An analytical framework for the numerical homogenization of monotone elliptic operators and quasiconvex energies
Texte intégral
(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. An analytical framework for the numerical homogenization of monotone elliptic operators and quasiconvex energies Antoine Gloria. N° 5791 – version 2 version initiale Décembre, 2005 – version révisée Janvier, 2006. ISSN 0249-6399. ISRN INRIA/RR--5813--FR+ENG. Thème NUM. apport de recherche.
(3)
(4)
(5) !" $#%&('
(6) #%)*+
(7) ,-!
(8) .%! /+
(9) 01
(10) 23045 !6"7 8 '69
(11) :;<+$*
(12) 2+6 =%>@?ACBD>E4FHGDACIJBLK MNPO2Q(RSUTWV/XZY\[^]`_aO2Q(Rb]+c\dPQ(ebfhgjikdPRb] lmfanpo`R2_qVsrhtVHuUt vqwyxPxzn{f|_q}^R!faRb~NR2f~NPRc k PyRbfa]hgn{c X{R2f]|gnyc%gcPg_hgjwpRWeb~RbQPfaRyyykX{R2f]|gnyc fae2\gj]|ebR5kwpc\\gR2fJ\yyy)y(xwyyRJ] ∗. †. 1$b2 P yy¡ u¢ckdQR2fny£-Q(R_aNPn^}P]Nw{R)R2Rbc¤xPfhn{xznk]|RJ}¥gc¦_hNPR§faRb~2R2ck_[yRJwpf]W_hn¨xzRbf|£©n{fhQ/_hNPR c\dPQ(Rbfhgj~2wyqNPnyQ(n{yR2cgªJw _hgnyc«ny£1¬©xn{]a]|gP[cPn{cPgcPRJwpf ®(R2gx^_ag~¥nyxR2fw _anyf]2¯°MNPRb]hRHQ(R_aNPn^}P]wpfaR dz]|dwy[¤}^R±cRb}²wp__hNR%}^gj]h~2fhR2_hR%RbyR2L¯¥Vsnk]`_)np£+_hNR2Q³~n{Q(xPd^_hR%wHc\dPQ(R2fag~bwp-nyxR2fw _hn{fb~n{]hRy gc¤w]hR2cz]|R _hnsR(Q§wy}^R§xPfaRb~2g]hRy´_hn_hNPR§NnyQ(ny{R2cPgª2RJ}HR2gx^_ag~(n{xzRbfawp_hn{fU£©n{f5_hNPR§xPfanyRbQ¯§MNPR xdPfhxn{]hRqny£@_hNPRWxPfhRJ]|Rbck_¶µ-n{fh· gj]m_an~jwpfag¸£©[ _aNPRW~2nyc]|_hfad~ _agn{c§np£@_aNPg]-nyxR2fw _anyfmgc§_aNPRW~2nyc\yR2¹(~bwy]hR \[!gck_hR2faxPfaR_agcPW_hNRQ(R2_hNPn^} w _$_hNPR+~n{ck_hgckdnyd] RbyRb\wyc}_hnR¹\_hRbc}g__hnW_hNPRcPn{c~n{c\yR¹]hR_h_hgcP¯ MNR }^gj]a~faR_hgªbwp_hgnycHny£¶_aNPg]cPRbµn{xzRbfawp_hnyfWQ§w[szR(xR2fh£©nyfaQ RJ}Hgc4]hR2{R2fwp$µw[^]bfaRb~2n yRbfhgcPwpcz} {R2cPRbfawygª2gcPw1 wpfagR_`[snp£-Q R2_hNPn^}P]b ]hd~N¦wy]W_hNPR(Q)dP¸_ag]a~2wyR)±cPg_hR§RbRbQ(R2ck_!Q R2_hNPn^}º¬»VH]a¼$½¶V¥® n{f_hNPR5NPR2_hR2fany{R2cPRbnyd]-Qd¸_ag]a~2wyR5Q(R_aNPn^}4¬D¾UVHV¥®¯Pr¿c¨wy}P}^g_hgnyc%_hn _aNPR!wpn yRy\µR5gck_hfan^}^d~2Rwyc n{fhgygcwyCwyc}%yR2cR2fwpCc\dPQ(R2fag~bwp´~n{fhfaRb~_hn{f-gc_hNPR~2nyc\yR2¹1~2wy]hRy¯ ÀsÁkÂCÃaÄÅ Æ$y¡ NPn{Q n{yRbcPgªJw _agn{c´ ~n{ck{R2fayRbc~R{+R2gx^_hgj~4nyxR2fw _hn{fbqikdw{]|gj~n{c\yR¹^g_`[yq¾WVHVÈ V¥]h¼$½¶VÈz~nyfafaRb~ _anyfJk~2n{wyfa]hRW{fawygcgcP ΓÇ. Ì!ÖæDÙøÉ@Ó¿êëÝÊÒäØÔÖaË9ÛCêìCÌ5æ»Ó`ÖÍ»âbÒÔÉ9ÒÔØÔß`ÛLÎpÓ`ÓmÚUÏJÉCÊæDÓ`Ð`Ó`ÚÑ ùpÓ¿ÒÔÑ Ó í5æ©× Õ$îJÐ`ÏJÖaÑ ×Dï æ»ØÔæLÑ æDÖÓ`Ù ÙJпÖ×DÐ|ÒÔÛ Ó ÓÏyÛDÚÑ ÖÓ`ú+ÙÛ´ÚÓ Ü^Ì5ú+Ñ ÙÍ»ØÔ×LÛDÉ@Û´ÛDÌ5×LÓ¿Öa×9×Lð$Ó`ÉCÉ$ú+Ý ÏJÓ`ÖÍ©Ù2ÞÕ$×DÛDÛCË´ÛLß`ÑÍëÓ`÷\ð-Ûh×DÏ Ï ÝàmË´Ó$Ó¿Ñ2û×9Ð|æDñ2á-Û»Þ×@Ö`Ó`ââÙÓ|Ó`Ð`æ»ÙbÑ ÛLÞØÔÞÑ Ómæ©ÙW×hãCò ÖÒäÉCÙÖØÔÚqÑ ÛDÙÓ$Ó¿×óíbÜPÖ×DпÖÓ`×|ÛDÙô@ÐhÚÖÖÛ@ÒjÙÏ ×D×LÉCÝÑ ÓØØÔ×LÙJß$Ù2ÓÞòå$õ ú+ÒÔÓ`Ñ Ó`ÛLæ»Ð|æDØäØÔÖÖÐ|æ»ö$Ö×DÒ{Ó`ØÔÙÛhÖÏæ»Ù Øäç ÖÖJçaÒò ÷üJèæ ÛDé ØÔé Û ∗. †. Unité de recherche INRIA Rocquencourt Domaine de Voluceau, Rocquencourt, BP 105, 78153 Le Chesnay Cedex (France) Téléphone : +33 1 39 63 55 11 — Télécopie : +33 1 39 63 53 30.
(13) §69& #*
(14) 6 5
(15) ('¨ $
(16) 8 ' 7+6"!0 5 +'%6 +
(17) 01
(18) 8 '+6 %! +6 7+6 $*
(19) 2+6 8 '%6
(20) :;§+6 ¡ WRcnyQfhRbd]|RJ]+Q(e_aNPn^}^Rb]Unyck_Ue_aexPfanyxn{]he2RJ]q~RJ]q}^R2facPgO2faRb]qwycPcPebRb]qxnydfUNPn{Q n{ye ce2gj]|Rbfc\dPQ(e2fagikdPRbQ Rbck_)}^Rb]nyxe2fw _aR2dPf]5R2gx^_agikdPRJ]gcebwpgfaRb]!n{dÈcPnycÈgcebwpgfaRb]b¯st-RJ]Q(e_aNPn^}^Rb]Ç ]hnyck_U]hnydP{R2ck_q}^e±zcPgRJ]Uwydcg{Rbwpds}^gj]a~faR_b ¯ 9w(xPdPxwyf|_U
(21) } Rbc{_afhR!RbRb]U~nycz]|gj]`_aR §~2wy~2dPRbfqc\dPQ(e2fag Ç ikdPRbQ(R2ck_ dPc(cPnydP{R2\nyxe2fw _aR2dPfJpikdPgwyxPxPfan¹\gQ(Ry Rbc)dPc§]hR2c] xPfaeb~2g]hR2fJ nyxe2fw _aR2dPf NnyQ(ny{e2cPebg]he }Pd%xPfanyObQ R{¯ ny^o`RJ~ _ag¸£}^R!~R5_hfw wpgCRb]|_+}PR~jwpfag¸±Rbfw(~n{c]|_hfad~ _agn{c%}^R~2R_+nyxe2fw _aR2dPfc\dPQ(e fagji{dRU}Pwpcz]¶RW~bwy]¶~2nyc\yR2¹^R+R2cgck_hfan\}PdPg]awpck_dPcPRU{R2f]|gnyc§~2nyck_hgc\dPRW}^RW nyxe2fw _aR2dPfJ{}PRUyebcPe2fwpgj]|RbfÇ wpxPxPfan^~NPRwpdH~bwy]Wi{dzwy]hg~2nyc\yR2¹\R5R2_W} R2cs£»wygfaR wycwp[^]|R{¯qt-RcPnydP{R2 nyxe2fw _hRbdPfqxR2d^_5wpnyf] _afhR c}Pgg] ]a~Q)faedP_ag_h]hgeWeb~}^NPRRb}^gRb] ebn{fhd(Rbc{_aw5Rb]-Q(£»ew _a2NPnyn^c}^]bRUkfh¾WR2_hVHfanyVÈdP¯ywp½¶ckc§_-nywyd^gcz_afh]|g@R{p}^cPg nyebdzfh] Rbckgck_h_aRbfh]-n^Q(}^dPegj_a]hNPnyn^c}^]RbdP]c~n{~2nyQ(faQ fhRJRW~ _aR2RJdP]f e2c\e2dPQ(Q(R2e2ckfa_agj]¶i{d± R Ç }wpc]R~2w{]~2nyc\yR2¹^RW{e2cPebfawyD¯ Å J à §¡ NPn{Q(nyyebcPe2gj]aw _hgnyc9 ΓÇ ~n{c\yR2fayRbc~R{pnyxe2fw _aR2dPfR2gx^_agikdPR{^i{dzwy]hg~2nyc\yR2¹\g_he{{¾WVHVÈ V¥]h¼$½¶VÈz~nyfafaRb~ _aR2dPfJ^~nkwpf]|RWyfwpgcPgc.
(22)
(23) !"#
(24) %$&'()* ,+- . . /. 9
(25) *Z. #³0$2q 7 6@ ! q+) #
(26) 43 69' 6 ¼nyf!_aNPR]awp·yR§ny£]hgQ xgj~g_`[4wyc}¤gc¤n{fa}^Rbf5_hnHfaR2jw _aR _hNPRxfhRJ]|Rbc{_µnyfa·s_ansR¹^g]|_hgcPHwyxPxPfan{wy~NRb]b µR¨±f]`_1~2nyc]hgj}^R2fwÈ]a~2wywyf Rbgx^_hgj~sxwyf|_agwyU}^g RbfhRbc{_agwyqRbikdwp_hgnyc_hNwp_1gj](_hNPR¥½mdPR2f Ç wpyfwpcyR RJikdw _agn{cHnp£-w%xPfanyPR2Q ny£¶Q(gcPgQ(gªJw _agn{c¥np£wpc¥RbcPR2fay[{¯!MNPR)RbcPR2fay[s}^Rbc]hg¸_`[¨gj]w{]h]hdPQ(Rb}H_hnzR ~2nyc\yR2¹4wpcz}¦_hnH wpfa[4wp_)wH]a~2wyR]hQ§wpmµ+g¸_aN²faRb]hxzRJ~ __ans_hNPR%]|gª2Rny£_hNPR1}PnyQ§wpgc´$µ+NPg~NÈQ§wy·yRb] }PgfaRb~_c\dPQ(R2fagj~2wpC]hgQ)dPwp_hgnyc]-gQ xn{]a]hgPRW_hn xzRbf|£©n{fhQ¢gc1xPfwy~_hgj~Ry¯ 5WdPfxPdPfaxn{]hRWgj]-_hngck_hfan^}^d~R xwyc%fhn w{{RR2_hfNzwpwy_RJ}(_hNPR2R§cPRbwyfh]a{]h[§n\~2}^gR2wpc_hRJ]h}sg_`[Q(µ+gcPNPggjQ(~N%gªJ}^w _an\gRbn{]c4cxPnpfa_nywyRbfh[Q w{g]]5Qwd~~nyNfafawyRb]-~ _5_hNPwpR5xPxn{fhfhgny¹^ggcQ(wpwp_hR2gcnyR2csfayny[£¶_h}^NRbR c]|n{g_`fh[gywpgcczwy} Q(gcPgQ gªbwp_hgnycxPfanyPR2Q"gc_hNPR]hR2cz]|R5np£ ΓÇ ~n{c\yR2fayRbc~R{¯ uU¸_aNPnydP{N]|_hfag~_ ~2nyc\yR2¹\g_`[¦R2c]hdPfaRb]!_aNPRR¹^g]|_hRbc~Rwpc}²dPcPgji{dR2cPRJ]h])np£q_aNPR%]hnyd^_agn{cÈ_hn4_hNPR ½¶dPRbf Ç 9wyyfwpcP{RmRbikdwp_hgnyc(wpc} wpn µq]´_an!d]|R+wWQ(nyfaR}PgfaRb~_ wyxPxPfan{w{~N1¬©R{¯ z¯ GÇ ~nyc\{R2fayR2cz~RJ®nydPf wyfh{dPQ(R2ck_a]wpfaRwy]hRb}1nyc1 wyfhgjw _agn{cwpCxPfagcz~gxPRJ]2¯ Vsnk]`_np£ _hNPR5R2¹\gj]|_hgcP)µnyfa·^]W¬D]|RbR5Ry¯ 7¯ 6 8L9 6b 8L 6 :;8©®$wy~ _adwp[5_hfaRbw __aNPRl¶U½U)¯ 5WdPf wpxPxfhnkwy~Nwyn µq]´_an!}^Rbwykµ+g_hN)_hNR~bwy]hR-ny£zcPn{cPgcPRJwpf$R2jwy]|_hgj~g_`[y w{]5µ+g¶R]hR2R2c9¯%MNPR(±f]|_)Y\Rb~_hgnyc¤g]}^Rb}^gj~2wp_hRJ}¥_an¨_hNPR§gck_hfan^}^d~ _agn{c¤np£_hNPR1w{R2fwp{Rb}sRbcPR2fay[ }PR2c]hg¸_`[gcº_hNR¨£©fawyQ Rbµ-n{fh·Èny£W_aNPRsNPn{Q n{yRbcPgªJw _agn{c²_hNR2nyfa[¤£©nyf%~nyc\{R¹²R2cR2faygRb]b¯«MNR2nyfaR2Q ]|_awp_hRJ]¶_aNPR ~n{ck{R2fayRbc~R+ny£9_hNR5w{R2fwp{Rb} R2cPRbfh{[(_hn)_hNPR5NPn{Q(nyyRbcPgª2Rb}R2cPRbfh{[(wyc}MNPR2n{fhRbQ gck_hfan^}^d~RJ](Γw4Ç cR2µyR2cR2fwp+~n{fhfaRb~_hnyf faRb]hdP¸_J¶µ+NPgj~N«}^Rb]a~fagRb] _aNPR¨±cPRs]a~2wyRJ] £©Rbw _adPfaRb] np£_hNPR ]hnyd^_agn{c¦ny£_hNPR%nyfag{gcwy Q(gcPgQ(gªJw _agn{cÈxPfhn{PR2Q3µ+g¸_aNPnyd^_ wy]a]|dPQ(x^_agn{cÈnyc¤_aNPRNPR2_hR2fany{R2cPRbg¸_agRJ]2¯ MNg]~n{fhfaRb~_hn{f gj]~2wpRb}wc\dPQ(R2fagj~2wp\~n{fhfaRb~_hn{f gcfaR£©RbfhRbc~R-_h'n 6b; 8ópµ+NPRbfhRg¸_ µwy] ±f]`_ }^RbfhgyRJ}gc _aNPR]`_an\~Nzwy]|_hgj~-wyc}]|_awp_hgnycwyfh[5~2w{]|R{¯$Y\RJ~ _hgnyc(qgj]}^Rb}Pg~bw _hRJ}_anU_aNPRxPfhn\ny£»] ny£P_hNRQ§wygc_hNPRbnyfaR2Q§]2¯ <Rb]hgj}^Rb]_aNPRwpczwp[\]hgj]-ny£~2nyc\yR2¹xPfanyPR2Q§]bPw){R2cPRbfawygªbwp_hgnyc%np£$_hNPgj]qwpxxPfhnkwy~N%g]gck_hfan\}Pd~RJ} gc1Y\RJ~ _hgnycs{¯ / _an}^RJwpPµ+g_hNcnyc~2nyc\yR2/ ¹R2cPRbfh{[}^Rbc]|g_hgRb]b¯ UR2±cPg_hgnycsJ!wpc}§MNPRbnyfaR2Q / wpfaR_hNPR czw _hdfawy ~n{dPck_hRbfhxwyf|_]qny£¶UR2±cPg_hgnyc wpc}¥MNPR2n{fhRbQ £©nyf5i{dzwy]hg~2nyc\yR2¹%/ R2cR2fay[¨}^R2c]hg_hgRb]b¯MNPR xfhn\np£$np£MNPR2n{fhRbQ 5g]qw{}PwpxP_hRb}1_hn _aNPRikdwy]hg~2nyc\yR2¹~2wy]hR5gcHY\RJ~ _agn{c¨P¯ ¯ ¼$gcwp[{Y\RJ~ _agn{c / faR2jw _aRb]_aNPR!xPfaRb]hR2ck_qµnyfa·§_hn]hnyQ(R!µR2 ·\cPn µ+c%c\dPQ(Rbfhgj~2wy´Q(R_hNn\}]q]|dz~N w{]-_hNPR!VH]a¼$½¶V&wpc}1_hNPR!¾WVHV _hn(µ+NPgj~N1_aNPR5xPfhRJ]|Rbck_wycwp[^]|gj]Ç wpxPxPgRb]np_hNgc1_aNPR~n{ck{R¹wpcz} ikdw{]|gj~n{c\yR¹]hR_h_hgcP{]b¯ =¦Rwy]a]hdPQ(R5_hNPR!faRbw{}^R2f+gj]£»wpQ(ggjwpfUµ+g¸_aN¨_hNPR!w{]|gj~5xPfanyxR2fh_hgRb]ny£$_aNPR ~n{c\yR2fayRbc~RW_hNR2nyfa[y¯ Y^NPnydPj}(_aNPRUcPRbRb}wyfhgj]hRy 6Ô> 8CxPfan kgj}^RJ]mµ+g_hN1w!{n\n\}(gck_hfan^}^d~ _agn{c1wpcz?} 6 8{gΓ{RbÇ ]mwQ n{fhRW]|[^]|_hR2Q§wp_hgj~ ]|_hdz}^[ny£W_aNPRH]|d^o`Rb~_b¯ ¼nyf~n{c]hg]|_hRbc~[{-R_d]fhRJ~2wpq]hnyQ(RscPnp_w _agn{cºwyc}ºxPfhn{xzRbf|_agRJ](np£_hNPR ~2nyc\yRbfh{R2c~2R!gcÈY\n{zn{RbH]|xw{~RJ]2¯!r¿c¥_hNPR§]hRbikdPR2L }^RbcPnp_aRb]5wpc4nyxR2c4zn{dPc}^RJ}4]hdP]hR_5np£ ΓÇ ¬}PnR±≥cPRJ1}® kW[ (Ω) }PR2cPny¯ _hRb]$_aNPRqY\nynyR2)]|xw{~R-£©nyf p ≥Ω 1 wyc} p }^RbcPnp_aRb]$_aNPRq~n{c o`dP{wp_hRR¹^xnycPRbRc{_ + =1 @)Á ACBCD D»%Å BFEHG " F : W (R ) → R I J'K LMNO!%
(27) " . &P"QSRTUP"V W$& F Γ(L ) X
(28) Y.()PVZ[P \9Q
(29) Y.()P ]^ _ `%aM bV U W$&J . 0. 1. n. 1,p 1 p. 0. 1 p0. . 1,p. n. L,c Ld(e\%\&"# #Γ(W PKfP")X . gPWh4aiQ. Ë ËÙjéçk>l. 1,p. F : W. 1,p. n. (R ) → R. . Ω. p.
(30) . # . Z[ ] G LT L-O L" b L .Ce"Y# ]T L W 1,p (Ω) u →u. u ∈ W 1,p (Ω). %a Y#'P""
(31) . u. P#
(32) $U W$&. u * u. Z[#P \ Q. . F (u) ≤ lim inf F (u ).. Z[ L ]
(33) Y;"# P""
(34) cT"Y;"# Z[P \9Q u¯ → u]' L W 1,p (Ω) a. . u∈W. 1,p. (Ω). L$ ' i gP .PK P"" "%
(35) . P#
(36) $J W$&. u ¯. u ¯ * u. . lim sup F (¯ u ) ≤ F (u).. W R±cg¸_agn{c4g]wy]hnfaR£©RbfhRJ}(_hn(wy]¶_hNPR!]hRbikdPRbc{_agwy ~2nyc\yRbfh{R2c~2Rq]|gc~2RWg_+g]]|_aw _aRb}1d]|gcP)_hNPR ~2nyc\yRbfh{R2c~2RUny£$]hRbikdPR2cz~Rb]b¯ =4R!faR£©Rbf_hn`6Ô\zY\RJ~ _agn{c4{Γ¯ Ç 8@£©nyf+RbikdPg wpR2ck_q}^R±zcPg¸_agn{c]b¯ MNPR ~n{c\yR2fayRbc~R(gQ(xPgRb])_aNPR~2nyc\yRbfh{R2c~2R§np£qQ(gcPgQ§w¦wpc}²Q gcPgQ(gªbR2f]ny£q£©dPc~_hgnyc](wy] ]|_awp_hRJ}1gcΓ_hÇ NPR5£©n{n µ+gc -Á Q W ² gE Pf,G " FOP"":KW
(37) (R L )d→PcReI L $ ffT- L _ \ 7,!(U
(38) PK W$& Γ(La) X
(39) "!* Y;
(40) ">."()#
(41) P' LY [K LF W$& .
(42). ΩL,cF Ld(. P" P. W$&"eKY"#. 1,p. . n. p. . W 1,p (Rn ). ∃. c > 0,. ∀v ∈ W 1,p (Ω), ∀ > 0,. c||∇v||pLp (Ω) ≤ F (v). u0 ∈ W 1,p (Ω) lim inf{F (v + u0 ), v ∈ W01,p (Ω)} = inf{F (v + u0 ), v ∈ W01,p (Ω)}. →0. %a "Y;"# P" "%
(43) `T L LT ,+>P N P# I P"" "%
(44) Z f", I ",ua] a^VK L* LK ,+>". L W 1,p (Ω) Q u *u . wyc}. r¿c§_aNPR+£©n{n µ+gc Γ }^R2cPny_hRJ] _hNR Γ(L )Ç ~n{ck{R2fayRbc~R{¯9¼Pn{f¶wpn{xzRbc§zn{dPc}^RJ}]|d]|R2_ O ny£ R zµ-R!}^RbcPnp_aR u ∈ L (O) Z p. n. 1. < u >O =. . W$&"V i gP# PJ P#
(45) $ L$ . 1,p inf{F (v + u0 ), v ∈ W0 (Ω)} N u inf{F (v + u0 ), v ∈ W01,p (Ω)}. 1 Ln (O). u(x)dx.. O. "!$#% &'(*)+&,"-./10%32546/879:#% /7. 9R_dz]!±f]`_)fhRJ~2wp¶~2w{]h]hg~bwpmNnyQ(ny{R2cPgªbwp_hgnyc¦fhRJ]|dP_a]£©nyf~2nyc\yR2¹4RbcPR2fay[¦}^R2c]hg_hgRb]1¬D]|RbRJ6 8L76 8. wyc} 6>:;8뮯 !
(46) W : R × R → R gPO W$ >a# !
(47) ? CY"# ξ ∈ R @)Á ACBCD D»Å%B g. f P ' P ! I ,K%a cK;P# 7L x ∈ R W (x, ·) gPK
(48) * . L* dP"Q W (·, ξ) utwyfawp_hNPebn\}Pnyfa[)£©dc~ _agn{cn{c R × R gj]qw'<nyfaR2£©dPc~_hgnyc¨nyc R × R ¯ -
(49) Á % a L² P G " W : R × R → [0, +∞) I P" NK!
(50) P`P . gPW Ld( L$ 'L,c L)( <;. n. *=. n. n. n. ; ?. . n. n. %>. n. . n. n. n. Í©Õ$Ë´Íëð. .
(51) .
(52) !"#
(53) %$&'()* ,+- . . W gPe =c W$%>>a#!%
(54) " Kb'P "Y;"# x ∈ R W (x, ·) gPK
(55) Y W$&"T i gP# 0 < c ≤ C a p ≥ 1 P
(56) $? W$& . n. . c|ξ|p ≤ W (x, ξ) ≤ C(1 + |ξ|p ). =. K;P# 7L x ∈ Rn a eL ξ ∈ Rn Q &P# a" Ω I !%aa \ P I P" 7 Rn %a P" *KL F (u) =. Z. > 0. eL u ∈ W 1,p (Ω, R) Q
(57) $&" >\ [ i
(58) " . W$&"e i gP# P !
(59) %a P#
(60) $? W$& O$&Y Ω. p. eL. -Á ² . u ∈ W 1,p (Ω, R). Q. ¬|J®. W (x, ∇u(x))dx. Γ(L ) − lim F (u) = →0. Z. Whom. P" gPW L)(. . Whom (x, ∇u(x))dx, Ω. #^aa L . _ a ," PP!e L$ p ≥ 2 W$& W(x, ·) gP
(61) L dP#fa " I ,C 'P# CL x ∈ Ω a a(·, 0) = ∂W (·, 0) ∂ξ. gP I !%aia %a L$ L$ L,c Ld(U'% _*
(62) L a
(63) L ! L \\ "# . P $&,a T;P# L x ∈ Rn , eL ξ , ξ ∈ Rn ∃ 0 ≤ α ≤ p − 1, C > 0 |.
(64). 1. |a (x, ξ1 ) − a (x, ξ2 )| ≤ C(1 + |ξ1 | + |ξ2 |). e;P# L.
(65) $ ( LY". |ξ1 − ξ2 |α ,. KL. 0. f ∈ Lp (Ω) . W$&P"! . u ∈ W01,p (Ω). _. −div (a (x, ∇u )) = f. W01,p (Ω) . >\V _ i .i
(66) [ L$ fP"b! . −div (ahom (x, ∇u)) = f,. ahom : Rn × Rn → Rn. gPf", _a _. Whom. ahom =. u ∈ W01,p (Ω). ai h%a L G "T'. Ë ËÙjéçk>l. ¬Dy® ¬/ ®. _. I. ∂Whom . ∂ξ. ^iaa L . a P . gPWh # P Z ] c L L$J L$ KP"
(67) > "
(68) * c %aU \ %" P"'T
(69) - "
(70) "hom C %a L$ K \&%" α/(β − α) L P#[ a α Q
(71). 2. ∃ 2 ≤ β < +∞, c > 0 | x ∈ Rn , ξ1 , ξ2 ∈ R n (a (x, ξ1 ) − a (x, ξ2 ), ξ1 − ξ2 ) ≥ c(1 + |ξ1 | + |ξ2 |)p−β |ξ1 − ξ2 |β .. >U
(72) Y (iPf L $ . p−1−α. β. a Z ] c L L$ W$&.
(73) . # . ! . . 0%/7 # 7. 9R_ R§w1£»wyQ(g[Hnp£-R2cPRbfh{[H}^R2c]hg_hgRb]!]aw _hgj]|£©[kgcP%_hNPR§w{]h]hdPQ(x^_hgnycz] wpc} ¯ MNPR xfhn{P(WRbQ")µR5~2nyc]hgj}^R2fgj] Z ¬©k® W (x, ∇u), u ∈ W (Ω) + BC . inf <[ W (Ω) + BC gc²¬©\® µ-RQ RJwpcswpc\[¨]|d]|xzwy~R!ny£ W (Ω) wy]a]|n^~2gwp_hRb}1_hn1d]|dzwp9nydc}Pwpfa[ ~2nyc}Pg¸_agn{c]2¯r¿c¨xwyf|_ag~2dPwyfb\µR5µ+g9~n{c]hg}^Rbf ¬|J® φ + W (Ω) £©nyfqwpc\[ φ ∈ W (Ω) ¬Ly® {u ∈ W (Ω) | u(x) = ξ · x + v(x), v ∈ W (Ω)} £©nyf wpc\[ ξ ∈ R µ+g¸_aN W (Ω) = gj] xzRbfhgn^}^gj~ } g£ R ~2wpc)zRnyP_awpgcPRJ}!\[5_hNPRxzRbfhgn^}^gj~¶faR2xPgj~2w _agn{c {v np£ Ω | v ∈ W (R ), v Ω− ¬ / ® {u ∈ W (Ω) | < ∇u > = ξ} £©nyfqwyck[ ξ ∈ R ¯ MNPRJ]|R(_aNPfaR2RnydPcz}Pwpfa[¥~n{c}^g_hgnyc]gc}PR2Rb}È}PnscPnp_gc dPRbc~R_hNPRRbcPR2fay[4}^R2c]hg_`[¥np£_aNPR gQ g_Uny£+¬»k®zwy]qPfhgR z[1faRb~2wyRb}%gc¨_hNPRuUxPxR2c}^g¹@¯+MNPg]qg]Uµ+Nk[%µ-R}Pn§cPnp_UQ§wy·yR5_hNR2Q ]|xRb~2g¸±zΓ~ Ç gc_hNPR]hRbikdPRbD¯ TUx%_an§R¹\_hfwy~_hgnyc´\xPfanyRbQ/¬©\® ΓÇ ~nyc\{R2fayRb]¶_hn Z ¬Dy® inf W (x, ∇u), u ∈ W (Ω) + BC . MNRW}^Rbc]|g_`[ W gj]¶cnp_-R2¹^xPgj~g_h[(·\cPn µ+c´¯ ´R2Q(Q§w)!nycP[)xfhn \gj}^Rb]¶µ+g¸_aN1wpcR2¹^g]|_hRbc~RqfaRb]hdP_b¯ ¼nyf+PfaR2\g_`[y^µR5}PR2cPny_hR!\[ . 1,p. . Ω. 1,p. 1,p. 1,p 0. 1,p. 1,p #. 1,p. 1,p loc. |Ω. 1,p. n. n. 1,p #. n. n. Ω. . . 1,p. hom. Ω. hom. Z. W (x, ∇u), I (u) = Z Ω Ihom (u) = Whom (x, ∇u),. £©n{f u ∈ W (Ω) ¯ =¦R!cPn µ gck_hfan^}^d~R!w cPny_hgnyc¨np£$R2cPRbfh{[1wyRbfawyyRb]¶nycwpj]2¯ @)Á ACBCD D»%Å B J η > 0 ai L)( I B(x, η) W$& I LN?a LdP Ta.h W$&T ( ai P L Ω. 1,p. . η.
(74) " _a \ L*. x ∈ Rn . . . Wη, (x, ξ) = inf < W (·, ∇v(·)) >B(x,η) | v ∈ W 1,p (B(x, η)), < ∇v >B(x,η) = ξ. Rn × R n. [. R. a W$&T;PP"-
(75) " _a "%".( !%
(76) " . % Iη, (u) =. Z. Wη, (x, ∇u),. KL. ¬»{®. u ∈ W 1,p (Ω).. Ω. Í©Õ$Ë´Íëð.
(77) .
(78) !"#
(79) %$&'()* ,+- . . %Á ²^ E RTe$ YdP"a I L P. ai h Ld( Y""()aJ ( P"Q LC W$& P!b P \ P"" _a W$d(>$ ! 7 W$d gPe$& aOB(x, ()η)"%"#
(80) \ ^%" b(;$ I $ ->a;P N (x, η) P . gPW Ld(? L$ 9 X "Y# x ∈ Ω W $&"` i gP# 0 < c ≤ C P
(81) $M L$ U"Y;"# η > 0 c|B(x, η)| ≤ |N (x, η)| ≤ Q C|B(x, η)|. 9R_Ud]qgck_hfan^}^d~2R5_hNPR~2nyc~2R2x^_qny£ RbikdPg´gj]hnyjw _hRJ}Q gcPgQ(gªbR2f] WR£©nyfaR]|_aw _agc(_hNPR!±zfa]|_qQ§wpgc faRb]hdP_b¯. LY OT L "%".( !
(82) P F W$& '" .#
(83) 'P.\
(84) (V, d) TP" @)Á ACBCD D»Å%B W$& 7T L fT L LT ,+>PKN . gPK "! c gP", [a f W$&"' i gP# P L . . . gP W$&f!*F N. . η→0 →0. eT(V,. L LTd) ,+-"e. N. . QI. . P#
(85) $? W$& u %a u F B e"Y# N > 0 B ⊂ V N N N 5WdPf±zfa]|_+fhRJ]|dP_qgj]_hNPR5£©n{n µ+gc W$&?"%".( ia P# L . P W P" gPW %a %aJ L$ U "%".( Á\Å Á E p > 1
(86) *Y;".()#PK [ η, a P a i ( _. Q
(87) $ . ' * " P "%
(88) p 1,p Iη, Γ(L ) L* LK ,+>Γ(W )X Ihom η 0 i gP# PK T K K " T P L. $ L. L. T ,. > + N 1,p uη, inf{Iη, (v) | v ∈P W (Ω, R) + BC} uhom
(89) ? $ W. & $ . inf{Ihom (v) | v ∈ W 1,p (Ω, R) + BC} >b L W 1,p (Ω, R), ¬óp® lim lim u = u uN ∈B. η,. hom. >\V [ i
(90) " . %Q = ##P \ %a L)( *UT L LT ,+> u N inf{I (v) | v ∈ W 1,p (Ω, R) + BC} L $ hom hom i gP# P'P" "%
(91) u fT L LT ,+>PKN inf{I P
(92) $? W$& 1,p (Ω, R) + BC} η, η, (v) | v ∈ W P )( L W 1,p (Ω, R). ¬.:{® lim lim u = u η,. η→0 →0. hom. aia L u gP fT L N ! 7 gP" _a T L LT ,+-"P L W$& P"&P" T h L . gP P" gP"η,, _a %a`Z
(93) ]e$ ,aP"Q u hom
(94) \ #
(95) ! . gP P# .#
(96) " T
(97) *Y; b'P "Y# $ e n L$ !* P" "%
(98) NOK L* LK ,+>"P u P#W .(x, )(·)
(99) *Y;".()#Pf L W 1,p (Ω, R) [ L$ f! " eKR L* LK ,+>" u Q η, hom %Á ²^ ;
(100) $&TaiKNf W$&fb LK L .P L Z ] gP L\&# [ %aU
(101) I T
(102) $&d(ia LU(),Q ¦= R~2wycwy]hn§}^R±cRw ]hR_+ny£$c\dPQ(R2fagj~2wp´~2nyfafhRJ~ _anyf] _hnwyxPxPfan¹^gQ§w _aR ∇u gc Lp(Ω) ¯ \ # L . M Ω L a gP " L* P# I a' L&P'NTa _"'N @Á)ACBCD »D Å% B FG " H,i i∈[[1,I ]] I a" H Q RTUa.h {Q UK }T. L (M N \\%? i L' . &PUN ai* . L Lp (Ω) ;PP"-
(103) " _a [ H) Ce"Y;"# % a Q w ∈ Lp (Ω) H > 0 W$&".
(104). H. H,i. MH (w) =. IH X i=1. Ë ËÙjéçk>l. < w >QH,i 1QH,i ..
(105) # : _\% L)( W$& % [ . N
(106) $&" ?a.h % W$& !#
(107) 7
(108) #
(109) " [P
(110) Y U ( ai P L PO W$&! " T L LT ,+>PeN. . N G. Á\"Å T Á' . inf ;. . (Z. W (x, ∇v) | v ∈ W QH,i. OPP#!
(111) $&. 1,p. JP# .#
(112) " . (QH,i ), < ∇v >QH,i = MH (∇uη, )|QH,i. * L . Q R' Z $ ]'Y %a Z ]Kc L L$. . H,i vη,. p ≥2%a K. h. lim lim ||∇u −. η,H→0 →0. %Á ²^
(113) $& a"'Ne W$& LK L .PK L. IH X. β≤p. ). ¬»{®. .. QRT i[\J W$& _ . ¬`J{®. H,i ∇vη, 1QH,i ||Lp (Ω) = 0.. i=1. a η L Z ]U gPK% L \ # _* aU'' _-i H Q Y " $ Y K [. h # P . ," () _e+>Q H =η→0 % Á ²^
(114) $&" $ ,aP Wη,(x, ξ) gPf_\%
(115) a I ∂B(x, η) . inf < W (·, ∇v(·)) >B(x,η) | v ∈ W 1,p (B(x, η)), v(y) = ξ · y
(116) $ " P"T$& a;P vH,i gPf_\%
(117) a I L$ !* fP"! L W 1,p(Q ) >Q ()Q. H,i. η,. inf. (Z. QH,i. W (x, ∇v) | v(x) = MH (∇uη, )|QH,i · x. . ∂QH,i. ). ..
(118) $ 7\-NPK 9
(119)
(120) I T;P# L ia\ [a [ L$ #P"T
(121) ;P"#P"Q I [ L%a I L$ \&"# -a
(122) %Á ²^
(123) W` W$& b(;$ I $ ->a N (x, η) gPJP#
(124) $H W$& R
(125) I ^ [\
(126) N (x, η) L$ %
(127) $ " $&,a;Pec L W$ n. ¬|yJ®. n o 1,p Wη, (x, ξ) = inf < W (·, ξ + ∇v(·)) >N (x,η) | v ∈ W# (N (x, η)) .. ¼fhn{Q/w%_hNPRbnyfaR_hgj~2wyxzn{gck_np£\gRbµ! Nzwy]_hNRwy}^ wpck_wpyR_hnH]|_hfanycP{[H~2nyc\yRbfh{R!_hn g c W (Ω) Cµ+NPR2faRbw{] u n{cP[¨~n{c\yR2fayRJu]+µRbwp·\[y¯qMNPgj]WNwy]WwpcHgQ(xzn{f|_wpck_W~n{c]hRbikdPR2cz~Rn{csu _hNPR xfaw{~ _hgj~2wy¶c\dPQ(R2fagj~2wp~n{Q xd^_awp_hgnyc²ny£ wpc} ¯¤uU] ]`_afhn{cPy[¤~nyc\{R2fayRb]_an _hNPR {faw{}^gR2ck_ny£ u }^n\Rb]5cPny_5nk]h~2gjw _hR(w _!un{fa}^Rbf( wpu__aNPR§xuR2fagn^} µ+NPR2c η wpcz} wyfhR(]hud~gR2ck_h[ _a]hNQ§wpwpc ´g£ _h∇u nHwpxxPfhn}P¹^nkgQ§RJ]+w _hcPR ny_qu n{]a~gµ+jg¸w _a_aN²R5wwp_W±]hcPQ(g_hwyR§ R2]hR2~bQ(wpRbRbc{]b_¯¶MQ(NPR2gj_h]+NPQ§n^}@w¯[%MwpNPng]µ g]!_hn(c_npwp_!·{_hRNw(RQ ~2wyRJ]h]|R(Nz£©]|nygª2f R!ujwp¯%fayr¿Rbc f £»w{~ _J$]|_awp_hRny£_aNPR1wyf|_Q)dP¸_ag]a~2wyR§Q(R2_hNPn^}P]£©nyfRbgx^/ _hgj~§Rbikdwp_hgnyc]d]|dzwp[¦~nyQ(xPdP_hRc\dPQ(R2fagj~2wp wyxPxPfan¹^gQ§w _agn{c]-np£ u wy]µ+g@R]hR2R2c¨gc¨Y^Rb~ _agn{c ¯ η,. 1,p. hom. . η,. . η,. hom. η,. hom. η,. . *-. η,. # /79. #9.
(128) # 6. . 6!379)+&,"-5)+!37. . 9R_d]PfhgR z[)faRb~bwp_aNPR~2nyfafhRJ]|xnycz}^gc!yR2f]hgn{c§np£ ´RbQ(Q(w 5£©n{fikdw{]|gj~n{ck{R¹ R2cPRbfh{[ }^Rbc]hg¸_agRJ]2¯ . Í©Õ$Ë´Íëð.
(129) .
(130) !"#
(131) %$&'()* ,+- . @Á)ACBCD D»Å%B. LY p ≥ 1 n ≥ 1 %a d ≥ 1 e!%
(132) " . ikd!w{%]|gja~n{ac\yPR ¹ P"Z_ OP# L \ i{cdz Lwy W$]hg~2nyc\yR2¹ L W$&P#P"
(133) "$` L"$ ]T : KL Ln (∂E) = 0 I I E Rn W (A) = min. %Á ²^. . 1 Ln (E). Z. gP. W : Rn×d → [0, +∞] W 1,p X W. & $ " K i gP# PeV\ n×d A∈R . W (A + ∇φ(x))dx | φ ∈ E. W01,p (E; R3 ). . .. " ;P#
(134) *Y; '!%
(135) " . M gP d U
(136) Y. W$& O gP "Y;"# d #. a eY# ζ ∈ Rn×d L$ !
(137) X R 3 t 7→ W (ζ + tξ) gPe
(138) Y dQ X. ξ ∈ Rn×d. MNPR!~Nwyfaw{~ _aR2fag]|_hgj~2]-np£i{dzwy]hg~2nyc\yR2¹\g_`[(g]_an(R2c]hdPfaRU_aNPR!µ-RJwp· Ç n µR2f]|RbQ gj~n{ck_hgckdg¸_`[§ny£ gc{_aR Ç {fawy£©dPc~ _agn{cwpj]b¯ -Á ² W 1 ≤ p < ∞ %a W : R → R gPe " ;P#
(139) Y O!%
(140) " . `P" gPL Ld( .
(141). n×d. . W$&"J W$&4!%
(142) " . %. 0 ≤ W (A) ≤ C(1 + |A|p ) Z. gPf- W (∇u) J : u 7→. . . eL. A ∈ Rn×d ,. ,OP""K X
(143) L dP. W 1,p (Ω). Q. @Á)ACBCD D»Å%B
(144) " . Q LY;"MJ
(145) * . L* dP4!
(146) f : R → R, ξ 7→ f (ξ) W$& P" . gPWh4P^Z ] L .P P
(147) Y *Y;",\ Qf gPJai h%a PU W$& ( [P# " ;P#
(148) Y !
(149) "eK" _ f Q
(150) #\ #
(151) "!, L gP( LY"KTL ξ ∈ R I Ω. n×d. n×d. Qf (ξ) = inf = inf. (Z. (Z(0,1). f (ξ + ∇u(y))dy | u ∈. W01,p ((0, 1)n , Rd ). f (ξ + ∇u(y))dy | u ∈. 1,p W# ((0, 1)n , Rd ). n. ). ). .. = W$%>>a#!%
(152) " . a.h a n Rn×d @Á)ACBCD D»Å%B G " (x, A) 7→ W (x, A) I T $!
(153) $ L$ gP# 'H L* [[() p ≥ 1 \ ; P# L LY;V
(154) P _* P c %a C P#
(155) $ L$ 4R× ;P# 'L % a e L x ∈ Rn A ∈ Rn×d (0,1)n. c|A|p ≤ W (x, A) ≤ C(1 + |A|p ). ¬|Jy®. gPO W$&"`P" a _'P" gPW ]|_awyc}Pwyfa}%yfan µ_hN~nycz}^g¸_agn{c Z ea" p]Q @Á)ACBCD D»Å%B
(156) $ !%
(157) " W : Rn × Rn×d → R, (x, A) 7→ W (x, A) gP ]`_wpc}Pwyfa}HR2cPRbfh{[ }PR2c]hg¸_`[ W gPe " ;P#
(158) *Y; = L$ >>a#!
(159) L$ gP?. gPfP#! I ,e L L .P hP 4Y;# I ,a
(160) * . L* dPO L L .PP"
(161) aY# I , W (·, ·). gP P#
(162) Y K;P# Y"# x ∈ Rn W (x, ·) %a W P . gPWh #PZ ]Q.
(163) $ !
(164) . Ë ËÙjéçk>l. W.
(165) J. # . =¦RwpfaR!cPn µ gcsxn{]hg_hgnyc_hnfaRb~bwp@_hNRNPnyQ(n{yR2cgªJw _hgnycfaRb]hdP¸_+£©n{fUikdwy]hgj~nyc\{R¹1RbcPR2fay[1}PR2c Ç ]hg_hgRb]b¯ ?P" NTP _aaJ"%".( -aiÁ P# L² . P7P . g PW
(166) Ld$ (e W$&"4 ( c Q W $
(167) G " aW L V: RZ ]e× R a" → [0, +∞) ! I #Tf L Q !aa . \&"JP# I "P . . Ω. . Rn . eL. . u ∈ W 1,p. %a (Ω, R ). I (u) =. n. n×d. d. Z. p > 1fP"" > 0. . I. W (x, ∇u(x))dx..
(168) $ W$&"J gP# .P? $ ()" ,+>a^P# [%aia .( a"&P# L . Whom : Ω × Rn×d → [0, +∞) P" . gPW L)( Z ]KafP#
(169) $ W$& >\` _e i
(170) Γ(Lp ) − lim I = I W 1,p (Ω, Rd ) $&" hom →0 Z Q W (x, ∇u(x))dx I (u) = Ω. -t n{c{_afawyfh[_an%_aNPR§~n{ck{R¹H~2w{]|R{´ikdwy]hg~2nyc\yR2¹^g¸_`[¨gj]5cPnp_5xPfaRb]hR2fayRb}¥\[¨_hNPR§wyRbfawyygcPny£WR± cg¸_agn{c / ¯-MNPRbfhR2£©nyfaR5µ-R!Nw{R_hn§d]hRikdwy]hg~2nyc\yR2¹R2c\yRbn{x]_hn§n{^_awygc¨faRb]hdP_a]q~2nyfafhRJ]|xnyc}PgcP)_hn Ç MNR2nyfaR2Q {¯ @)Á ACBCD D»%Å BFE η > 0 C(x, η) gP W$&f$d\ "
(171) I ' R
(172) " _a L x ∈ R a N,")( L$ Q RT L$ ^a.h f W$&TY-(iaU ( ai P L I η n o ¬| / ® W (x, ξ) = inf < W (·, ξ + ∇v(·)) > | v ∈ W (C(x, η), R ) hom. Ω. . . η,. Rn × Rn×d. %Á ²^ . n. . [. . R. C(x,η). n. 1,p #. d. %ae L$ "%".(!%
(173) " %%PP">
(174) [aK _f L .P8 P
(175) Y " Y \. Iη, (u) =. Z. QWη, (x, ∇u),. TL. QWη,. u ∈ W 1,p (Ω, Rd ).. " ;P#
(176) *Y; i h4
(177) a
(178) #PT [
(179) Y i h
(180) . af#T!,^Z ]? gP "! LY;," [#Td !=,`1Z ]Q L$ "%".( ai P L P QW ?P# _aa ( a"&P# L #P?P" gP X Á\Å Á > 1 %a Ld( Z ] %a I p Γ(L
(181) Y.()P [η, P a η () _ 0 Q
(182) $ . p 1,p ) Γ(W ) X η, *?P"
(183) u 'T L LT ,+>P inf{I (v) | v ∈Ihom W$&"U i gP# P WP 1,p (Ω, R d ) + BC} η, T L LT ,+> u η,
(184) ? $ W. & $ inf{Ihom (v) | v ∈ W 1,p (Ω, Rd ) + BC} hom >b L W 1,p (Ω, Rd ), ¬|2k® lim lim u = u . Ω. η→0 →0. η,. >\V [ i
(185) " . %Q = ##P \ %a L)( e TT L LT ,+-" i gP# P'P" "%
(186) u fT L LT ,+>PKN η,. hom. lim lim uη, = uhom. η→0 →0. W$&". . uhom inf{Ihom (v) | v ∈ W 1,p (Ω, RdP) +
(187) $UBC} W$& inf{Iη, (v) | v ∈ W 1,p (Ω, Rd ) + BC}. P# .)(' L. W 1,p (Ω, Rd ).. W$&". ¬|Jy®. Maa L . u gP'fK LJN'"! 4 gP", _a?K L* LK ,+>"PK L W$&TP""&P" N T. h L η, g. P " P . g " P , _. a % ` a Z T ] & $ , ; a " P Q u
(188). . hom. Í©Õ$Ë´Íëð.
(189) {.
(190) !"#
(191) %$&'()* ,+- . . MNPRbnyfaR2Q / g]wy]`_afaw{~ _+]|gc~2Rikdw{]|gj~n{ck{R¹^g±z~2wp_hgnyc§gj]-cnp_+R¹^xPg~2g¸_gc%{R2cPRbfawywyc}{R2fa[§Nwpf} _an~n{Q(xPd^_hRUgcxfaw{~ _hgj~R{¯ ¾Un µ-RbyR2fJywycwp_hR2facwp_hgyRq~n{c]hg]|_a] gc1~2nyc]hg}PR2fagcP5_aNPRqRbcPR2faygRb] W n{c ww{]+±]`cP_gw _h_aR(Rb}%}^ggQ cRb_hcN]hR5gn{£©cnywpn µ+]|xzgcPwy~ R±f]`_5wpc}sxzwy]a]|gcP_an_aNPR)gQ g_5nycH_aNPR(}^gQ(Rbc]|gnyc¥gc¦w1]hRb~n{c}H]|_hRbx Á\Å Á G
(192) &P# a"OeP"" Ch L _fa L"&P# %P#P#
(193) I $P \ L$
(194) # Pf W (Ω, R ) %a'JPP" X
(195) " _aK "!
(196) L dPK LbK C\ "
(197) " [P η,. . . %a. . X. ∪ H VH. 1,p. . d. (VH , PH ) VH ⊂ V H Q T R W. & $ " J a . h O L. $ L , c L. ) ( L. _. [ ( !%
(198) " . %PO 1,p = W (Ω) 1. 1,p. H Iη, (u) =. %aiK L$ ;PP#! \ . &PTN
(199) $&. Q
(200) $&"? gP# K L* LK ,+>"P. H. . P#
(201) $ P"" "%
(202) ' W$&" gP# .PT?uη, T L ∈ LTV ,H+> W$&. Z. 0 ≤ H2 ≤ H1 W 1,p (Ω, Rd ). ¬|b{®. Wη, (x, ∇PH u). Ω. H Iη, Γ(W 1,p ) X. N. 2.
(203) Y.()Pe [. Ihom. ;P. η. a. H. ()U [. a P
(204) $. H inf{Iη, (v) | v ∈ W 1,p (Ω, Rd ) + BC} uhom inf{Ihom (v) | v ∈ W 1,p (Ω, Rd ) + BC}. lim lim lim uH η, = uhom. - L. W 1,p (Ω, Rd ),. ¬|p®. >\V [ i
(205) " . %Q = ##P \ %a L)( e TT L LT ,+-" W$&" d ∈ W 1,p (Ω, R ) + BC} hom (v) | v i gP# P'P" "%
(206) uH inf{I H (v) | v ∈uWhom1,p (Ω,inf{I L. # P
(207) ? $ W. & $ d R ) + BC} V H→0 η→0 →0. η,. η,. lim lim lim uH η, = uhom. P# .)(' L. H→0 η→0 →0. ¬|-:{®. H. W 1,p (Ω, Rd ).. L LT ,+>P L ^ i _aaP""&P"T T.h X
(208) Vaia L . H gPKK LbU K"!. gP" _a T * L . Z f $d
(209) $ uη, P & $ a [ \ ,
(210) a I I B ∩ V ] W$&" Z
(211) #]T$ , aP"Q B. M NPR§gj}^RJw%R2Ngc}¤MNPRbnyfaR2Q ¨gj]5_hNwp_5_hNPR~2nyQ(xwy~_hcPRJ]h]ny£Q(gcPgQ(gª2Rbfa]5gj]!}^dPR _hn¨_aNPR ±cPg_hR }PgQ(R2cz]|gnycnp£$_hNPR!Q(gcPgQ gªbwp_hgnyc¨]|xzwy~R!wp_+±cPg_hR wyc} η wpc}%_hn§ikdw{]|gj~n{ck{R¹^g_`[(wp_+_hNPR!gQ g_b¯ %Á ²^ W W gPe
(212) *Y; W$&" KL ξ ∈ R H. . inf.
(213). (Z. (0,1)d. . . n×d. 1,p W (ξ + ∇u), u ∈ W# ((0, 1)d , Rn ) (Z. . ). W (ξ + ∇u), u ∈ W. = inf. (0,1)d. 1,p. d. n. ). ((0, 1) , R ), < ∇u >(0,1)d = 0 ..
(214) $! gP`" L ^ai-P?% $ ,a P#
(215) Y ( a"&P# L #PJ;P`
(216) P# LVP" W$& \ b
(217) Y !%
(218) " . ξ 7→ | det(ξ) − 1| R2×2 \ L ξ = 0 Q
(219) $ .I W$&fY;""()a "%".( ai P# L .KN T h L . gP cfP#! L _ I ,Oa.h L U LU L$ " ;P#
(220) *Y;
(221) ;P" $d
(222) $f\&"# -a
(223) L#
(224) $d I !a#
(225) %a L P $&Y;K _ I K
(226) P# a"aQ. Ë ËÙjéçk>l.
(227) . # . %Á ²^
(228) $&^
(229) Y".(i"%
(230) ^ J L$ JT L LT ,+>Pe P#
(231) Y "%".( a"&P# L #PJ L
(232) $& X ". gP > L 1,p PJ\\ ;P"a _ W$&`P# .#
(233) " .bM
(234) Y
(235) ;P">Q
(236) $! gPJ LK L _ gP. _
(237) $!
(238) aU L$ e$d\ W W$&P# gPN!. gP", [a?T L LT ,+>PTZ[ $d
(239) $
(240) *% I
(241) $&
(242) a? L \
(243) "
(244) ] gPT LMU`\ . L,Q'
(245) !* ["K ) \, gPf( LY" I P$& I %aU L P# [ I L L P7' $ ()" ,+>a "%".(' $!
(246) $` gP cP# #
(247) T! X %e
(248) Y ;P
(249) &P# aia' L %a'*!'" X.
(250) L #\,a L Q. $ § #
(251) +69' 6 % a L"&P# & L ny¼£nyf9¬»k_hNP® ¯§R]hr¿c4wy·y_hR Nnyg]£Pg]hgdzQ ]`x_afaR§wp_h~2gnyw{c´]|R{9µwpRm±f_h]`N_R~~n{nycQ(]|gj}^xPRbdPf9_aw gc)_agn{Y\cRb~]5_hQ§gnyc)w[sP¯zmR§_hNPxR R2fh£©nyfaQ(X Rb}¦wpcwy[k_ag~bwp[{¯K=¦yR(Rbfa_h]hN\gnydc ] {R_]|n{Q RWd]hR£©dPCgc]|gyNk_nyc_aNPRUgck_hRbfhRJ]`_np£ u wyc}n{c§_hNRWgcPyfaRb}PgRbc{_]mcPRbRb}^RJ}§£©n{f¶_aNPRWxfhn\np£´ny£ MNR2nyfaR2Q/yµ+Ng~N¨gj]+gc¨_adPfacxR2fh£©nyfaQ(Rb}gc¥Y\Rb~_hgnycH^¯Ô(wpc}¨w{}Pwpx^_aRb}%_hn(_aNPRikdw{]|gj~n{ck{R¹1~bwy]hR gcHY\Rb~_hgnycs^¯ / ¯ MNPR!xPfan\np£$np£$MNR2nyfaR2Qgj]_aNPR2c_hNR!xPdPfaxznk]|Rnp£mY\Rb~_hgnycH^¯ ¯ . . η,. .
(252) 6. 48/87 6!. . . 61!. 0. )+!37 . r¿c_aNPR5nycPR Ç }^gQ(R2c]hgnycwy@gcPRbwyf+~2wy]hRy^xfhn{PRbQ/¬©\®fhRJwy}P] inf. µ+NR2faR. W (x, ξ) =. Z. 1 0. ¬|b{®. W (x, u0 (x))dx, u ∈ H 1 (0, 1), u(0) = 0, u(1) = 1 ,. 1 a (x)ξ 2 2. wpc} a g]+w)£»wyQ g[1np£$£©dPc~_hgnyc]gc . L∞ (0, 1). ]hd~N%_hNwp_. 0 < c ≤ a (x) ≤ C < +∞. £©n{fwpQ(n{]|_5RbyRbfh[ x ∈ R ¯1M n±P¹¥_aNPR§gj}^Rbw{]2´µR§Nw{R)Q§wy}PR(_hNPR§nydPcz}Pwpfa[¥~n{c}^g_hgnyc]!]hxRb~g±z~ gcȬ`J{® ¯ MNPgj]q~NPn{g~2RWgj]qwyfhPg_hfwpfa[§wpc}xPjw[^]cPn(Rb]a]|Rbc{_agwy@fhn{R{¯ MNPR!dcPgikdPR!Q(gcPgQ gª2Rbf+np£+¬`J{®-g] u (x) = C. µ+g_hN. Z. x 0. 1 dy, a (y). ¯ ½ ¹\_afaw{~ _hgcP§w ]|dPz]|RJi{dR2c~2Ry^µRWQ§w[1w{]h]hdPQ(RU_aNw _ 1 µ-RJwp·\[~nyc\{R2fayRb]m_hn§]hnyQ(R 1 gc L ¯ a b Z t-n{c]hRbikdPR2ck_a[{ C ~nyc\{R2fayRb]W_an C = gc R ¯1MN\d] u µ-RJwp·\[H~2nyc\yRbfh{Rb]gc 1 dx b (x) _an u : x 7→ C Z 1 dy ¯ H (0, 1) C =. Z. 1. 0. 1 a (y). −1. ∞. 1. . 0. x. 1. hom. 0. ∗. −1. ∗. . b∗ (y). Í©Õ$Ë´Íëð.
(253) /.
(254) !"#
(255) %$&'()* ,+- . . r¿c_hNg]+n{cPR Ç }^gQ(R2c]hgn{cwp@~bwy]hRy\_hNR}^R±cg¸_agn{c¤¬D{®-np£ wpc¨wyRbfawyyRJ}§R2cPRbfh{[}^R2cz]|g_`[1faRbw{}P] Wη, (x, ξ) =. 1 inf 2η. Z. x+η x−η. W (y, v 0 (y))dy, v ∈ H 1 (x − η, x + η), Z x+η 1 0 v (y)dy = ξ . 2η x−η. Y\_hfwpgyNk_|£©n{fhµwpf}~2wy~2dPjw _hgnycz]-{g{R_hNPR!R2¹\xgj~g_£©nyfaQ Wη, (x, ξ) =. ró_+£©n{n µq]_hNwp_+_hNPR!Q(gcPgQ gª2Rbf+np£ inf. gj]_hNR5£©dPc~ _agn{c. Z. 1. 1 2. . 1 a (·). 0. −1. ξ2.. (x−η,x+η). 1. Wη, (x, u (x))dx, u ∈ H (0, 1), u(0) = 0, u(1) = 1 0. uη, (x) = Cη,. Z. x 0. . 1 a (·). . . ¬Dy{®. dy, (y−η,y+η). ! µ+NR2faR C = Z 1 dz ¯ 9R_ C = Z 1 ¯4=.NPR2c dz gj]·yRbx^_±P¹^RJ}@ Zu a]`_a(·)fhn{cPy[~2nyc\yRbfh{Rb] gc H (0, 1) ¬ë£©nyf1_hNPR¥b xP(·)fhRb\gn{d]§R¹\_afaw{~ _hgnyc«gc ®(_hn η \[s_aNPR}^n{Q gcwp_hRb}¦~n{c\yR2fayRbc~R_hNPRbnyfaR2Q¨¯ ´R2_cPn µ 1 dy u (x) = C b (·) η {n²_hnª2Rbfhnz¯¼Pn{f%RbyR2fa[ ´R2Rb]hydR¥xnygc{_ y ny£ 1 ∈ L (0, 1) ¬ë_hN\d]wpQ(n{]|_%RbyR2fa[\µ+NPR2faRsnyc b ® 1 1 ¯¦MNPR}^nyQ(gcw _aRb}²~2nyc\yRbfh{R2c~2R(_hNPRbnyfaR2Q _hNPRbc]hNPn µq]!_aNw _ → (0, 1) b (·) gc H (0, 1) ¯mt-b n{(y)c]hRbikdPR2ck_a[ u →u gc H (0, 1). ¬DPJ® lim lim u = u 1. η,. . 0. z−η,z+η. !−1. η. η. (z−η,z+η). 1. ∗. 0. ∗. 0. η, x. η,hom. −1. 1. (y−η,y+η). 1. ∗. ∗. η,hom. hom. (y−η,y+η) 1. ∗. η→0 →0. η,. 1. hom. \Y gc~2R_hNR%~n{c\yR2fayRbc~R(n{^_awygcPRJ}Ègj]]|_hfanycPHgc H (0, 1) _aNPR1RbcPR2fay[ Z W (x, u ) wpj]hn ~2nyc\yRbfh{Rb]¶_hn Z W (x, u ) gc R ¯ MNPR~2nyc\yRbfh{R2c~2RUny£ u _hn u gj]+]`_afhn{cP^gcs~nyck_afaw{]`__hn _hNzw _qnp£ u ¯ %Á ²^ E
(256) I iaY* [-(idP#e _ae _c#T!, Z ] $
(257) P L$ W$&U
(258) \&i
(259) %PP &P#, P L, P [JK W$& LT L P L %a P \& L _aV! L 1. 1. 1. 0. η,. . Ë ËÙjéçk>l. hom. . . %a LLdP# . [a I " ,Q. 0 η,. 0 hom. hom. . η,. 0. . . η. .
(260) b. # . 9R_!d]5Q§wp·{R_aNPR§wpn yRn{cPR }^gQ(R2c]hgn{cwpxPfanyRbQ R2yRbc¥Q(nyfaR(]|xRb~2g¸±z~ \[H~n{c]hg}^RbfhgcP%_hNPR R2¹PwpQ(xPRqnp£9wycnyxR2fw _anyfmny£@_hNPRU£©nyÇ faQ a (x) = a( x ) µ+NPRbfhR a(·) g]W Ç xR2fagn^}^gj~+nyc R ¯r¿c_aNPgj]-~bwy]hRy . . 1 a (·). . 1 = 2η. Z. z+η z−η. . 1 dy = a(y/) 2η. Z. z+η z−η . 1 dy = a(y). . 1 a(·). . + O( ). η (0,1). M Ng](]hNPn µq])_hNwp_ lim u = u gc H (0, 1) g£ lim η() = 0 wyc} lim = 0 ¶µ+NPgj~Ngj] Q(n{fhR)xPfhRJ~gj]|R_hNwyc¬L^J®¯MNPR(]hwyQ R)xPfanyxR2fh_`[Nnyj}P]W£©nyf5]`_an\~Nzwy]|_hgj~NPn{Q n{yRbη() cPgªJw _agn{cs£©nyfµ+NPgj~N µR5fhR2£©R2f_an)_aNPR!xPfaR2gQ(gcwpfagRb]+wpcz}_aNPRY\Rb~_hgnyc§np£ 6J;8L¯ r¿c _hNPR¦xPfankny£np£)MNR2nyfaR2Q¥£©nyf_hNPR¦nycR }^gQ(Rbc]|gnyczwpW~bwy]hRy+_aNPR¦Q(wygc gcyfaRb}^gR2ck_%gj]%_hNPR xnygck_`µ+g]hR¨~nyc\{R2fayR2cz~Rny£Ugck_aR2yfwpcz}P])}PdPR_hn4Ç _hNPRswyRbfawyygcPsnp£µ-RJwp·\[¤~2nyc\yRbfh{gcs£©dPc~_hgnyc] n{c wyj]b¯ Y\gcz~R4_hNPR¤R¹^xPfaRb]a]hgn{c.np£_aNPR¦Q gcPgQ(gªbR2f¨gc¢¬Dy{®1g]¨wycwp[k_hgj~2wpL+_aNPRÈ~nycz~d]|gnyc g] _aw{n¨~NP_hgNPR2yR%RJ}^}¦gQ(k[¤Rbcd]|g]hnygcPcÈswy_hcN}4R%_h}^nsn{Q _aNPgR§cwpg_hcPRbRJ}wpfa~g¸ny_`[{c\¯{R2=.fayR2NPczRbc²~R }^_hRbNPwyRbnygcPfaR2sQ¨µ+¯¥g_hMN¤NP_hgjN] R§]|xQRbdP~2g¸_h±zg ~1Ç }PRgQ(¹^xPR2faczRb]|]ag]|nygcnywyc¤¶gj~2]w{]|gR{cP9·ycPRJ}n ]hd~Nwpcwy[k_hgj~2wy£©n{fhQ)dPjwNPn{}P]¶£©nyf_aNPRWQ(gcPgQ gª2Rbfb¯m¼nyn µ+gcP_hNPRgcPR5np£9_hNPRxPfankny£´£©nyf-_hNRn{cPR Ç }PgQ(R2cz]|gnycwy ~bwy]hRyµR~bwpc¨£©n^~d]Unycs_aNPR faR2Rbc¨£©nyfaQdPjw(£©nyfU_hNPR ]|n{dP_hgnycHwpcz}¨d]|R_hNPR wp]|_hfwy~_ ~n{c\yR2fayRbc~Rm_hNPRbnyfa[y¯$r¿c_aNPRxPfhRJ]|Rbck_µnyfa·!NPn µR2{R2fJJµR-£©n\~2d] n{c_hNR+Q gcPgQdPQ gc]|_hRJwy}np£_hNPR GÇ Q(gcPgQ gª2Rbfwyc}sd]hR ΓÇ ~nyc\{R2fayR2cz~R!wpfaydPQ(Rbc{_]+_an1gcP·¨_aNPR)~2nyc\yRbfh{R2c~2R!np£m_aNPRRbcPR2faygRb]q_hn%_hNPR ~2nyc\yRbfh{R2c~2Rqnp£ _hNPR!Q(gcPgQ gª2Rbfa]b¯ MNPgj]+wpxPxPfan{w{~Ngd]|_hfw _hRJ]-_hNwp_qgc¨w µw[ _hNR!Q n^}^g±RJ}1RbcPR2fay[ ny£WR±zcPg¸_agn{c / gj]Uw(faR2jw ¹^Rb}R2cR2fay[ygc¨_aNPR]hxPgfag_Unp£$_hNPRNPnyQ(ny{R2cPgªbwp_hgnyc¨np£Qd¸_agxPRgck_hRbyfwpj] }PRbwp_+µ+g¸_aN¨g^c 6 8ó¯ (z−η,z+η). →0. . . 0% (. (. 1. hom. η(),.
(261) 10%. →0. →0. . MNR%£©n{n µ+gcP¦_hNPfaR2R¨R2Q(Q§w _aw¦faR2jw _aR%_aNPR¨xzn{gck_`µ+gj]|Rs~2nyc\yRbfh{R2c~2R1np£5R2cR2fay[²}^Rbc]|g_hgRb] _an¤_hNPR ~2nyc\yRbfh{R2c~2Rqnp£$_hNRwy]a]|n^~gjw _aRb}1R2cPRbfh{[(£©dPc~ _agn{cwpj]b¯ ΓÇ -Á ² T \ ;P# L !%Q
(262) " . G %Q *X>\=\ ;RP"T W$&B = B(x , R) x ∈ X a Q Rc>&0P" "aO ," ~2nyc\yR2¹ Q F :B →R I F (x) = M Q <
(263) +∞ $&" G 0 < r < R %a K =sup(M − m)/(R F (x) = m > −∞ inf − r) ¬Ly{® |F (x) − F (y)| ≤ K||x − y||. n. . . R. R. 0. 0. =. . . x∈BR. x∈BR. eY"#. L W$&T
(264) ",;P#!. x y. ¯r B. . Q. ór _+gj] 9dR2Q(]hRbQ§}%w gc% _hNPgj]+R!w(uU~xPjwyxz]aRb]|cgj~2}^wyg¹1fa_aRbn ]hdPxfh_+n np{£$Rq~2_hnyNc\RWy±zR2¹1fa]|_+wpcnpwy£9_a[^NP]hgR]b¯£©nyMNPn µ+R5gfhcPRJ wy}^RbRbQ(fgjQ(]wpfa_aRw£©R2^faµ+Rb}1NPg_h~?n N¨6 wp fa8@RW£©n{~2fqfhdzw)~gjxwpfhn\£©npny£`f ¯ _aNPR]hRbikdPR2LPwpcz}%wy]hn§]`_w _hRJ}%wyc}xPfan yRb}1gcV6 8ó¯ - W$&Á (²c W$J
(265) a L \ ;P# L iQ Q G W˜ : R ×R → R I P" N "!%
(266) " &PP . gPW Ld( ¬D / ® ˜ (·, ξ) ≤ C(|ξ| + 1) 0≤W. . Br. . . n. n. p. Í©Õ$Ë´Íëð.
(267) .
(268) !"#
(269) %$&'()* ,+- . . %aP
(270) $? W$& %Kb'P "Y;"#. gP ~2nyc\yR2¹ n Q G 4dPfPP#! W$& 4 W$&" n ˜ i gP# P`\ " I !aaP I P" ωxN ∈RRn P#W
(271) $(x, W$&·) dL ξ ∈ RnR W˜ (·, ξ)
(272) Y".(iP \ L c gP" ;P# "Y;"# $&" ω _UO!
(273) W˜ (·, ξ) Q
(274) W˜ gP' !!%
(275) " ^ L$ P" . gPWh4P Z ] %a ;P# "Y# x ∈ ω W˜ (x, ·) gP ~2nyc\yR2¹ Rn W$&" I˜ : u 7→ Z W˜ (x, ∇u)dx Γ(Lp ) X
(276) Y.()PO _. - Á ² PC . _. ˜ (x, ∇u)dx W. . W 1,p (ω). Q. . . ω.
(277) $&" Q Q G " (X, d) I ' '" .#
(278) eP \
(279) -Q G (F ) I 'P""
(280) 'N !%
(281) X [ R Q&
(282) W (F ) g P ! X "P""K
(283) * . L* dP (X, d) L$ F Γ(d) X
(284) Y.()P X ω. . . . %aU* F
(285) *Y;".()#Pf _ F \ L c gP"K L X Q 9R2Q(Q§w :Wd]hRb]$_aNPRcPnp_agn{c)ny£ ny£WR±cg¸_agn{c!¬D]|RbR Ô6 ^kY\Rb~_hgnycyΓ¯ Ç 8P~Ryn{¯ck{¯Ô® R2yfayg¸_Rb]cxP~fhRmn\gnyc £zQ gj]R2fa_hwpfa_hg~NPRb]hfxwy}P~2gfaRbRb]b~¯9_blm¯Cfa=4n \R+g}PfaRRb£©}!Rbf_aNP_hn'Rfh6 PgyyNklm_faRny¹\x_hn{Rb]hcg¸]h_aggn{n{cc P¯ @8 £©nyfq}^R2_awygj]2¯ ¼$gcwp[¥µ-R fhRJ~2wy w%xwpfh_hgj~dPjwpf!~bwy]hRnp£w_aNPR2n{fhRbQ np£ 6 8 µ+Ng~N¦fhRbwp_hRJ]U_aNPR§~n{ck{R2fayRbc~Rny£. F. L. I˜ : u 7→. Z. X. . Q(gcPgQ gª2Rbfa]+_hn(_hNR ~nyc\{R2fayR2cz~Rq£©n{fqcPn{c Ç ~2n\R2f~gyRW£©dPc~ _agn{cwpj]b¯¶M9n(_aNPgj]UwpgQ¨PR_qd]+gck_hfan^}^d~R _aNPR5£©nyn µ+gcP cPny_hgnyΓcÇ ]b¯ @)Á ACBCD D»%Å BFE*EHG " F I !%
(286) " % W$& '" .#
(287) `P \&i
(288) (X, d) [ R Q RT a"% [ I W$& \ ;PP# `" \ . P" TLC L$ eT L LT ,+>P' Q " P" "%
(289) . G I (X, d) _ R Q RTKa"% _ I K − limF M (FX) L$ \ ;PP# I (F b )I\% 'P" 7 O W$&. L (X, d) Q b LK L .Pe eL L$ fP""
(290) P fT L LT ,+>P u ∈ M (F )→0 =¦R!fhR2£©R2f_an`6 ^Y\Rb~_hgnyc]§wpcz}s;8@£©nyfq}^R2_awygj]nyc%_hNPRJ]|R!cnp_hgnycz]+wpc}%nyc_hNR5£©nyn µ+gcP
(291) %$ " ;Q OPP!U W$& ) gP`VP" " "%
(292) ` O!%
(293) " % P? $d
(294) $ Γ(d) X -Á ²
(295) Y.()P^ W$& '" #
(296) VP \&i
(297) (X, d) _ `(F !%
(298) " % F W$& gP ai* .
(299) L +∞ Q
(300) W % V a W. & $ ? L. > h V J .. _. L. aM X Z[ W$& gP M (F ) 6= ∅] L$ M (F ) = lim inf F = inf F →0 X X Q
(301) # \&# .
(302) "!, T L. L. T ,. + " W$&" i gP# .P P" " "%
(303) K − lim M (F ) u ∈ M (F ) →0 P#
(304) $U W$& lim u = u L (X, d) Q u ∈ M (F ). M Nc(F !%)
Documents relatifs
C'est aussi le résidant pas chez leurs parents fréquentent deux fois parativement à la province, les étudiants de Nanterre ne lement ce moindre attachement à la ville
In this paper, we study the performance of an enhanced channel estimation technique combining estimation using an autocorrelation based method and the Expectation-
Selon Laeven et Valencia (2012) , le coût budgétaire de la crise bancaire s’est respectivement élevé à 44 et 41 points de PIB en Islande et en Irlande, ce qui place ces deux
Dans ce travail, nous comparerons les entreprises coopératives, détenues par leurs membres, dont le produit risqué est la ressource critique apportée et dont l’objectif est
À cet effet, un nombre considérable de recherches démontre cliniquement que les enfants de parents souffrant de problème de santé mentale sont plus à risque d’un retard dans leur
usual route to school and provide us with some details about their travel routine (e.g., schedule, mode of transportation, people accompanying them, etc.); 3) recent interventions
La question de l’extension des limites de Paris jusqu’à cette ligne avait été discutée mais ne fut tranchée que par un décret impérial du 9 janvier 1859 qui décidait de