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1 . Introduction to Systematically Pass to the L Imit in a Product Of ISSN sub w ww divided by hline to the power of B to the power of hline from hline to Banach a n sub : sub period 1 e 735 m i s hyphenn begin 8787f a l i periodg n ∗g sub d e slash j open parenthesis o electronic u r n al s slash B closing parenthesis sub J M A slash to the power ofISSN a c h ^ Jf subB J ^f period n r u l Math e f3em ogf u r0.4 n apt periodg ^f sub n r u l l Anale f3em ogf f period0.4 pt 5g M g af tBanach open parenthesisgg a 20 n h eg 1f mw 1 an closingf r a c f parenthesisww gf n r u l e f3emgf0.4 pt g gg^f a tc sub h i to the J powerf J g of comma. Math sub c nof ao l period u 1 r A comma n a subg n. 1 subf al 0g l 1Anal y endashf subo s i f 1g s 35. Capital 5 Sigmaf M hyphen a g t ( 20 f h e g 1 f m g 1 f a g ) f t g^f , g f ; i g f c g no f a l g . 1 f A g CONVERGENCE B Banach ana chJJ:Matho urna:l Analof :5Mat(20he1m1a)ticnoal: 1A;n1a0l1y −−si1s35 ISSNw ww , GABRIELf n g 1 NGUETSENGf a g 0 tof thel powerg ::1e1735 ofmisf 1−y AND8787g:de NILS −−=j(oelectronic SVANSTEDTf surnals i =Bg)J to1MA thef powers =g of35 2 *g f : f . g 1 f e g 735 f m iCommunicated s g − 8787 by Cf period. g Badeaf d e / j g ( f o g e l e c t r oΣ n− i cCONVERGENCEf u r n a l s / B g ) Abstractf J g periodMA/ .. We discussgnn n twoSigma new concepts− CONVERGENCE of convergence in L to the power of p hyphen spaces comma the so hyphen 1 2∗ nendcalledf a l i weak g n ∗g Capital Sigma hyphenGABRIEL convergence NGUETSENG and strongAND Capital NILS Sigma SVANSTEDT hyphen convergence comma which are intermediate between classical weak convergence and strongCommunicated convergence by period C . Badea We also introduce n centerlinethe conceptf ofGABRIEL CapitalAbstract Sigma $ . NGUETSENG hyphenWe discuss convergence ^f two1 newg$ for concepts ANDRadon NILS measures of convergence $ SVANSTEDT period in OurLp− basic ^spacesf 2 tool , isthe∗ the g so$ classi - g hyphen cal Gelfand representationcalled weak theory Σ− convergence period Apart and from strong being Σ− a naturalconvergence generalization , which are of intermediate n centerlinewell hyphenf knownCommunicatedbetween two hyphen classical by scale C weak convergence . Badea convergenceg theory and comma strong theconvergence present study . We lays also the introduce foundation of the mathematicalthe frameworkconcept of Σ that− convergence is needed to for undertake Radon measures a systematic . Our study basic tool is the classi - n centerlineof deterministicf Abstract homogenizationcal Gelfand . nquad representation problemsWe discuss beyond theory the two . Apart usual new fromperiodic concepts being setting a natural of period convergence generalization in of $ L ^f p g − $ spaces , the so − g A few homogenizationwell - known problems two are - scale worked convergence out by way theory of illustration , the present period study lays the foundation n centerline1 period .. Introductionf calledof the weak mathematical $ nSigma framework− $ that convergence is needed to undertake and strong a systematic $ nSigma study − $ convergence , which are intermediate g To systematically passof deterministic to the l imit homogenization in a product of problemstwo weakly beyond convergent the usual s e hyphen periodic setting . n centerlinequences onef classicallybetweenA requires classical few homogenization that open weak parenthesis convergence problems at are least worked and closing outstrong parenthesis by way convergence of illustration one of the two . We s equences also introduce converges g strongly period .... More precisely comma ....1 .let CapitalIntroduction Omega be an open set in the N hyphen dimensional numerical n centerlinespace R tof thetheTo power systematically concept of N open of parenthesis $ passnSigma to the N greater l− imit$ equal in convergence a product 1 closing ofparenthesis for two Radon weakly comma measures convergent let open . parenthesis Our s e - basic u sub tool epsilon is the classi − g closing parenthesisquences sub one epsilon classically greater requires0 be a sequence that ( in at L least to the ) power one of of the p open two parenthesis s equences Capital converges Omega closing parenthesis openn centerline parenthesisstronglyf cal Capital . Gelfand OmegaMore precisely representationprovided with , let LebesgueΩ be theory an open . Apart set in from the N being− dimensional a natural numerical generalization of g N p measurespace closingR parenthesis(N ≥ 1); andlet let(u" open)">0 parenthesisbe a sequence v sub in epsilonL (Ω)(Ω closingprovided parenthesis with sub Lebesgue epsilon greater 0 be a s equence in L n centerline f w e l l − known two − scale convergencep0 theory , the present1 study laysp the foundation g to the powermeasure of p to the ) and power let of( primev")">0 openbe a parenthesis s equence Capital in L (Ω) Omega; where closing1 < parenthesis p < 1 and commap 0 =1 where− 1 less1: p less infinity and p p0 p 1 dividedIt by i prime s a classical = 1 minus fact hline that from if pu to" ! 1 subu in periodL (Ω) ( strong ) and v" ! v in L (Ω)− weak n centerline f of the mathematical framework1 that is needed to undertake a systematic study g It i s a classicalas " ! fact0; then that ifu u"v sub" ! epsilonu0v0 in rightL arrow(Ω)− weak u in L to. the power of p open parenthesis Capital Omega closing parenthesis open parenthesisHowever strong closing , in parenthesis a great number and v sub of s epsilon ituations right aris arrow ing v in in mathematicalL to the power of analysis p to the it power i of prime open parenthesisn centerlines Capital oftenf of Omegacrucial deterministic closing to investigate parenthesis homogenization the hyphen l imiting weak behaviorsproblems of beyond products the of usual the preceding periodic form setting . g as epsilonin right spite arrow of the 0 factcomma that then u none sub epsilon of the vtwo sub sequences epsilon right i s arrow allowed u sub to0 strongly v sub 0 in L to the power of 1 open parenthesisn centerline CapitalfA few Omega homogenization closing parenthesis problems hyphen weak are period worked out by way of illustration . g However comma .. in a great number of s ituations aris ing in mathematical analysis it n centerlinei s often crucialf1 .ton investigatequad Introduction the l imiting behaviorsg of products of the preceding form in spite of the fact that .. none of the two sequences i s allowed to strongly n hspacehline ∗fn f i l l gTo systematicallyDate : Received pass : 4 Mayto the 20 1l 0 ; imit Accepted in :a 9 product July 20 1 0 of . two weakly convergent s e − Date : Received : 4 May 20 1 0 semicolon Accepted∗ Corresponding : 9 July 20 author 1 0 period . nnoindent* Corresponding201quences 0 Mathematics author one period classically Subject Classification requires . thatPrimary ( at46 J least 1 0 ; Secondary ) one of 35 B the 40 , two 28 A s 33 equences . converges 201 0 MathematicsKey words Subject and phrases Classification . Homogenization period .. Primary , homogenization 46 J 1 0 semicolon algebras Secondary; Σ− convergence 35 B 40 comma , Gelfand 28 A 33 period nnoindentKey wordstransformations and t r o n phrases g l y . periodn h f i l.. l HomogenizationMore precisely comma , n homogenizationh f i l l l e t $ algebrasnOmega comma$ be Capital an open Sigma set hyphen in convergence the $N −comma$ dimensional Gelfand numerical 1 1 transformation period nnoindent1 1 space $ R ^f N g (N ngeq 1 ) , $ l e t $ ( u f n varepsilon g ) f n varepsilon > 0 g$ be a sequence in $L ^f p g ( nOmega )( nOmega $ provided with Lebesgue nnoindent measure ) and let $ ( v f n varepsilon g ) f n varepsilon > 0 g$ be a s equence in $ L ^f p ^f nprime gg ( nOmega ) , $ where $ 1 < p < n infty $ and $ p n f r a c f 1 gf nprime g = 1 − n r u l e f3emgf0.4 pt g ^f p g f 1 g f . g$ nnoindent It i s a classical fact that if $ u f n varepsilon g nrightarrow u $ in $ L ^f p g ( nOmega ) ($ strong )and $v f n varepsilon g nrightarrow v $ in $ L ^f p ^f nprime gg ( nOmega ) − $ weak nnoindent as $ n varepsilon nrightarrow 0 , $ then $ u f n varepsilon g v f n varepsilon g nrightarrow u f 0 g v f 0 g$ in $ L ^f 1 g ( nOmega ) − $ weak . However , nquad in a great number of s ituations aris ing in mathematical analysis it i s often crucial to investigate the l imiting behaviors of products of the preceding form in spite of the fact that nquad none of the two sequences i s allowed to strongly n begin f a l i g n ∗g n r u l e f3emgf0.4 pt g nendf a l i g n ∗g n centerline fDate : Received : 4 May 20 1 0 ; Accepted : 9 July 20 1 0 . g n centerline f $ ∗ $ Corresponding author . g n centerline f201 0 Mathematics Subject Classification . nquad Primary 46 J 1 0 ; Secondary 35 B 40 , 28 A 33 . g Key words and phrases . nquad Homogenization , homogenization algebras $ , nSigma − $ convergence , Gelfand transformation . n centerline f1 1 g 1 2 .. G period NGUETSENG comma N period SVANSTEDT nnoindentconverge period1 2 n ..quad ForG example .
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