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Stochastic homogenization and random lattices

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(1)Stochastic homogenization and random lattices Xavier Blanc, Claude Le Bris, Pierre Louis Lions. To cite this version: Xavier Blanc, Claude Le Bris, Pierre Louis Lions. Stochastic homogenization and random lattices. 2007. �hal-00140076�. HAL Id: hal-00140076 https://hal.archives-ouvertes.fr/hal-00140076 Preprint submitted on 4 Apr 2007. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published 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..

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(419) k?j ‚f

(420) n,ghfnj“ijk4jn;m

(421) ]@vjb b;_Zk?ivY¼¢_`n;v bän;jg“mƒn;§b;e?mRbb;e4j v f]Z}6b;_Zî fk Ú Ó fƒ‚ á B

(422) Y q)â|_dvA’f}4k?6j–_ZA k ‘ ø ä4mƒk?¨bet}?vAcf

(423) ktÊ

(424) jnijvä}?a¨b;fmƒk–j¤tbn8m

(425) cb_`f

(426) k=ä6”~jm

(427) §t]Zo“_`kA‘ ø Y@st_`gu_Z]dmƒn;]`o

(428) ä B& Ì  c

(429) f t k

(430) Ê  j  n  i  j | v | ”  j ƒ m t § ` ] “ o Z _ k Ò.ÔÕh¡ Ø Ù,Ú Ó  Ù,Ú Ó Í ¢ Ù,Ú ’ _Zk B Ö á B

(431) Y â ҕòóõôpí ¾•e4jk=ä4be4j"vf

(432) ]`}6b_`f

(433) k. €|ò. bf. Õî. ïpÙ.Ú Ó Í ¢¤£ ’ _`k. B . á B

(434) Y ¥â à. á B

(435) Y¹

(436) â•v;mRb_dv ë?jv ف€ òí. Õî. á BYZB¨â. ï Í ‚¢¦æف€ ò:§ Ýwv4Ö. á ”Ae4jnj ¦ž© § 6jk4fƒb;jv•b;e4j"mÊ

(437) jn8mƒij3_`k–be4j"ajn;_Zf66_dcpv jb b;_Zk4itâ ä6b;et}?vÌ }. ã k¼m46_Zb_`f

(438) k=ä Ò òˆ óõô í. Õî. ÿyف€|òí. Õî. ï Í ‚¢. }¥_`k. Õî ïí } ÿyف€|òí < ï ï Í ‚¢. B C. á BYZBBâ à. C ÔWÒ6‚؈ }¥_`k B. á BYZBr

(439) â à. £j¤tbä3af

(440) _`kb;_Zk?i³f}6b¨b;e?mRb. á BY )âm

(441) k? á B

(442) Y`B

(443) BâÈmƒn;j c }4n]Zåæ‚njj

(444) ä3m

(445) k? á B

(446) Y ¥âm

(447) k? á B

(448) Y`Brâmƒn;j 4_ZÊjn;i

(449) jkcjåæ‚n;jj

(450) ä•b;e4j cf

(451) guajk?v;mRb;j³cf

(452) gua?mc b;k4jv;vha4n_`k?c_Za4]`jµéZB¹täABzê á f

(453)  n ä@gufnj–a4n;jc_dvj]`o e4jn;j

(454) ä@be4j¥cj]Zj’4n8mRbjG/˜— (

(455) Dªdz2z{3)â mƒ]`]ZfR”£v|bfua?mvv~b;fube4j"]`_`gh_Zb£_`k–’fƒb;e¥v _d6jv•f

(456) ‚ «. Ò òˆ óõô í. Õî. Õî ïí } ÿ³Ù€|ò,í < ï ï?¬ ˆ Ù,Ú Ó Ý. Õî í } ÿyف€|ò?Ô (Ø ï ˆ. «. ҕòóõôpí. Õî. ïpÙ,Ú Ó ¬•Ö. i

(457) jbb_`k4i «. ñ£]`]<be4_dvA_dv•ÊRmƒ]`_`¨‚f

(458) n£m

(459) kto#} ý. ÔWÒ6‚ØVˆ } ¬ ˆ Ù.Ú ’ Ý. @ºúY^¾•e4_dvämƒ]`f

(460) k4ih”A_5b;e. }. ˆ £ ’ à. ͼÎ4ÏZÐ=Ô£ ’ Ø^ÝìÞ ä6i_ZÊjv~be?j7e?f

(461) guf

(462) ijk4_`ljÈjÉ}mRb_`f

(463) k›Y. P ftf

(464) §t_Zk?ih’m

(465) c8§mRbAb;e4j"a4n;ftfƒ‚I”~jpe?mÊ

(466) jD5(}?v b£f

(467) }6b;]Z_`k4j<ä4”~jpvjjpb;e?mRb•b;e4j"g“mƒ_`k¨b(”~fu_Zk?i

(468) n;j6_`jk)b;v~e?mÊjp’jjk. á _dâÈbe?j"”|jmƒ§Ècf

(469) ktÊ

(470) jnijk?cj3fƒ‚njvcmƒ]`jÈ‚}4k?cb_`f

(471) k?v á ‚f

(472) n ÒAòó(ô mƒk? á _`_dâÈbe?j"”|j]Z]Zåæafvj4k4jv;v~fƒ‚be4jcf

(473) n;njc bfn|a?nf’4]Zjg. ف€|ò âä. á b;e?mRb£_dv á B

(474) Y¹

(475) â âä. mƒ]`f

(476) k4i"”A_Zbe“b;e4j3cf

(477) guajk?v;mRbj“cfgham

(478) c b;k4jv;vºa4n;_`k?c_`a4]`j

(479) Yºñ~b|]`jmv(b^‚fng“m

(480) ]Z]`o

(481) ä”|j£g“mohvmo.be?mƒb|”Ae4jk4jÊjn^”~j£g“mo 6jë?k?jpmuvjbb_`k4ih‚f

(482) n•e4f

(483) gufi

(484) jk?_ZlmRb_`f

(485) k‚f

(486) nA”Ae4_dc8eÈb;e4j"mƒ’fRÊ

(487) j:b(”|fha4n;f

(488) ajnb_`jv á _+âÜå á _`_+â•mƒn;j7v;mRb;_`v ë?j=ä6”|j7”A_Z]`]=’jp_Zk af)v _Zb_`f

(489) k–bfm

(490) a4a4]`ob;e4) j  *+ -,/.z”•1ƒämƒk¨ajn‚f

(491) n;g m

(492) k¨j¤6a4]Z_dc_Zb3e4fguf

(493) i

(494) jk4_`lmRb;_ZfkÈf

(495) ‚f}4nAjÉ)}?mƒb_`f

(496) k=Y|¾•e4_`v£”A_Z]`] ’j"j¤tjgua4]Z_Zë?jÈ’toÈf}4n£mƒn;i

(497) }?ghjk)b~_`kstjcb_`f

(498) kçB

(499) Y €4Y. ¹.

(500) š ´I‰ßu o K±7}4jb;vjk4i é` B ¨êÜäAmƒk? O ­.1 Q ®;ŠU

(501) ž4¬‰ìU

(502) Q+ N ¯‰)PR®‰)NU

(503) ‰©«¶‰)O´IQ=SD° ¾•e4_`vugujbe4f6½”•m

(504) vhë?n8v(b_`k)bn;f66}?cjy’t\ ‚}4n b;e4jn4jÊ

(505) j]Zfajç’)o ñ£]`]`m

(506) _Zn;j¼éZBêæY ã k cf

(507) k)bn8m

(508) v b:b;f¼be4jm

(509) ’fRÊj,jk4jn;i

(510) oèghjbe4f6<ä_Zb”~mvpf

(511) n;_Zi_Zk?m

(512) ]Z]`o¥_`kb;nf66}cj bfȁ6jmƒ]I”A_5b;e¥be4j.ajn_`f66_`c.v jb b_`k4iY ã k¥be4_dv3vjbb_`k4i?äbe?j.cn;}?c_dmƒ]›bftf

(513) ] á ”Ae4_dc8e¥_`kèv fgujv jk?v j.a4]`mo6v•b;e4j.n;f

(514) ]`j"fƒ‚ be4j3cf

(515) guajk?v;mRb;j“cfgua?m

(516) cbk4jvvºa?n_`k?c_`a4]`j£_`kube4j£a?njcj4_Zk4i.ghjbe4f6?âº_`v^be?mƒb~mƒkto,’f}4k?6j“v jÉ}?jk?cj ÚÓ _` k ‘ ø vmƒb_dv(‚o“b;e4jp‚f

(517) ]`]`fR”A_Zk4iucfktÊ

(518) jn;i

(519) jk?cjv á }4a–bf“j¤)b;n;mc b;_ZfkÈfƒ‚ºmuv }?’?v jÉ)}4jk?cjâÌ. û ² ý. ² B .  Ž ÔL Ö B  ³ ´<µ Ô Ø ØÖ¶„Ÿ·–Ù,ÚÓ. Ú ’ _ZkA‘. Ú Ó Í ¢ í ÕIÖ. Õî. ². ø Ö. ï ‹ ÕÍ ¸¹„Ÿ·U„ ÔÙ,Ú ’ ÔÕ%ØIÿ³ÙºÚ ÔW՛ÖV~6ØØ ø. ÔW՛Ö~4Ø ‹ ~ ‹ ՛Ö. ‚f

(520) n¼v ² fghj. i f)jv¨mv‚f]Z]`fR”£vÌ ”|j ý B& Ô L Ö ‘ ø Ô Ž Ø Ø YiN:v _`k4i³bî e4_dv–njv }?]5bäAbe4jça4n;ftfƒ‚fƒ‚,e4f

(521) guf

(522) ijk4_`lmƒb_`f

(523) k  Ú ø ë?n8v(b£]`_Zk?jfƒ‚ ³´<Bµ Y qtâ~’)o g.}4]Zb_`a4]`oÈb;e4jp A ” 4 e  j ; n j ‘

(524) m ? k  ‘ ý ’ ÔWÕØIÿ ’ ý á q ÕI֟r Ó s Ö ø ÔL Ø ø Ô L Ö ‘ ³ø ´<µ Ô Ž Ø Ø Ö mƒk? ø e4jmƒ’ øv }?v j ` _ ¨ k ; b R f

(525) Ê p j  c  f t k

(526) Ê  j ; n

(527) i  j ? k  c  j I Y • ¾ 4 e d _ A v ` _ u g 4 a Z ] ` _  j J J J ’ ÔÕØÿyÙ º J ÔW՛ÖV~6ØØ ÔW՛Ö~4Ø@ÝìÒ.Ô~4ØÔÙ ø J. „ ·. J. „ ¥ÔæÙ,Ú ’ ÔÕØÿyÙ. º Ú. ÔW՛Ö~4ØØ6Ò.Ô~6ØÔÙ ø. ’ ÔÕØIÿyÙ J. º J. ø. ÔW՛ÖV~6ØØ ‹ ~ ‹ Õ¥Ý. „ ·. Þ. ’ à. á BYZB€â. J. ã bA‚f

(528) ]`]ZfR”£v Í¥Î6Ï`Ð º Ñ Ò.Ô~4ØÔÙ,Ú ’ ÔWÕ؛ÿ³Ù º Ú _ZkMŽE”A_5b;e–ajn;_`ft4_`c7’f

(529) }4k??mƒn;oÈcfk?6_Zb_`f

(530) k=Y^¾•e4_dv•_`gha?]Z_`jv•b;e?mRb ú Ú ø. ÔÕIÖ~4Ø@Ý. ø. ÔÕIÖ~4Ø Ø ‹ ~)Û<Ýwv4Ö. x Ú ’ x Õ $ WÔ ÕØ<€ ó<‡ ÔÕ%Ø à. F $4». ø ã k?v jn b;_Zk?i,b;e4_dv•jÉ)}?mRb;_Zfk¨_`k)bf á B

(531) Y`Bۉ@i

(532) _`Ê

(533) jv~be4j"e?f

(534) guf

(535) ijk4_`lja4n;f

(536) ’4]`jg¼Y ñ3im

(537) _Zk=ät”~j:vjj3be?mƒb|b;e4j7cf

(538) ktÊ

(539) jnijk?cjAfƒ‚=n;jv;cm

(540) ]Zjh‚}4k?c b;_Zfk?v~a4]dmotv~m§joun;f

(541) ]`j

(542) äbf

(543) ijb;e4jn|”A_5b;ebe?j76jë?k4_Zb_`f

(544) k fƒ‚^b;e4jhcf

(545) n;njc b;f

(546) n3a?nf’4]Zjg á _`kè‚Wmc bp_Zgua4]`_`c_5b;]Zo cf

(547) k)b8mƒ_`k4j¥_`k á B

(548) Y`B€)â âY¾•e4_`vpvf

(549) guje?fR”©ve4fR”£v3be?mƒb"f

(550) }4n7i

(551) jk4jn8mƒ] ’j]Z_`j‚f

(552) k¨be4j"§jo_`k4inj6_Zjk)b;v•e4f]`Èbn;}4j

(553) ämRbA]Zjm

(554) v b•‚f

(555) n;gum

(556) ]Z]`o

(557) Y ñ:v”~j“af_Zk)b;jçf

(558) }6bhmƒ’fRÊ

(559) j

(560) ä=be4j“b(”~fƒåõv;cmƒ]`jucf

(561) ktÊjn;i

(562) jkcjhghjbe4f6µ”•m

(563) v"mƒbë?n8v(b.6jv _`i

(564) k?j bf6jmƒ]^”A_Zbeµb;e4j ajn_`f66_`c"vjbb_`k4i?Y™¼£fR”~jÊjnä6_5b:”~mv~be4jk6jÊj]`f

(565) aj¨‚}4nbe4jn£bf4jmƒ]›”A_5b;e¥g.}?c8e¥guf

(566) n;jpi

(567) jk4jn8mƒ]=cm

(568) vjv á v jj“éZBBƒä Brtä6B€6ä4Bqƒê+âä

(569) ”Ae4_dc8e,a?nfRÊt_d6j~m7k4_dcjäƒn8mRbe?jnbjc8e4k4_dcmƒ]æäR‚n8mƒguj”~f

(570) n;§p‚f

(571) na4}4b b_`k4ipb;e4j£mƒ’fRÊ

(572) j|‚f

(573) n;g“mƒ]6cfk?v _d6jn;mƒb_`f

(574) k?v _Zk)bf:g“mRbe?jg“mRb;_`cmƒ]bjn;g“vY ã bn;jg“m

(575) _Zk?î v›bemRb_5b_Zk)b;n_`k?v_`cmƒ]`]Zopj¤6a4]`f

(576) _Zb;v›be?j@‚Wmc b›b;e?mRb”~j|e?mÊjºb(”~f:6_Z—jnjk)bv;cmƒ]`jvÌ m"gu_`cnfhvcmƒ]`j

(577) ä

(578) ”Ae?_`c8e”~j:4jk4f

(579) bj3’to ätm

(580) k?m"g“mcn;f.vcmƒ]`j

(581) ä)”Ae4_dc8e”|j:v jb|jÉ)}?m

(582) ]?b; f ½

(583) Yº¾•e4_dv|j¤6a4]dmƒ_`k?vºb;e4j:”~f

(584) n84v ¾ b(”~fƒåõvcmƒ]`j ¾ cfktÊ

(585) jn;i

(586) jk?cjY ã k¼v fguj7vjk?vj

(587) ät”Ae?mƒb£6jajk??v•f

(588) kb;e4jpgu_`cnfuvcmƒ]`j:_dv•v jbAf

(589) k¨vf

(590) guj7}4k4_Zb3cj]Z] ”Ae?_`c8e á _`v3be4jh}4k4_Zb"cj]`]fƒ‚ºb;e4j,ajn;_`ft4_`c.]`mƒb b_dcjh_`kbe?j.ajn;_Zf66_dc,cm

(591) vjâäi_ZÊt_`k4i¨m

(592) k ¾ j¤ta?]Z_dc_Zb ¾ cf

(593) n;n;jc b;f

(594) n£jÉ)}?mRb;_Zfk=Y Ç j bet}?v7e?mÊ

(595) j._Zkèbe4_dvpcm

(596) vj,m¨guf

(597) n;jj¤6a4]`_dc_Zb7”•mo¼fƒ‚|cf

(598) gua4}6b;_Zk4i¨e4f

(599) gufi

(600) jk?_Zljcf)jŒc_`jk)b;v3be?m

(601) kè”A_Zbeèbe4jujk?jn;i

(602) o ghjbe4f6<Y Ç ju”A_Z]`]k4fƒbpfRÊ

(603) jnÊt_`j” be?j,”~f

(604) n;§6v.é`B

(605) BƒäBrtäB€6äBqƒê^_Zkèbe4jha4n;jvjk)b7_`k)bn;f66}?c b;f

(606) n;o–vjcb_`f

(607) k=ä<’jcmƒ}?vjh”|j,”A_Z]`] cf

(608) gugujk)bAf

(609) k¨be?jg _Zk¥stjcb_`f

(610) k¹6Y QR|¿. ÀÁ]&^_bÃÂ=Y(]ÄZ1[egÂ=YWVZÂŸbhYŒY(WVcO`&Â. ¾•e4jpa4njv jk)b•vjcb_`f

(611) k–_Zk)bn;f66}?cjv•m,4_`v;cn;jb;jpm

(612) k?–m,cf

(613) k)b_`kt}4f

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