Asymptotic behaviour of equicoercive diffusion energies in dimension two

Asymp o ic beha iou o equicoe ci e di usion ene gies in dimension wo Ma c BRIANE Juan CASADO-D´ IAZ Cen e de Ma h´ema iques Dp o. de Ecuaciones Di e enciales y An´alisis Num´e ico I.N.S.A. de Rennes & I.R.M.A.R. Uni e sidad de Se illa m[email p o ec ed] jcasado[email p o ec ed] No embe 3, 2006 Abs ac In his pape , we s udy he asymp o ic beha iou o a gi en equicoe ci e sequence o di usion ene gies Fn,n∈N, de ined in L2(Ω), o a bounded open subse Ω o R2. We p o e ha , con a y o he h ee dimension (o g ea e ), he Γ-limi o any con e gen subsequence o Fnis s ill a di usion ene gy. We also p o ide an explici ep esen a ion o mula o he Γ-limi when i s domains con ains he egula unc ions wi h compac suppo in Ω. This compac ness esul is based on he uni o m con e gence sa is ied by some minimize s o he equicoe ci e sequence Fn, which is speci ic o he dimension wo. The compac ness esul is applied o he pe iod amewo k, when he ene gy densi y is a highly oscilla ing sequence o equicoe ci e ma ix- alued unc ions. So, we gi e a de ini i e answe o he ques ion o he asymp o ic beha iou o pe iodic conduc ion p oblems unde he only assump ion o equicoe ci eness o he wo-dimensional conduc i i y. 1 In oduc ion This pape deals wi h he asymp o ic beha iou o sequences o di usion ene gies in a bounded open subse Ω o R2. The p o o ype o he di usion ene gy is gi en by he ollowing quad a ic unc ional de ined in L2(Ω): Fn(u):=     ZΩ An∇u·∇u dx i u∈H1 0(Ω). +∞i u∈L2(Ω) H1 0(Ω), o n∈N,(1.1) whe e Anis a symme ic posi i e de ini e ma ix- alued unc ion in L∞(Ω)2×2. The knowledge o he limi beha iou o Fnis c ucial in he homogeniza ion heo y applied o conduc ion p oblems (see e.g. [1] o an in oduc ion), since An hen ep esen s he conduc i i y ma ix o a gi en he e ogeneous medium. In his con ex , Spagnolo [23], wi h he G-con e gence heo y, and Mu a & Ta a [25], [20], wi h he H-con e gence heo y, p o ed he compac ness o he sequence Fn, when Anis assumed o be bo h equicoe ci e and equibounded. A ew imes la e , Bu azzo & Dal Maso [10] and Ca - bone & Sbo done [12] ex ended he esul o compac ness by only assuming ha he sequence Anis bounded and equiin eg able in L1(Ω)2×2. A he same pe iod, Fenchenko & Kh uslo [15] showed ha he equiin eg abili y condi ion canno be elaxed since high conduc i i y egions in h ee dimension may induce nonlocal e ec s which co espond o a lack o compac ness in he homogeniza ion p ocess (see also [2], [9], [4] o di e en app oaches). 1 Nonlocal e ec s na u ally appea in he limi beha iou o he di usion ene gy. Indeed, using he Beu ling-Deny [3] heo y Mosco [18] p o ed in pa icula ha any sequence Fn Γ-con e ges, up o a subsequence, o he s ong opology o L2(Ω) (see De ini ion 3.1) o a Di ichle o m (see De ini ion 3.3). Acco ding o he Beu ling-Deny o mula any Di ichle o m can always be spli up in o h ee e ms: a s ongly local o m ( he di usion pa ), a local o m and a nonlocal one. In e sely, Cama Eddine & Seppeche [11] p o ed ha any Di ichle o m in L2 loc(R3) can be ob ained as he Γ-limi o a sequence o di usion ene gies o ype (1.1) wi h a sui able iso opic conduc i i y An. The nonlocal e ec s ob ained in he p e ious wo ks a e based on h ee-dimensional mic os uc u es whose model example is a medium ein o ced by a pe iodic la ice o high conduc i i y hin ibe s. Then, i is na u al o ask i he appea ance o nonlocal e ec s is speci ic o he h ee dimension (o g ea e ). Recen ly, in [6] o pe iodic mic os uc u es and mo e gene ally in [7], we showed ha he answe is posi i e. Assuming ha he sequence Anis bo h equicoe ci e and bounded in L1(Ω)2×2, we p o ed ha he Γ-limi o any sequence o ype Fnis a s ongly local Di ichle o m. The e o e, he dimension wo p ese es he compac ness in he homogeniza ion p ocess. The p oo in [6], [7] is based on wo-dimensional di -cu l ype lemmas which ex end he one o Mu a & Ta a [26], [21]. No e ha he equicoe ci eness assump ion is essen ial o ob ain s ongly con e gen sequences in L2(Ω). Howe e , he use o di -cu l lemmas is s ic ly limi ed o conduc i i y sequences which a e bounded in L1(Ω)2×2. In his pape , we s udy he asymp o ic beha iou o he sequence o di usion en- e gies (1.1), wi hou assuming any boundedness assump ion on An. Ou app oach is comple ely di e en o he one used in [6] o [7] o Anbounded in L1(Ω)2×2. The key- ing edien o he me hod is a uni o m con e gence esul sa is ied by some ene gy mini- mize s (see Sec ion 2). Mo e p ecisely, we p o e (see Theo em 2.1) ha o any bounded ene gy (wi h espec o Fn) sequence in H1(Ω) which s ongly con e ges in L2(Ω) o a con inuous unc ion, he e exis s a smalle ene gy subsequence which s ongly con e ges o he same limi in L∞ loc(Ω). The p oo o his esul uses ha he p-capici y, o p∈(1,2), o a con inuous cu e is posi i e (see Lemma 2.8), which is speci ic o he dimension wo. This combined wi h he con inui y o he limi and he maximum p inciple allows us o cons uc a uni o mly con e gen sequence. Up o ou knowledge, he p e ious esul p o ides new uni o m es ima es on solu- ions o uni o mly ellip ic pa ial di e en ial equa ions wi hou any con ol om abo e on he coe icien s. He e, we gi e an example (see Co olla y 2.5) o such an es ima e o A-ha monic unc ions, whe e Ais any uni o mly ellip ic (bu no necessa ily uni o mly bounded) ma ix- alued in L∞(O), o a bounded open subse Oo R2. This uni o m es ima e is used in he las sec ion o he pape . Mo e gene al cases a e he subjec o a wo k in p og ess [8]. On he o he hand, hanks o he uni o m con e gence esul o Sec ion 2 and unde he only assump ion o equicoe ci eness o An, we p o e (see Sec ion 3 and Theo em 3.6) ha he di usion ene gy FnΓ-con e ges (up o a subsequence) o he s ong opology o L2(Ω) o a s ongly local Di ichle o m F. Mo eo e , i he domain o he Γ-limi F con ains he space C1 c(Ω) o he C1- egula unc ions wi h compac suppo in Ω, we ob ain (see Theo em 3.4) he ollowing ep esen a ion o mula F(u) = ZΩ A∇u·∇u dµ, ∀u∈C1 c(Ω),(1.2) whe e µis a Radon measu e on Ω and Aa ma ix- alued unc ion in L∞ µ(Ω)2×2. In o he wo ds, he sequence o he di usion ene gies Fnis ela i ely compac o he L2(Ω)- s ong Γ-con e gence opology in he se o he uni o mly coe ci e di usion ene gies. In pa icula , he compac ness esul implies ha he limi o he ene gy densi y Andx is s ill a densi y o ype A dµ. 2 The compac ness esul o Sec ion 3 has a ema kable applica ion in he pe iodic homogeniza ion amewo k. In his con ex , he conduc i i y Anis a highly oscilla ing sequence de ined by An(x) := Bn(x εn), whe e Bnis an equicoe ci e sequence o (0,1)2- pe iodic ma ix- alued unc ions in L∞(R2)2×2and εnis a posi i e sequence con e ging o ze o. Associa ed wi h Bn he cons an ma ix A∗ n(see o mula (4.4)) ob ained, o a ixed n, om he pe iodic homogeniza ion o Bn(x ε) as ε→0 (see e.g. [1]), plays a undamen al ole in he homogeniza ion p ocess. Indeed, ex ending [6] we p o e (see Theo em 4.1) ha he asymp o ic beha iou o he di usion ene gies (1.1) is comple ely de e mined by he limi beha iou o he ma ix A∗ n, acco ding o he ollowing al e na i e: •i he spec al adius ρ(A∗ n) o A∗ nis bounded, A∗ ncon e ges, up o a subsequence, o a ma ix A∗and he Γ-limi Fo Fnsa is ies (1.2) wi h he cons an densi y A∗dx, in he whole space H1 0(Ω); •i ρ(A∗ n) ends o +∞, he domain o he Γ-limi educes o {0}. As an immedia e consequence, he ques ion on he asymp o ic beha iou o he wo- dimensional pe iodic conduc ion p oblem (−di Bn(x εn)∇un= in Ω un= 0 on ∂Ω, o ∈H−1(Ω),(1.3) is now de ini i ely sol ed unde he only assump ion o equicoe ci eness o Bn: •i ρ(A∗ n) is bounded, (1.3) con e ges, up o a subsequence, o he conduc ion p oblem wi h he cons an conduc i i y lim n→+∞A∗ n; •i ρ(A∗ n) ends o +∞, he po en ial uno (1.3) s ongly con e ges o ze o in H1 0(Ω). The pape is o ganized as ollows. Sec ion 2 is de o ed o he uni o m con e gence esul s and Sec ion 3 o he Γ-con e gence o sequences o di usion ene gies o ype (1.1). In Sec ion 4 we apply he esul s o Sec ion 3 o he pe iodic amewo k. No a ions •N∗:= N {0}deno es he se o he posi i e in ege s; •a∨b, esp. a∧b, deno es he maximum, esp. he minimum, o a, b ∈R; •B(x0, δ) deno es he disk o cen e x0∈R2and o adius δ > 0; •χEdeno es he cha ac e is ic o he se E; • ∃ lim means ha he limi does exis ; •Ω deno es an open subse o R2and ¯ Ω he closu e o Ω in R2; •Hloc(Ω) means locally in he space H(Ω); •C(Ω) deno es he space o he con inuous unc ions in Ω, C0(Ω) he subspace o C(Ω) composed o he unc ions which a e ze o on he bounda y o Ω, Cc(Ω) he subspace o C0(Ω) composed o he unc ions wi h compac suppo in Ω, and Ck c(Ω), o k∈ N∩{+∞}, he subspace o Cc(Ω) composed o he k- h con inuously di e en iable unc ions in Ω; •D0(Ω) deno es he se o he dis ibu ions on Ω; •M(Ω) deno es he se o he Radon measu es on Ω; 3 •a sequence µnin M(Ω) con e ges o µ∈M(Ω) in he weak ∗sense o he measu es in Ω i lim n→+∞ZΩ ϕ dµn=ZΩ ϕ dµ, ∀ϕ∈C0(Ω), and he con e gence is deno ed by µn* µ in M(Ω) ∗; •q.e. means quasi-e e ywhe e in he sense o he 2-capaci y in R2, and a.e. means e e ywhe e in he sense o he Lebesgue measu e in R2; • o any p∈(1,2) and o any subse Eo R2,Cp(E) deno es he p-capaci y o E wi h espec o R2, which is de ined by Cp(E) := in ZR2|∇u|pdx :u∈D1,p(R2), u ≥1 a.e. in a neighbou hood o E, whe e D1,p(R2) is he space o he unc ions uin L 2p 2−p(R2) such ha ∇u∈L2(R2)2. 2 Uni o m con e gence esul s 2.1 S a emen o he esul s Le Ω be an open subse o R2. In his sec ion, we conside a gi en sequence o symme ic ma ix- alued unc ions An∈L∞(Ω)2×2,n∈N, which sa is ies he ollowing equicoe - ci eness p ope y in Ω ∃α > 0 such ha ∀n∈N,∀ξ∈R2, Anξ·ξ≥α|ξ|2a.e. in Ω.(2.1) Fo any unc ion u∈H1(Ω)∩C(Ω), we will s udy some ques ions ela ed o he exis ence o sequences unin H1(Ω) which bo h con e ge uni o mly o uin Ω and sa is y he ollowing minimiza ion p ope y ∃lim n→+∞ZΩ An∇un·∇undx ≤lim in n→+∞ZΩ An∇ n·∇ ndx, o any sequence nin H1(Ω) (some bounda y condi ions can be added), which s ongly con e ges o uin L2(Ω). Ou main esul in his way is he ollowing heo em: Theo em 2.1. Le ube a unc ion in H1(Ω) ∩C(Ω) and le ˆunbe a sequence H1(Ω) which s ongly con e ges o uin L2(Ω) and sa is ies ∃lim n→+∞ZΩ An∇ˆun·∇ˆundx < +∞. Then, up o a subsequence o n, s ill deno ed by n, he e exis s un∈H1(Ω) which sa is ies lim sup n→+∞ZΩ An∇un·∇undx ≤lim n→+∞ZΩ An∇ˆun·∇ˆundx, (2.2) and un−→ us ongly in L∞ loc(Ω).(2.3) Mo eo e , i he suppo o uis con ained in a compac se Ko Ω, hen we can ake un such ha un= 0 q.e. in Ω K, o any n∈N. Rema k 2.2. I in Theo em 2.1 he sequence ˆunis in H1 0(Ω) and uin H1 0(Ω) ∩C0(Ω), hen we can choose unin H1 0(Ω), which s ongly con e ges o uin L∞(Ω). To his end, i is enough o conside a bounded open se ˜ Ω such ha ¯ Ω⊂˜ Ω and o apply he second pa o Theo em 2.1 o he sequences ˜unand ˜ Ande ined by ˜un(x):=un(x) i x∈Ω 0 i x∈˜ Ω Ω,and ˜ An(x):=An(x) i x∈Ω I2i x∈˜ Ω Ω. 4 Co olla y 2.3. Conside ˆun∈H1(Ω) and u∈H1(Ω) ∩C(Ω) such ha ˆun* u weakly in H1 loc(Ω) and di (An∇ˆun) = 0 in D0(Ω).(2.4) Then, we ha e he ollowing uni o m con e gence ˆun−→ us ongly in C(Ω).(2.5) Rema k 2.4. No e ha in Co olla y 2.3 each unc ion unis con inuous in Ω by he De Gio gi-S ampacchia heo em (see e.g. [16] Chap e 8). Co olla y 2.5. Fo any open subse Ωo R2and any compac subse Ko Ω, he e exis s a cons an C > 0which only depends on Ωand Ksuch ha , o any ma ix- alued unc ion A∈L∞(Ω)2×2sa is ying he uni o m coe ci eness (2.1) and any unc ion u∈H1(Ω) solu ion o di (A∇u) = 0 in D0(Ω), he ollowing es ima e holds ue kukC(K)≤CkukH1(Ω). Rema k 2.6. Co olla ies 2.3 and 2.5 can be deduced om [8] whe e mo e gene al esul s, o non-necessa ily homogeneous equa ions, a e p o ed. 2.2 P oo o he esul s Le us now gi e he p oo o he uni o m con e gence esul s s a ed in he p e ious sec ion. We will need he wo ollowing lemmas: Lemma 2.7. Le Obe a bounded open subse o R2and le u∈H1(O)∩C(¯ O). Deno e M:= max ∂O uand m:= min ∂O u. Then, o any unc ion such ha −ubelongs o H1 0(O), he unc ions ( −M)+and (m− )+belong o H1 0(O). P oo . Conside ϕε∈C∞ c(O), o ε > 0, which s ongly con e ges o −uin H1 0(Ω) as ε→0. Since uis con inuous in ¯ O, he unc ions Uε:= (u+ϕε−M−ε)+and uε:= (m−ε−u−ϕε)+ha e compac suppo in O. The unc ions Uεand uεbelong o H1(O), hence hey also belong o H1 0(O). The e o e, using ha Uεand uεs ongly con e ge espec i ely o ( −M)+and (m− )+in H1(O), yields he esul . Lemma 2.8. Fo any p∈(1,2) and any con inuous cu e Lo ex emi ies a, b, we ha e Cp(L)≥Rp|a−b|2−p,(2.6) whe e Rp>0is he p-capaci y o a uni segmen in R2. P oo . Using a ansla ion, a o a ion and a homo he y, we can educe he p oo o he case whe e a= (0,0), b= (1,0). Then, conside a cu e Lo ex emi ies (0,0), (1,0) and ake a unc ion ϕ∈C∞ c(R2) such ha ϕ>χL. By he P´olya-Szeg¨o inequali y [22] ex ended o any powe p≥1 (see e.g. [24] and Chap e I.4 o [19]), i is known ha he S eine symme iza ion ϕ∗o ϕwi h espec o {x2= 0}, de ined by i s le el se s (x1, x2)∈R2:ϕ∗(x1, x2)> c=(x1, x2)∈R2:|x2|<1 2{y∈R:ϕ(x1, y)> c}, 5 belongs o W1,p c(R2) and sa is ies ZR2|∇ϕ∗|pdx ≤ZR2|∇ϕ|pdx. Mo eo e , since Land ϕa e con inuous, i is clea ha ϕ∗>1 in [0,1] ×{0}, hence Cp([0,1] ×{0})≤ZR2|∇ϕ∗|pdx ≤ZR2|∇ϕ|pdx. Taking he in imum in ϕ, we ge he desi ed es ima e (2.6). P oo o Theo em 2.1. Using he densi y o H1(Ω) ∩C∞(Ω) in H1(Ω) (see e.g. [16]), we can also assume ha ˆunis con inuous in Ω. Fo any δ > 0, we de ine Ωδby Ωδ:= {x∈Ω : d(x, ∂Ω) > δ}. Since uis con inuous in ¯ Ωδ, o any l∈N∗, he e exis s δl>0, wi h lim l→+∞δl= 0, such ha |u(x)−u(y)|<1 2l,∀x, y ∈¯ Ωδl,wi h |x−y| ≤ δl.(2.7) Le p∈(1,2). Since ˆunweakly con e ges in H1(Ω), he e exis s (see e.g. [14]) a subse- quence o ˆun, s ill deno ed by ˆun, which con e ges o u Cp-quasi uni o mly in e e y open se ω⊂Ω, wi h ¯ω⊂Ω (ωcan be chosen as Ω i Ω is smoo h). Thus, we can choose his sequence in such a way ha o any l∈N∗, he e exis s a ela i ely closed subse Klo Ω sa is ying Cp(Ω Kl)< Rpδ2−p l,(2.8) |ˆun(x)−u(x)|<1 2l,∀x∈Ωδl∩Kl,∀n≥l. (2.9) Then, we de ine unby      un:= ˆunin Kl, un−ˆun∈H1 0(Ω Kl), ZΩ Kl An∇un·∇undx ≤ZΩ Kl An∇ ·∇ dx, ∀ , −ˆun∈H1 0(Ω Kl).(2.10) Clea ly, unsa is ies (2.2). Le us p o e ha uns ongly con e ges o uin L∞ loc(Ω). To his end, we ix δ > 0. We ha e |un(x)−u(x)|=|ˆun(x)−u(x)|<1 2l,∀x∈Ωδ∩Kl,∀n≥l, wi h δl< δ. (2.11) Conside a connec ed componen Oo Ω Klsuch ha O∩Ωδ6= Ø. Since Ois opened and connec ed, i is connec ed by cu es. Thus, o any y1, y2∈O, he e exis s a cu e L⊂Owhich connec s y1, y2. By Lemma 2.8 and (2.8), we ha e Rp|y1−y2|2−p≤Cp(L)≤Cp(O)≤Cp(Ω Kl)≤Rpδ2−p l, hence diam (O)≤δl. Then, aking lla ge enough such ha 2δl< δ, we ge ha ¯ O⊂Ωδl, and in pa icula , ∂O ⊂Ωδl∩Kl. Deno e mn:= min ∂O ˆunand Mn:= max ∂O ˆun. By Lemma 2.7 (un−Mn)+and (mn−un)−belong o H1 0(O), and by de ini ion (2.10) un is An-ha monic in O. Then, he maximum p inciple yields mn≤un≤Mn,q.e. in O, ∀n∈N.(2.12) 6 On he o he hand, since ∂O ⊂Ωδl∩Kl, we ha e by (2.9). mn≥min ∂O u−1 2land Mn≤max ∂O u+1 2l,∀n≥l. Mo eo e , ¯ O⊂Ωδl, diam (O)≤δland (2.7) imply ha min ∂O u≥u(x)−1 2land max ∂O u≤u(x) + 1 2l,∀x∈O, ∀n≥l. The e o e, (2.12) combined wi h he wo p e ious es ima es yields |un−u| ≤ 1 2l,q.e. in O, hence he sequence unsa is ies he uni o m con e gence (2.3). Now, assume ha he suppo o uis con ained in a compac subse Ko Ω, and conside an open se ˜ Ω which con ains Kand is s ic ly con ained in Ω. Fo any ε > 0, le Sεbe he unc ion de ined by Sε(s) := (s−εsgn(s)))χ{|s|>ε}, o s∈R. Since un s ongly con e ges o uin L∞(˜ Ω), he sequence εn:= kun−ukL∞(˜ Ω) ends o ze o. The e o e, he sequence ˜un:= χ˜ ΩSεn(un) sa is ies condi ions (2.2), (2.3) and anish q.e. in Ω K. P oo o Co olla y 2.3. Since ˆunsa is ies di (An∇ˆun) = 0 in D0(Ω), i is H¨olde con inuous in Ω by he De Gio gi-S ampacchia heo em. Then, he a gumen used in he p oo o Theo em 2.1 p o es ha he sequence unde ined by (2.10) s ongly con e ges o uin L∞ loc(Ω). Howe e , we ha e un= ˆunby cons uc ion, which yields he desi ed esul .  P oo o Co olla y 2.5. We eason by con adic ion. I he esul does no hold ue, hen, o any n∈N, he e exis un∈H1(Ω), An∈L∞(Ω)2×2and γn>0 such ha Anξ·ξ≥γn|ξ|2,∀ξ∈R2,a.e. x∈Ω, and kunkC(K)> nkunkH1(Ω).(2.13) Up o eplace Anby An/γnand unby un/kunkC(K), we can assume ha γn= 1 and kunkC(K)= 1. Then, Anis equicoe ci e and by (2.13) uns ongly con e ges o ze o in H1(Ω). The e o e, by Co olla y 2.3 uncon e ges uni o mly o ze o in K, in con adic- ion wi h kunkC(K)= 1.  3Γ-limi o equicoe ci e di usion ene gies 3.1 Γ-con e gence and Di ichle o ms In his sec ion we i s ecall he de ini ion o he De Gio gi Γ-con e gence and some o i s p ope ies which will be used in he sequel. We e e o [13] o an exhaus i e p esen a ion o he Γ-con e gence. De ini ion 3.1. A sequence o unc ionals Fn:L2(Ω) −→ [0,+∞] is said o Γ-con e ge o F:L2(Ω) −→ [0,+∞] o he s ong opology o L2(Ω) i , o any uin L2(Ω), (i) he Γ-limin inequali y holds ∀un−→ us ongly in L2(Ω), F(u)≤lim in n→+∞Fn(un),(3.1) 7 (ii) he Γ-limsup inequali y holds ∃¯un−→ us ongly in L2(Ω), F(u) = lim n→+∞Fn(¯un).(3.2) Any sequence sa is ing (3.2) will be called a eco e y sequence o Fn, o limi u. In he sequel, we will always conside he Γ-con e gence wi h espec o he s ong opology o L2. Consequen ly, his opology will be no necessa ily men ionned. P ope ies 3.2. a)Since L2(Ω) is sepa able, any sequence o unc ionals Fn:L2(Ω) −→ [0,+∞]has a subsequence which Γ-con e ges wi h espec o he s ong opology o L2(Ω). b)Le Fn:L2(Ω) −→ [0,+∞]be a sequence o quad a ic o ms which Γ-con e ges o F. Then, Fis a quad a ic o m on L2(Ω) which is semi-lowe con inuous wi h espec o he s ong opology o L2(Ω). c)Le Fn:L2(Ω) −→ [0,+∞]be a sequence o quad a ic o ms which Γ-con e ges o F. Le Φn,Φbe he pola o ms espec i ely associa ed wi h Fn, F on hei domains. Then, o any u∈L2(Ω), wi h F(u)<+∞, a sequence unin L2(Ω) is a eco e y sequence (3.2) o Fn, o limi u, i and only i ∀ n−→ s ongly in L2(Ω),wi h Fn( n)≤c, lim n→+∞Φn(un, n) = Φ(u, ),(3.3) o equi alen ly, (3.3) wi h = 0. Now, we ecall some no ions abou Di ichle o ms, which will be used in he s a emen o Theo em 3.8. We e e o [18] o mo e de ails in connec ion wi h he Γ-con e gence. De ini ion 3.3. Le Xbe a Hausdo , sepa able, locally compac space, and le mbe aσ- ini e nonnega i e Radon measu e on X. Le Hbe he space L2 m(X) endowed wi h i s Hilbe no m k·kH. Le F:H−→ [0,+∞] be a quad a ic o m o domain D(F) := {u∈H:F(u)<+∞}, whose pola o m Φ is a bilinea o m de ined in D(F)×D(F). (i) The o m Fis said o be closed i i is semi-lowe con inuous wi h espec o he no m k·kH. The o m Fis said o be closable i he e exis s an ex ension ˜ Fo F in Hsuch ha D(F)⊂D(˜ F). The closu e o a closable o m is i s smalles closed ex ension in H. (ii) The o m Fis said o be Ma ko ian i ∀u∈D(F), := (u∨0) ∧1∈D(F) and F( )≤F(u). (iii) A Di ichle o m on His a closed Ma ko ian quad a ic o m de ined in H. (i ) The o m Fis said o be egula i he e exis s a subse o D(F)∩C0(X), which is dense bo h in C0(X) wi h he uni o m no m and in D(F) wi h he no m (F+k·kH)1/2. ( ) The o m Fis said o be local i Φ(u, ) = 0,∀u, ∈D(F),wi h supp (u)∩supp ( ) = Ø. The o m Fis said o be s ongly local i Φ(u, ) = 0,∀u, ∈D(F),wi h u= cs in supp ( ).(3.4) Thanks o he Beu ling-Deny heo y [3] any egula Di ichle o m Fon L2 m(X) can be spli up on i s domain in o h ee speci ic o ms: a s ongly local o m Fd, a local o m and a nonlocal one. Mo e p ecisely, he ollowing ep esen a ion o mula holds o any u∈D(F), F(u) = Fd(u) + ZX u2(x)k(dx) + ZZX×X diag u(x)−u(y)2j(dx, dy),(3.5) whe e Fdis called he di usion pa o F,k he killing measu e and j he jumping measu e. 8 3.2 S a emen o he esul s As in Sec ion 2, le us conside a bounded open subse Ω o R2, and a sequence o symme ic ma ix- alued unc ions An∈L∞(Ω)2×2which sa is y (2.1). Fo any n∈N and any open subse ωo Ω, we de ine he quad a ic o m Fn(·, ω) in L2(ω) by Fn(u, ω):=     Zω An∇u·∇u dx i u∈H1 0(ω). +∞i u∈L2(ω) H1 0(ω). (3.6) The o m Fn(·,Ω) is simply deno ed by Fn. Assume ha FnΓ-con e ges o some quad a ic o m F o he opology o L2(Ω), which holds ue o a subsequence in i ue o P ope ies 3.2 a). Since Fnis clea ly Ma ko ian, he p ope ies (3.1), (3.2) o he Γ-con e gence imply ha Fis also Ma ko ian. Mo eo e , hanks o P ope ies 3.2 b)Fis closed. The e o e, Fis a Di ichle o m in he sense o De ini ion 3.3 (iii). The ollowing esul gi es a necessa y and su icien condi ion o ha e F egula wi h C1 c(Ω) ⊂D(F). When his condi ion is sa is ied, Fis a s ongly local (3.4) Di ichle o m whose in eg al ep esen a ion o egula unc ions is independen o he domain. Theo em 3.4. The domain D(F)o Fcon ains C1 c(Ω) i and only i , o any x0∈ Ω, he e exis s δ > 0, wo unc ions w1, w2in C1(B(x0, δ)) and wo sequences w1 n, w2 n in H1(B(x0, δ)),n∈N, such ha              B(x0, δ)⊂Ω, ∇w1(x0),∇w2(x0)a e linea ly independen , wi n−→ wis ongly in L2(B(x0, δ)) , o i= 1,2, An∇wi n·∇wi nis bounded in L1(B(x0, δ)) , o i= 1,2. (3.7) Assume ha C1 c(Ω) is con ained in D(F). Then, he e exis a nonnega i e Radon mea- su e µon Ωand a nonnega i e ma ix- alued unc ion Ain L∞ µ(Ω)2×2such ha he egula pa A o Adµ wi h espec o he Lebesgue measu e sa is ies A ξ·ξ≥α|ξ|2,∀ξ∈R2,a.e. in Ω,(3.8) and such ha , o any open se ωo Ω, he Γ-limi F(·, ω)o Fn(., ω)wi h espec o he s ong opology o L2(ω)does exis on C1 c(ω)and eads as F(u, ω) = Zω A∇u·∇u dµ, ∀u∈C1 c(ω).(3.9) Mo eo e , o any u∈C1 c(ω)and any un∈H1 0(ω)which s ongly con e ges o uin L2(ω) and such ha Fn(un, ω) ends o F(u, ω), he sequence An∇un·∇uncon e ges o A∇u· ∇u dµ in he weak ∗sense o he measu es in ω. Rema k 3.5. In he second pa o Theo em 3.4 he Γ-con e gence o Fn(·, ω) o F(·, ω) holds ue up o a subsequence which does depend on he open se ω. Howe e , he in eg al ep esen a ion (3.9) o F(u, ω), which is alid on C1 c(ω), is independen o ω. In ac , he in eg al exp ession (3.9) holds ue o any u∈C1 0(ω), wi h A∇u·∇u∈ L1 µ(ω). Indeed, i is easy o check ha hese unc ions can be app oxima ed by unc ions in C1 c(ω) in he s ong opology o D(F(., ω)). Theo em 3.4 p o ides an in eg al ep esen a ion o F, assuming ha D(F) con- ains C1 c(Ω). The ollowing esul gi es a co ec o esul : 9 he closed se s. Using he compac ness o ¯ Ω, he cha ac e iza ion o he open se s implies ha ¯ Ω/Ris compac and hus, he se Ω∗:= ¯ Ω/R {[∂]}is locally compac . Using ha he Γ-limi Fo Fn(3.6) is closed o he s ong opology o L2(Ω) and he de ini ion o he measu e m, he es ic ion o F o D(F)∩C0(Ω) is a closable o m in L2 m(Ω∗), whose closu e is deno ed by F∗. Le us p o e ha F∗is a egula Di ichle o m acco ding o De ini ion 3.3. Se H(s) := (s∨0) ∧1, o s∈R. Then, o any u∈D(F) and any eco e y sequence un∈H1 0(Ω) associa ed wi h uand Fnby (3.2), we ha e F(H(u)) ≤lim in n→+∞Fn(H(un)) ≤lim Fn(un) = F(u). In pa icula , his holds o any u∈D(F)∩C0(Ω), hence he es ic ion o F o D(F)∩ C0(Ω) is Ma ko ian. Since D(F)∩C0(Ω) is an algeb a which sepa a es poin s, he S one-Weie s ass he- o em shows ha he unc ions o he o m u+c, wi h u∈D(F)∩C0(Ω) and c∈R, a e dense in C(¯ Ω/R). Now, conside ∈Cc(Ω∗), n∈D(F)∩C0(Ω) and cn∈R, such ha n+cncon e ges o in C(¯ Ω/R). Since and n anish in [∂], he sequence cncon e ges o ze o and hus ncon e ges o in C(¯ Ω/R). This p o es ha D(F)∩C0(Ω) is dense in C0(Ω∗), which implies ha F∗is egula . The e o e, F∗is a egula Di ichle o m. I emains o p o e ha F∗is s ongly local, i.e., he pola o m Φ o Fsa is ies (3.4). Le u, ∈D(F)∩C0(Ω) and c∈R, such ha u=ccons an in supp ( ). Fi s , we assume ha c≥0. Taking in o accoun Rema k 2.2, we conside wo eco e y sequences un, n which s ongly con e ge espec i ely o u, in L∞(Ω) and such ha Fn(un), Fn( n) end espec i ely o F(u), F( ). We also choose nsuch ha supp ( n)⊂supp ( ). Le Hε, o ε > 0, be he unc ion de ined in Rby Hε(s):=   c c−εsi s < c −ε ci c−ε≤s≤c+ε s+εi s > c +ε, i ε < c, (no e ha Hε(0) = 0), and Hε(s) := (s−εsgn(s)) χ{|s|>ε}i c= 0. The sequence εn:= kun−ukL∞(Ω) con e ges o ze o. Then, he sequence Hεn(un) sa is ies he same p ope ies han un, bu we also ha e Hεn(un) = cin supp ( )⊃supp ( n), o εnsmall enough. The e o e, he P ope ies 3.2 c) o he eco e y sequence nyields Φ(u, ) = lim n→+∞ZΩ An∇(Hεn(un)) ·∇ ndx = 0. In he case c < 0, we simply use he equali y Φ(u, ) = −Φ(−u, ) = 0.  4 Applica ion o he pe iodic case 4.1 S a emen o he esul s In his sec ion we conside a sequence Bno symme ic ma ix- alued unc ions in L∞(R2)2×2, which sa is ies he ollowing assump ions: Bnis Y-pe iodic, whe e Y:= (0,1)2, i.e., ∀n∈N,∀κ∈Z2, Bn(·+κ) = Bn(·) a.e. in R2,(4.1) Bnis equicoe ci e in R2, i.e., ∃α > 0 such ha ∀n∈N,∀ξ∈R2, Bnξ·ξ≥α|ξ|2a.e. in R2.(4.2) 16 Le εnbe a sequence o posi i e numbe s which ends o 0. F om he sequences Bnand εn we de ine he highly oscilla ing sequence o ma ix- alued unc ions Anby An(x):=Bnx εn,a.e. x∈R2.(4.3) In i ue o (4.1) and (4.2) Anis an equicoe ci e sequence o εn-pe iodic ma ix- alued unc ions in L∞(R2)2×2. Le A∗ nbe he cons an ma ix de ined by A∗ nλ·λ:= min ZY Bn(y)(λ+∇ϕ(y)) ·(λ+∇ϕ(y)) dy :ϕ∈H1 #(Y), λ ∈R2,(4.4) whe e H1 #(Y) deno es he se o Y-pe iodic unc ions in H1 loc(R2). The ma ix A∗ nis symme ic and posi i e de ini e wi h A∗ n≥α I2. By he classical esul o pe iodic ho- mogeniza ion (see e.g. [1]) A∗ n, o ixed n, is he homogenized ma ix associa ed wi h he oscilla ing sequence An(x ε) as ε ends o ze o. No e ha in he de ini ion (4.3) o An he oscilla ions pe iod εndepends on he sequence n. In his pe iodic amewo k we a e in e es ed in he asymp o ic beha iou o he di u- sion ene gy Fnde ined by Fn(u):=     ZΩ An∇u·∇u dx i u∈H1 0(Ω) +∞i u∈L2(Ω) H1 0(Ω), (4.5) as well as he conduc ion p oblem (−di (An∇un) = in Ω un= 0 on ∂Ω,(4.6) o a gi en in H−1(Ω). The ollowing esul shows ha he asymp o ic beha iou o he di usion ene gy Fn(4.5) only depends on he limi o he spec al adius ρ(A∗ n) o he ma ix A∗ n(4.4). Theo em 4.1. Le Ωbe a bounded open se o R2. Conside a highly oscilla ing sequence o ma ix- alued unc ions Ansa is ying (4.1),(4.2) and (4.3). Then, we ha e he ollowing al e na i e: I ρ(A∗ n)is bounded, he e exis s a subsequence, s ill deno ed by n, and a posi i e de ini e ma ix A∗such ha A∗ n(4.4) con e ges o A∗in R2×2and Fn(4.5) Γ-con e ges o he s ong opology o L2(Ω) o he quad a ic o m Fassocia ed wi h A∗by F(u):=     ZΩ A∗∇u·∇u dx i u∈H1 0(Ω) +∞i u∈L2(Ω) H1 0(Ω). (4.7) I ρ(A∗ n) ends o +∞, he sequence FnΓ-con e ges o he s ong opology o L2(Ω) o he quad a ic o m Fwhose domain is D(F) = {0}.(4.8) In e m o he conduc ion p oblem (4.6) Theo em 4.1 implies he ollowing esul : Co olla y 4.2. Le Ωbe a bounded open se o R2. Conside a highly oscilla ing sequence o ma ix- alued unc ions Ansa is ying (4.1),(4.2) and (4.3). Then, we ha e he ollowing al e na i e: 17 I ρ(A∗ n)is bounded, he e exis s a subsequence, s ill deno ed by n, and a posi i e de ini e ma ix A∗such ha A∗ n(4.4) con e ges o A∗in R2×2and, o any in H−1(Ω), he solu ion uno (4.6) weakly con e ges in H1 0(Ω) o he solu ion uo he conduc ion p oblem (−di (A∗∇u) = in Ω u= 0 on ∂Ω.(4.9) I ρ(A∗ n) ends o +∞, he sequence uns ongly con e ges o 0in H1 0(Ω). P oo . Co olla y 4.2 is a immedia e consequence o Theo em 4.1 using he ac ha he solu ion uno (4.6) is he minimize o he unc ional u∈L2(Ω) 7−→ 1 2Fn(u)−ZΩ u dx, and he minimize s con e gence p ope y o he Γ-con e gence (see e.g. Co ollo y 7.24 p. 84 o [13]). Rema k 4.3. Co olla y 4.2 is an ex ension o he simila homogeniza ion esul ob ained in [4] (by a comple e di e en app oach) unde he assump ion ha he sequence o pe iodic ma ix- alued Bn(4.3) is bounded in L1(Y)2×2. This condi ion is mo e es ic i e since i is easy o check ha he boundedness o Bnin L1(Y)2×2implies he boundedness o ρ(A∗ n). We can also build a pe iodic wo-dimensional mic os uc u e such ha ρ(A∗ n) is bounded while kBnkL1(Y)2×2is no . The e o e, Co olla y 4.2 p o ides a comple e answe o he pe iodic homogeniza ion o he conduc ion p oblems wi h equicoe ci e sequences o symme ic conduc i i ies. 4.2 P oo o Theo em 4.1 The case whe e ρ(A∗ n)is bounded Le Xi n,i= 1,2, be he unique unc ion in H1 #(Y), wi h ze o Y-a e age alue, solu ion o ∀ϕ∈H1 #(Y),ZY Bn∇Wi n·∇ϕ dy = 0,whe e Wi n(y):=yi+Xi n(y),(4.10) o equi alen ly, di Bnei+∇Xi n= 0 in D0(R2),(4.11) whe e (e1, e2) deno es he canonic basis o R2. Le wi nbe he highly oscilla ing sequence de ined by wi n(x):=εnWi nx εn=xi+εnXi nx εn, o x∈Ω.(4.12) By (4.11) and he de ini ion (4.3) o An he unc ion wi nis clea ly An-ha monic. Mo eo e , by he Y-pe iodici y o Bn∇Wi n·∇Wi n, by (4.10) and he de ini ion (4.4) o A∗ n, we ha e o any bounded open subse ωo R2, Zω An∇wi n·∇wi ndx ≤cωZY Bn∇Wi n·∇Wi ndy =cωA∗ nei·ei≤c < +∞. This combined wi h he equicoe ci eness o Animplies ha he sequence wi nis bounded in H1 loc(R2) and hus weakly con e ges o xiin H1 loc(R2). Then, hanks o Co olla y 2.3 he sequence wi ns ongly con e ges o xiin L∞(Ω). On he o he hand, by he boundedness assump ion on ρ(A∗ n) he sequence A∗ ncon e ges, up o a subsequence, o some cons an 18 ma ix A∗in he space R2×2. Mo eo e , he εn-pe iodici y o ∇wi nimplies ha , o any i, j = 1,2, An∇wi n·∇wj n=Bn∇Wi n·∇Wj nx εn* A∗ei·ejweakly in M(R2)∗. So, as he g adien s o he unc ions xi,i∈ {1,2}, a e independen a each poin o Ω, he sequences wi nsa is y (3.12) o any open disk con ained in Ω. The e o e, since wi n a e An-ha monic and con e ge uni o mly in Ω, he cons uc ion o Lemma 3.7 yields he measu e µand he ma ix- alued Aby dµ = (A∗e1·e1+A∗e2·e2)dx and Aei·ejdµ =A∗ei·ejdx, hence A∗= (A∗e1·e1+A∗e2·e2)A. Then, in i ue o Theo em 3.4 he Γ-limi Fo he sequence Fn(4.5) sa is ies C1 c(Ω) ⊂D(F) and F(u) = ZΩ A∗∇u·∇u dx, ∀u∈C1 c(Ω).(4.13) Le us conclude. On he one side, he equicoe ci eness o Anand he lowe semi- con inui y o he H1 0(Ω)-no m gi e D(F)⊂H1 0(Ω). On he o he side, he densi y o C1 c(Ω) in H1 0(Ω), combined wi h he ac ha D(F) is a Hilbe space and ha om (4.13) a sequence o C1 c(Ω) which s ongly con e ges in H1 0(Ω) also s ongly con e ges in D(F), we ge D(F) = H1 0(Ω) and equali y (4.13) holds ue in H1 0(Ω). No e ha FnΓ-con e ges o F o he whole sequence such ha A∗ ncon e ges o A∗in R2×2. The case whe e ρ(A∗ n) ends o +∞ We p oceed by con adic ion. We assume ha he domain D(F) o he Γ-limi Fdoes no educe o {0}. Then, we p o e ha ρ(A∗ n) is necessa ily bounded. To his end, we p oceed in wo s eps. In he i s s ep, we p o e ha he e exis s a con inuous unc ion in D(F) {0}. The second s ep is de o ed o he p oo o he boundedness o ρ(A∗ n). Fi s s ep : D(F)∩C(Ω) 6={0}. Up o an ex ac ion o a subsequence we can assume ha he sequence Fnde ined by (4.5) Γ-con e ges o some quad a ic unc ional F:L2(Ω) −→ [0,+∞]. The s a ing assump ion is ha D(F)6={0}. Le u∈D(F) {0}. By he equicoe ci eness o An he unc ion u belongs o H1 0(Ω). The e exis s a sequence unin H1 0(Ω) which s ongly con e ges o uin L2(Ω) and such ha Fn(un) ends o F(u). Up o enla ge he domain Ω and o ex end he unc ions o H1 0(Ω) by 0 ou side Ω, we may assume ha he suppo s o u, una e con ained in a ixed compac Ko Ω. Fi s ly, le us p o e ha , o any τ∈R2o small enough no m, he ansla ed unc ion u(·+τ) belongs o D(F) and F(u(·+τ)) = F(u). We ollow he p ocedu e gi en in he p oo o Theo em 24.1 o [13]. Le τ∈R2and le κnbe a sequence in Z2such ha τn:= εnκn ends o τ. I τhas a small enough no m, hen we ha e K−τn⊂Ω o any n∈N. Then, using successi ely he ac ha un(·+τn) is equal o 0 in Ω (K−τn), he change o a iable y=x+τnand he εn-pe iodici y o An, we ob ain F(un(·+τn)) = ZK−τn An(x)∇un(x+τn)·∇un(x+τn)dx =ZK An(y)∇un(y)·∇un(y)dy =Fn(un). Mo eo e , he sequence un(·+τn) s ongly con e ges o u(·+τ). The e o e, he Γ-limin inequali y implies ha F(u(·+τ)) ≤lim in n→+∞Fn(un(·+τn)) = lim in n→+∞Fn(un) = F(u), 19 which also yields F(u)≤F(u(·+τ−τ)) ≤F(u(·+τ)), and hus F(u(·+τ)) = F(u). Secondly, le δbe a small enough posi i e numbe and le δbe he unc ion de ined on Ω by δ(x) := 1 δ2ZδY u(x+y)dy, o x∈Ω, which is con inuous on Ω. Le (Qj k)1≤j≤k, o k∈N∗, be a co e ing o he se δY by ksqua es o side δ √kand le yj kbe he cen e o Qj k. Then, he sequence o con ex combina ions o ansla ed o ude ined by k δ:= 1 δ2 k X j=1 |Qj k|u(·+yj k) s ongly con e ges o δin L2(Ω) as k→+∞, o ixed δ. Then, he lowe semi-con inui y and he con exi y o Fyield F( δ)≤lim in k→+∞F( k δ)≤lim in k→+∞ 1 δ2 k X j=1 |Qj k|Fu(·+yj k)=F(u)<+∞. The e o e, δbelongs o D(F)∩C(Ω). Since δs ongly con e ges o u6= 0 in L2(Ω), δ is a non-ze o unc ion in D(F)∩C(Ω) o δsmall enough, which concludes he i s s ep. Second s ep : Boundedness o ρ(A∗ n). Le be a non-ze o unc ion in D(F)∩C(Ω) and le nbe a sequence in H1 0(Ω) which s ongly con e ges o in L2(Ω) and such ha Fn( n) ends o F( ). By Theo em 2.1 he sequence nuni o mly con e ges o in Ω. Since is a non-ze o con inuous unc ion on Ω, he uni o m con e gence o n o implies ha he e exis s a non-emp y open subse ω0 o Ω and a cons an c0>0 such ha | n(x)| ≥ c0a.e. x∈ω0.(4.14) Le λnbe a uni no m ec o in R2such ha A∗ nλn·λn=ρ(A∗ n). Le wnbe he highly oscilla ing sequence de ined by wn(x):=λn·x+εnXnx εnx∈R2,whe e Xn:= (λn·e1)X1 n+(λn·e2)X2 n(4.15) and Xi n,i= 1,2, a e he Y-pe iodic solu ions o (4.11). Se ˜ Y:= (−1 2,3 2)2. Since he unc ion Wn(y) := λn·y+Xn(y) is Bn-ha monic in R2, by Co olla y 2.5 he e exis s a cons an C > 0 such ha o any n∈N, kWnkL∞(Y)≤CkWnkH1(˜ Y). Then, using he pe iodici y and he ze o Y-a e age alue o Xn, as well as he Poinca ´e- Wi inge inequali y yields kXnkL∞(Y)≤1 + kWnkL∞(Y) ≤1 + kλn·ykL∞(˜ Y)+CkXnkH1(˜ Y)≤1 + 3 √2+ 4 CkXnkH1(Y) ≤C0+C0k∇XnkL2(Y)≤2C0+C0k∇WnkL2(Y). Mo eo e , he coe ci eness o Bnand he de ini ion (4.4) o A∗ nimply ha αk∇Wnk2 L2(Y)2≤ZY Bn∇Wn·∇Wndy =A∗ nλn·λn=ρ(A∗ n), 20 which combined wi h he p e ious es ima es gi es kXnkL∞(Y)≤C0+C0 √αpρ(A∗ n). The e o e, by he Y-pe iodici y o Xnand he de ini ion (4.15) o wn he e exis s a con- s an c > 0 such ha o any n∈N, kwnk2 L∞(Ω) ≤c+c ε2 nρ(A∗ n).(4.16) On he o he hand, using he An-ha monici y o wn(4.15) and he Cauchy-Schwa z inequali y yields ZΩ An∇wn·∇wn 2 ndx =−2ZΩ An∇wn·∇ n nwndx ≤2ZΩ An∇wn·∇wn 2 ndx1 2ZΩ An∇ n·∇ nw2 ndx1 2 , hence he inequali y ZΩ An∇wn·∇wn 2 ndx ≤4ZΩ An∇ n·∇ nw2 ndx. (4.17) Le us conclude. On he one side, hanks o he uni o m es ima e (4.14) and he εn-pe iodici y o An∇wn· ∇wn he le hand-side o (4.17) is bounded om below by a posi i e cons an imes Zω0 An∇wn·∇wndx ≥cω0ZY Bn∇Wn·∇Wndy =cω0ρ(A∗ n), whe e cω0is a posi i e cons an only depending on ω0. 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