MATHEMATICS. Received February 26, 2016

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1 IN Doklady Mathematics 206 Vol. 94 No Pleiades Publishig Ltd Published i Russia i Doklady Akademii Nauk 206 Vol. 469 No MATHEMATIC Homogeizatio of the -Lalacia with Noliear oudary Coditio o Critical ize Particles: Idetifyig the trage Term for the ome o mooth ad Multialued Oerators J. I. Diaz a * D. Gómez-Castro a ** A. V. Podol skii b *** ad T. A. haoshikoa b **** Preseted by Academicia of the RA V. V. Kozlo February Receied February Abstract We exted reious aers i the literature cocerig the homogeizatio of Robi tye boudary coditios for uasiliear euatios i the case of microscoic obstacles of critical size: here we cosider oliear boudary coditios iolig some maximal mootoe grahs which may corresod to discotiuous or o-lischitz fuctios arisig i some catalysis roblems. DOI: 0.34/ Paers [2 0] were deoted to the study of asymtotic behaior of the solutio to the boudary alue roblem for the -Lalacia ( [2 )) i -eriodically erforated domai with oliear Robi-tye boudary coditio that cotais fuctio σ ( xu ). It was suosed there that σ ( xu ) is a smooth fuctio of it s argumets mootoe by ariable u. I this aer we exted the method itroduced i [ ] to deal with the roblems with more geeral coditios o the fuctio σ ( xu ). As i all aers i which the holes are of critical size ad the aortio arameter has a critical ower of (we will recise this later) we obsere a chage i the ature of the oliearity. Our aim is to reset this chage i the case σ ( u) = C u u 0 < < which is ot differetiable at 0 ad i the case whe σ is the maximal mootoe oerator for the Heaiside fuctio which is a multialued oerator ad [2 ). I a further aer [2] we exted the argumets to the case of geeral maximal mootoe grahs σ ad ( ). The article was traslated by the authors. a Comlutese de Madrid. Ist. Mat. Iterdisciliar & De. Mat. Alicada. Fac. Mat. Plaza de las Ciecias Madrid ai b Moscow tate Uiersity Faculty of Mechaics ad Mathematics Moscow 9992 Russia *idiaz@ucm.es **dgcastro@ucm.es ***origialea@ya.ru ****shaosh.ta@mail.ru Let be a bouded domai i R 2 with a smooth boudary ad let Y = (. Deote by 2 2) G0 the uit ball cetered at the origi. For δ>0ad 0 <! we defie sets δ = { x δ x } ad α = { x ρ ( x ) > 2 }. Let a = C 0 where α>ad C0 is ositie umber. Defie = = 0 ϒ ϒ where ϒ = { Z ( ag 0 ) } ϒ d d = cost > 0 Z is the set of ectors z with iteger coordiates. Defie Y = Y where ϒ ad ote that G Y ad ceter of the ball coicides with the ceter of the cube Y. Defie G = G = G =. Cosider the roblem Δ u = f x ν u σ ( ) u x u = 0 x () where Δ u di( u u) [2 ) ν u u ( uν ) ν is the outward uit ormal ector to γ=α( ). We suose that f L ( ) ' =. G ( a G ) G 387

2 388 DIAZ et al. Fuctio σ( λ) [0 ] is the maximal mootoe cotiuous maig [] that dee o arameter 0 λ < 0 σ( λ ) = λ λ (0) (2) λ >. We ote that σ0( λ) is the maximal mootoe maig for the Heaiside fuctio i.e. multialued fuctio 0 λ < 0 σ0( λ ) = [0] λ = 0 (3) λ > 0. oudary coditios of this tye corresod to the resece of so called chemical reactio of order o the boudary of caities [5 6]. The motiatio to trucate the owers comes from the chemical modelig i which cocetratios imose rage i [0 ] but it also correso to the case f L ( ) for which the solutio is bouded. Alyig mootoicity tools (see e.g. []) it is easy to see that roblem () is euialet to ask for u W ( ) satisfyig the itegral ieuality u u ( φu ) dx ( ψ( φ) ψ( u )) f( φ u ) dx (4) for ay arbitrary fuctio φ W ( ) where ψ( λ) for (0 ] is the rimitie of σ. For (0 ] we hae that 0 λ < 0 ψ λ ( λ ) = λ (0) (5) λ λ> ad if = 0 the 0 λ 0 ψ0( λ ) = (6) λ λ > 0. ace W ( ) is the closure i W ( ) of the set of ifiitely differetiate fuctios i that aish ear the boudary. It is well kow that roblem () has uiue weak solutio (see. e.g. [ Theorem 8.5]). The followig estimatio is alid L ( ) L( { xu > }) u u K (7) where costat K here ad below is ideedet of. Let H ( λ) be the fuctio gie through fuctioal euatio 0 H H σ( λh) (8) where 0 = cost > 0. Note that for ay rescribed λ euatio (8) has uiue solutio. I the case if = 0 H 0( ) λ > 0 ad if (0 ] the 0 λ < 0 λ = λ < λ < (9) (0) where b( s) is the strictly mootoe fuctio gie for s 0 by Note that i both cases Lischitz cotiuous. 0 λ < 0 H( λ ) = ( b) ( λ ) 0 < λ < 0 0 λ > 0 0 b () s s s = λ. H ( λ) () is bouded ad Deote by ũ a W -extesio of fuctio u (see [0]). Usig estimate (7) we get followig ieuality u. (2) K W ( ) Therefore there exists a subseuece (deoted as the origial seuece) such that 0:. u u weakly i W (. ) (3) The followig theorem gies a descritio of the limit fuctio u. Theorem. Let 3 [2 ) [0 ] α= ad is the weak solutio of γ=α( ) u the roblem (). uose that H ( λ) is the fuctio gie by euatio (8) i which 0 = C0. The the limit fuctio u itroduced i (3) is the weak solutio of the roblem Δ u A( ) H( u) H( u) = f x (4) u = 0 x. which is uderstood as a fuctio u W 0 ( ) satisfyig the ariatioal ieuality DOKLADY MATHEMATIC Vol. 94 No. 206

3 HOMOGENIZATION OF THE -LAPLACIAN 389 ( udx ) A ( ) H ( ) H ( )( udx ) for a arbitrary fuctio 0 ( ). Here ( ) A ( ) = C0 ω with ω the surface area of the uit shere i R. Proof. (a) Cosider the case = 0. Deote = 0. Note that the itegral ieuality i the case whe = 0 has the form (5) where φ is the ositie art of fuctio φ φ=φ φ. From (5) we coclude Usig the mootoicity of fuctio > we derie ieuality for u that is alid for a arbitrary fuctio ϕ W0 ( ). Let us take a test fuctio i ieuality (7) (6) for (7) where C0 ( ) H 0 ( λ) is gie by the formula (9) fuctio W( x) is defied as follows Here T is the ball of radius /4 ceter of which coicides with the ceter P of G f( u) dx u u ( φu ) dx W ( φ u ) f( φ u ) dx u K. L ( ) φ φ ( φ u ) dx ( φ u ) f( φ u ) dx ( x) ( x) W ( x) H ( ( x) ) φ = 0 λ 2 w x T G ϒ W( x) = x G (8) 0 x R T. ϒ w ( ) /( ) ( ) /( ) x P ( /4) ( x) ( ) /( ) ( ) /( ) a (/4) =. λ Note that W 0 weakly i W ( ) as 0. Usig that = H ( ) [0 ] (9) (20) ( u ) u. ubstitutig itroduced test fuctio i ieuality (7) ad usig (9) ad (20) we get that the limit as 0 of the left-had side of the ieuality (7) does t exceed the limit of (2) The limit of the right-had side of ieuality (7) is eual to f( u) dx. (22) Cosider the itegrals oer exressio (2): 0 < H 0 ( ) < 0 > H ( ) > 0 ad φ = ( H = if 0( )) H 0( ) > H 0 ( ) we get u ( φ ) { > } { (0 )} ( u) dx ( u ) { > } u w νw H0 ϒ G { (0 )} H ( H ( ) u ) w w H H ( u ). ϒ T ϒ G 0 0 ν 0 0 icluded i the ν w w H H ( H ( ) u ) ( ) { > } { (0 )} u u = α ( ( ) ( ) 0 H0 H0 H0 { > } u ) ( u ) { (0 )} u DOKLADY MATHEMATIC Vol. 94 No. 206

4 390 DIAZ et al. = ad α 0 if ( ) 0( ) ( α) H H u H ( ) H ( )( H ( ) u ) 0 α 2 α H ( ) H ( )( H ( ) u ) { (0 )} { > } u ( u ) = 0 α 0( ) 0( ) ( ) (23) (24) Usig that u K we get that the limit of L ( ) the exressio (2) does t exceed the limit of the followig exressio H H u 0 0 { > } 0 u { (0 )} ( u ) α0 2 α H ( ) H ( )( H ( ) u ) ( ) { > } { (0 )} u u = 0 α 0( ) 0( ) ( ) H H u 0 { (0 )} u ( ( x)) α0 2 α H ( ) H ( )( H ( ) u ) α0 2 α = J H ( ) H ( )( H ( ) u ) where J H H u 0 0( ) 0( ) ( α) u ( 0 ) 0 { (0 )} (25) w w H ( ) H ( )( u ). ϒ T Usig a euality roed i [3] we get lim w w H ( ) H ( u ) 0 ϒ T (26) It follows from (20) (26) that u satisfies ieuality for ay W0 ( ) ( udx ) A ( ) (27) Takig = u λw with w W0 ( ) arbitrary ad makig λ 0 sice H 0is Lischits cotiuous ad bouded we get that u is a weak solutio of roblem (4) for = 0 i the usual sese. (b) Now we cosider the case (0 ]. I this case we make similar reasoig. We set φ= WH ( ) i ieuality (7) as a test fuctio where H ( λ) is defied by (0). Further we oly eed to exlai the method of the comariso of the itegrals oer icluded i the obtaied ariatioal ieuality. Note that i this case ariatioal ieuality has the form where ( udx ) ν 0 0 ν = A ( ) H ( ) H( )( udx ). 0 0 H ( ) H ( )( u) dx f( u) dx. (28) ( H ) u (29) φ φ( φ u ) dx ( ψ ( φ) ψ ( u )) f( φ u ) dx ( ( ) ( u )) ψ ϕ ψ { (0 )} { u (0)} { (0 )} { u 0} { (0 )} { u> } { > } { u (0)} ( H ( )) u ( H ( )) u H DOKLADY MATHEMATIC Vol. 94 No. 206

5 HOMOGENIZATION OF THE -LAPLACIAN 39 H We substitute the test fuctio i (7) ad we cosider the remaiig itegrals oer i the left-had side of ariatioal ieuality (7): Note that { > } { u 0} { > } { u > } H u. ϒ G ν w w H H ( H ( ) u ) 0 = H ( )( H( ) u ) 0 α α H ( )( H( ) u ). α u 0 H H 0 ( )( H( ) u ) = { (0 )} { u> } { (0 )} { u (0)} H ( ( ) u ) ( H ( )) (30) ( H ( )) ( H ( ) u ) ( H ( )) { (0 )} { u 0} { > } { u 0} { > } { u (0)} (3) Next we comare itegrals oer the same subsets of i the left-had side of ieuality (7). We hae ( ) ( u ) ( u ). { > } { u> } I ( H ( )) u M H H u ( ( )) ( ( )) ( H( )) u = M u( H( )) (32) where M = { (0 )} { u (0)}. Usig Youg s ieuality we get ( H( )) u u( H( )). Therefore I 0. (33) (34) The remaiig itegrals oer subsets of are cosidered i the similar way ad we establish that the sum of all itegrals oer the corresodig subsets of is o-ositie. Therefore the limit fuctio u satisfy ariatioal ieuality ( udx ) A ( ) H ( )( u) dx f( u) for a arbitrary fuctio W0 ( ). Agai takig = u λw with w W0 ( ) arbitrary ad makig λ 0 we obtai that u is a weak solutio the usual sese. ACKNOWLEDGMENT The research of the first two authors was artially suorted as members of the Research Grou MOMAT (Ref ) of the UCM. The research of J.I. Diaz was artially suorted by the roect ref. MTM of the DGIPI (ai). The research of D. Gómez-Castro was suorted by a FPU Grat from the Miisterio de Educació Cultura y Deorte (ai). The results of this aer were started durig a isit of the last author to the UCM o Noember 205. This author wats to thak this suort as well as the receied hositality from the Istituto de Matematica Iterdiscilar of the UCM. DOKLADY MATHEMATIC Vol. 94 No. 206

6 392 DIAZ et al. REFERENCE. J.-L. Lios Quelues methodes de resolutio des roblemes aux limites o lieires (Duod Paris 969; Mir Moscow 972). 2. M. Gochareko GAKUTO It. er. Math. ci. Al (997). 3. M. N. Zuboa ad T. A. haoshikoa Differ. Euatios 47 () (20). 4. W. Jäger M. Neuss-Radu ad T. A. haoshikoa Noli. Aal. Real World Al (204). 5. C. Coca J. I. Diaz A. Lia ad C. Timofte Electro. J. Differ. Euatios (2004). 6. J. I. Diaz D. Gómez-Castro ad C. Timofte Proceedigs of the 24th Cogress o Differetial Euatios ad Alicatio 4th Cogress o Alied Mathematics (Cadiz 205) D. Gómez M. Lobo M. E. Pérez ad T. A. haoshikoa Al. Aal. 92 (2) (203). 8. D. Gómez M. E. Pérez ad T. A. haoshikoa Asymtot. Aal (202). 9. A. V. Podol skii Dokl. Math. 82 (3) (200). 0. A. V. Podol skii Dokl. Math. 9 () (205).. H. rezis J. Math. Pures Al (972). DOKLADY MATHEMATIC Vol. 94 No. 206

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