORQPY[_a]bORQm\3Wf_gWfj®Y$`YKNQ&²~fWN]PY_a`^Wo\3\^lxQPgdÅU§Y ³ Q"]"`^ORQ"ll_gObwfguEYKNQ k`^g~fORQ|`3YKNQel$Q"]PY_a`^W \3[$Qm\É`^[,aQPWfjiYK¢¿`^[YKNQbY§`3cx\3WNn¥`^WNQDcnG_gORQPWNl_a`^Wo\3]m\^lQ"l DeUXY ¤
ç
ë0ì ¨z©ÏÎ Ý
Z
(KN)Z
KN
Y
U(X,Y)
X
Ê_gji~f[$Q&-n JLKNQ£LQPgY[\3Ob_ ¤ `mkQPa`]P_gYXu?\^lt\Rwf[$`3r Q"]PY_a`^W±`3 YKNQORQm\3W±]P~f[i\3Y~f[$Q`^WEYKNQ nN\3Y\('Í,\C9_al
ABADCFE#GHJIKMLONPRQST
m
1µ¼DïGÄDÀ#V9WÄDÂD¾¿¼DïGÄXZY¦µ[fÄ]\
(
U
_alL_gObw7`^[Y\3WZY `^[YKNQ°l$Y~NnGu ³ QPa`m Y$`ÅKo\mkQ°YKNQQDGwfg_a]P_gYL`^[O `3HYKNQx~fWN]PY_a`^Wo\3alL¿`^[ ¤ \3Y nf`^Oe\3_gW?_gW®YKNQ°l$]m\3*\3[\3WNnÉkQ"]PY$`^[_*\3H]m\^l$Q"l+n Á*. ) Á,-+ ê ç © ¹ Á*).Á*+.- æ © ½)½ ¾ ½)½ © ² l$]m\3*\3[ · `F,C}c >*iG ë6 &J°9ØKo\3WÉ\3WNn|}Nf×KNQPWAhk0è ¯\3[_*\3Y_a`^Wo\3[$Q"lY$`^[\3Y_a`^W`3AWN`^WGc ¤ \3YL_gOe\3jkQ Qm\3Y~f[$Q"l n8`9nfQPal \3WNn®\3gjk`^[_gYKfORl éh1×9UúôÐ8 }Nfôwfwfd8±\3YKAg=h 5-( iiig Õ(mSiFS *CMc (mFS 5-(i wyxzAiw|{
õZö3÷ø÷úùçûüý>þÿ1ù>÷Xü ý Cÿ Dù
SiS
ëS bØtLØKNQD¿næ KN`^Y$QPdh ã J¦]$K~fObw1QP[aQihxv ã QP[_a]$KNQih°s&Ê \3~fjkQP[\^lmhLè Ø`^WNlY[\3_gWNQ"nªÊa`ml `3 8±\3Y[_9 c ¯\3g~NQ"nÚÊf~fWN]PY_a`^WNlnôwfwfg_a]m\3Y_a`^W5Y$` ã _«1~Nl$_a`^WJ¦QPWNl$`^[v QPji~f*\3[_g\3Y_a`^Wö^h ´Â µ¼Ä$Ä ì ¾²ÀÆ3î,µ ,Ç Im½²ï,JÈ9 µR[fĹ^ÀLKµ^ÀmÇmÄDÂÄPÀ ¼Ä°µ^À+KµiÃ8[1È9½§ÄDÂNM¾;î>¾¿µ^ÀohØ`^w7QPWfKo\3jkQPWAh ã QPWGc Oe\3[1h58±\"u®iii > &ê ØKNQPWfjNhqL£~f[$]Ko\3[$nhx£t£8QP[[_gOe\3W \3WNnË×1stlKNQP[mhèè8E`^Y_a`^W `3bØ~f[kQ"l?Ø`^WGc lY[\3_gWNQ"n|`^W®×~f[ d\^]"Q"l :tl_gWfjb\,ê¦QPkQPA×9QPYLôwfwf[$`k\^]$Kö^ho×9QPwfY$QPO ³ QP[ iiiÆ :xØêAôØôÐ8 J¦Q"]$KfWf_a]m\3v QPw7`^[Y iCcXSig D `68¥CØaQP[$]ihZs&CÊo\3~fjkQP[\^lmhivL`°QP[_gkQPWAhi}N `xu ³ _a]ih^J°3q\3wo\^nf`^w7`^~fa`fh3èXôÝaQPkQPlQPY¦ORQPYKN`9n `^[bYKNQÉ_gWkQP[$l$fQ åå à pÛ8 å à wf[$` ³ aQPOBö^hLq`ilY$QP[|Ùt`f) nÉ(m- (mSiSGhLÙQP~f[$`iUXOe\3jkQ®Ö~fOe\3W £[\3_gW:8\3wfwf_gWfj|iiiO8EQ"QPY_gWfj 56&v, ã QP[_a]KNQÅ\3WNn¢s&1Êo\3~fjkQP[\^lmhoèXêAQ"Õ l å ã qªQPW±JA[\3_gY$QPORQPWZYnfQ"ltUXOe\3jkQ"ltQPOY t_al_a`^WEwo\3[ s[$nG_gWo\3Y$QP~f[ éh UúÙvUXôÍv Q"l$Qm\3[$]KEvQPw1`^[YF 5F,g9hoÙ`1B (+,F,F`GNô al`e\3wfw1Qm\3[$Q"n?_gW?JA[\3_gY$QDc ORQPWZY nG~×_gjiWo\3dih (mS 5g¾²ÀÆ `F, `gFS 5F`CcXgS ^o (+,F,g9 *6&v,À`x_gObORQP°\3WNnv,98±\3g*\^nG_t\3WNnÙ×9`9]$KNQPWAhèXUXOe\3jkQ"l|\^! l åO ³ Q"nfnG_gWfj8\3wNl\3WNn 8?_gWf_gOe\3,×~f[ d\^]"Q"+l n8E`_aQ"lmhxØ`^a`^[mhxJ¦QDGY~f[$Qiht\3WNU n `^g~fORQPY[_a]_8Q"nG_a]m\3°U§Oe\3jkQ"l ö^h U§WY$QP[Wo\3Y_a`^Wo\3H}k`^~f[Wo\3`3 Ø`^Obwf~fY$QP[ _al_a`^W?FS , g iiig Ç(F(F(DMc (mF ,G ,6&Ùf×9`]KNQPW?\3WNnv,`x_gObORQPdhè ×Y$QP[$Q"`^ji[\3wfKf_a]s[_aQPWZY\3Y_a`^W ã _«7~Nl_a`^WGéh9_gWwf[$`]"Q"Q"nG_gWfjkl `3YKNQxiYK?UXWZYm1Ø`^WG f`^W×9]m\3aQDcú×wo\^]"Qi h ¯\3WN]"`^~fkQP[xØL\3Wo\^nN\Gh7st]PY$` ³ QP[ii- (i )(m6 &qHS``^[Wfwf[$` ³ lYmhv, ã QP[_a]$KNQih\3WNn à ô~ ³ QP[YmèXÙt`^Wfg_gWNQm\3[¥`^w7QP[\3Y$`^[$l¢_gW _gOe\3jkQ [$Q"lY$`^[\3Y_a`^WGé¯UXWq[$`9]"Q"Q"nG_gWfjkl`3tYKNQEU§WY$QP[Wo\3Y_a`^Wo\3°Ø`^WG¿QP[$QPWN]"QE`^WªØ`^Obwf~fY$QP"[ t_c l_a`^WR\3WNnbq\3YY$QP[Wv Q"]"`^jiWf_gY_a`^Wbcwo\3jkQ"l¯SiF `CcXSi-S (iç}U ååå Ø`^Obwf~fY$QP[L×9`9]P_aQPY§uk×G\3WR}k~o\3WAh q~NQP[Y$`Rv_a]"`fhf}k~fWNQ (+,F,g9 )(F(R¡B åHê¦`^[$QPWNl$QPWAhÖB åØg_gWNQihHèè8\3[$]Kf_gWfj±]P~ ³ Q"+l nbô Kf_gjiK=[$Q"l$`^g~fY_a`^W S ã l~f[ d\^]"Q ]"`^WNlY[~N]PY_a`^WE\3gjk`^[_gYKfOÅéhØ`^Obwf~fY$QP[ à [\3wfKf_a]"lmho- ( Ü DhNwo\3jkQ"l (+5iS È(+5F,Gih (+,F*g9 )(m6 &Ê8QPOR`^g_7\3WNn à G×G\3wf_g[$`Å\3WNn®×1fstlKNQP[mhZè ×9`^g_gWf, j ¯\3[_*\3Y_a`^Wo\3Aq[$` ³ aQPORl\3WNnq\3[ c Y_*\3 ã _«7QP[$QPWY_*\3_ å ~o\3Y_a`^WNlmhþ8±\3wfwf_gWfjå_gWZY$` à QPWNQP[\3J\3[jkQPY(8±\3Wf_`^anflLö^hL}Z\3W~o\3[u iii :xØê¦ô Øô8 JHQ"]$KfWf_a]m\3v QPw7`^[Y iCcX3Ü D )(m6S &}k`^KfWËÙt\^lKAh öPJLKNQE_gO ³ Q"nfnG_gWfjÝwf[$` ³ aQPO ¿`^[v _aQPOe\3WfWf_*\3WªOe\3Wf_`^anflLö^hô WfWo\3al`3 8\3YKNQPOe\3Y_a]"lmh 5iS (+,F5F`gDhfwfwEiCc 5iSG 6
ABADCFE#GHJIKMLONPRQST
S3
1µ¼DïGÄDÀ#V9WÄDÂD¾¿¼DïGÄXZY¦µ[fÄ]\ U
P &ô*8=kq`^guZ\3k`m7hèWV~o\3WZY~fO jkQ"`^ORQPY[uR`3 ³ `il`^Wf_a] lY[_gWfjkl éhG´ïZ·3î>¾¿¼Dî YHÄP½¿½§ÄDÂ>îDhYX)Z\[ ] (+,F*-(ig cX (mG )(+`6&ê Av~NnG_gWAh×1HstlKNQP[,\3WNâ n åÊ \3Y$QPOb_dhèRÙ`^W¢êA_gWNQm\3[J¦`^Y\3¯\3[_*\3Y_a`^W¥£\^l$Q"nÌÙt`^_al$Q v QPOR`mi\3¦ôgjk`^[_gYKfORl éh ´ï·CîD¾¼¥¹ W_^A`baWced&d
wyxzAiw|{
Unité de recherche INRIA Sophia Antipolis 2004, route des Lucioles - BP 93 - 06902 Sophia Antipolis Cedex (France) Unité de recherche INRIA Futurs : Parc Club Orsay Université - ZAC des Vignes 4, rue Jacques Monod - 91893 ORSAY Cedex (France) Unité de recherche INRIA Lorraine : LORIA, Technopôle de Nancy-Brabois - Campus scientifique 615, rue du Jardin Botanique - BP 101 - 54602 Villers-lès-Nancy Cedex (France) Unité de recherche INRIA Rennes : IRISA, Campus universitaire de Beaulieu - 35042 Rennes Cedex (France) Unité de recherche INRIA Rhône-Alpes : 655, avenue de l’Europe - 38334 Montbonnot Saint-Ismier (France) Unité de recherche INRIA Rocquencourt : Domaine de Voluceau - Rocquencourt - BP 105 - 78153 Le Chesnay Cedex (France)
Éditeur INRIA - Domaine de Voluceau - Rocquencourt, BP 105 - 78153 Le Chesnay Cedex (France)
{}|~|