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LIMITING JUMP CONDITIONS FOR JOSEPHSON JUNCTIONS IN GINZBURG-LANDAU THEORY ∗ AYMANKACHMAR 8 0 Abstra t. We onsider a S-N-S Josephson jun tion modeled through the 0 Ginzburg-Landau theory. When the normal material is su(cid:30) iently thin and 2 the applied magneti (cid:28)eld is below the riti al (cid:28)eld of vortex nu leation, we n prove to leading order that jump boundary onditions of the type predi ted a bydeGennes are satis(cid:28)eda rossthejun tion. J 2 1 Introdu tion The super ondu ting proximity e(cid:27)e t in a normal metal adja ent to a super- ] P ondu tor has re eived a lot of attention by the physi s ommunity, see [10℄ for A a review of this phenomenon. This is also the setting of the Josephson tunneling . e(cid:27)e tforsuper ondu ting-normal-super ondu tingjun tions(SNS),whereasuper- h urrent (cid:29)ows through the normal layer provided that it is su(cid:30) iently thin. t a The physi s literature ontains several approa hes to model the Josephson e(cid:27)e t m in the frame work of the Ginzburg-Landau theory of super ondu tivity. The (cid:28)rst [ modeling in this ontext is perhaps due to the physi ist deGennes [12℄. In the 3 setting of [12℄, the omplex-valued wave fun tion (whose modulus measures the v densityofsuper ondu tingele trons)anditsderivativearerelatedlinearlyonboth 0 sides of the normal material, in su h a manner that the super urrent is onserved 9 through the jun tion. 2 Inthispaper,weuseageneralizedGinzburg-Landauenergyfun tionalpresentedin 1 . [9℄,whi hhasprovedtoa ountrigorouslytovaκriousphysi alaspe ts( .f. [19,20℄). 0 Byworkingin theLondonsingularlimit(high -regime),wejustify asymptoti ally 1 the modeling of [12℄ provided that the applied magneti (cid:28)eld is below the riti al 7 0 (cid:28)eld of vortex nu leation, see Theorems 1.1 & 1.5. : We also mention in this dire tion that another justi(cid:28) ation of the deGennes mod- v eling is present in a paper of Rubinstein-S hatzman-Sternberg [23℄, who deal with i X geometri jun tions (weak links) in the frameworkof the Ginzburg-Landautheory. r We hope to arry out in a forth oming work a deeper analysis valid for higher ap- a plied magneti (cid:28)elds and whi h provides more details on erning the super urent (cid:29)ow and the distribution of vorti es in the jun tion. 1. Main results Ω = D(0,1) We move now to thRe2mathemaRti a]0l,s1e[t-up ℓof ]t0h,eRp[roblem. Let denotestheunitdis in . Given ∈ and ∈ ,weintrodu ethefollowing Ω partition of , Ω=S N, ∪ Date:February 2,2008. 2000 Mathemati s Subje t Classi(cid:28) ation. Primary 35J60; Se ondary 35J20, 35J25, 35B40, 35Q∗55,82D55. UniversitéParis-Sud,Départementdemathématique,Bât.425,91405OrsayFran e. E-mail: ayman.ka hmarmath.u-psud.fr. 1 2 AYMANKACHMAR where N = x Ω : dist(x,∂D(0,R))<ℓ , (1.1) { ∈ } S =D(0,R ℓ), S =D(0,1) D(0,R+ℓ), S =S S . 1 2 1 2 (1.2) − \ ∪ S We shall suppose that is the ross se tion of a ylindri al super ondu tor with N in(cid:28)niteheightandthat isthatofanormalmaterial. Bythisway,wegetaS-N-S Josephson jun tion. InGinzburg-Landautheory[13℄,thesuper ondu tingpropertiesaredes ribedbya ψ ψ 2 omplex valued wavefun tion , alled the `order parameter',whose modulus | | ψ 0 measures the density of the super ondu ting ele tron Cooper pairs (hen e ≡ A = (A ,A ) 1 2 orresponds to a normal state), and a real ve tor (cid:28)eld , alled the `magneti potential',su hthattheindu edmagneti (cid:28)eldinthesample orresponds curlA to . Sin e, the super ondu ting Cooper ele tron pairs an di(cid:27)use from the super ondu ting to the normal material in a normal-super ondu ting jun tion, we (ψ,A) Ω then have to onsider pairs de(cid:28)ned on . (ψ,A) Thebasi postulateintheGinzburg-Landautheoryisthatthepair minimizes the Gibbs free energy, whi h has in our ase the following dimensionless form [9℄ : 1 (ψ,A) = ( iA)ψ 2dx+ (1 ψ 2)2dx (1.3) Gε,H | ∇− | 2ε2 −| | ZΩ ZS a + ψ 2dx+ curlA H 2dx. ε2 | | | − | ZN ZΩ 1 = κ S H > 0 Here, ε is a hara teristi of the super ondu ting material ((cid:28)lling ), a > 0 is the intensity of the applied magneti (cid:28)eld and is related to the riti al N a temperature of the material in . The positive sign of means that we are above N the riti al temperature of the material (cid:28)lling . Minimization of the fun tional (1.3) will take pla e in the spa e =H1(Ω;C) H1(Ω;R2). H × Wewill beinterestedin the analysisoftheasymptoti behaviouroftheminimizers ε 0 N of (1.3) as → (London Limit) and when the thi kness of the ring is small by ℓ=ℓ(ε) 1 ε 0 taking ≪ as → . (ψ,A) A ording to [15℄, the fun tional (1.3) admits a minimizer in the spa e H. Our main result is the leading order asymptoti expansion of the jump of ( iA)ψ N ∇− a ross the jun tion, i.e. a ross the boundary of , provided that the ψ order parameter is not possessing vorfti esH. 1(Ω;C) f In order to (cid:28)x ideas, given a fun tion ∈ , we introdu e the jump of N a ross by [f] (θ)=f (R+ℓ)eiθ f (R ℓ)eiθ , θ [0,2π[. N (1.4) − − ∀ ∈ (cid:0) (cid:1) (cid:0) (cid:1) Our(cid:28)rstresult on ernsthe aseofverythinrings,ofthi kness omparablewith ε . Ω = D(0,1) S N (ψ ,A ) ε ε Theorem 1.1. Let , and as in (1.1), (1.2), and be a d > 0 a > 0 λ > 0 minimizer of (1.3). Given and , there exists su h that, if the applied magneti (cid:28)eld satis(cid:28)es H λ lnε , (1.5) ≤ | | ℓ=dε ψ >0 ε and if , then | | and we have n(x) ( iA )ψ exp(2√ad) 1 lim ε · ∇− ε ε 2√a − =0, (1.6) ε→0(cid:13) (cid:20) ψε (cid:21)N − exp(2√ad)+1(cid:13)L2(S1) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) lim [ψ ] =0. (cid:13) (1.7) ε 0 ε N L2(S1) → (cid:13) (cid:13) (cid:13) (cid:13) JUMP CONDITIONS FOR JOSEPHSON JUNCTIONS 3 x n(x)= x R2 0 Here, x for all ∈ \{ }, is the unit outward normal ve tor of any dis R2 | | in . H Remark 1.2. The regime on erning the applied magneti (cid:28)eld in Theorem 1.1 orresponds to that below the (cid:28)rst riti al (cid:28)eld : When (1.10) is satis(cid:28)ed, the order ψ Ω ε parameter has no vorti es in . λ > 0 ′ On the other hand, it is well known ( .f. [26℄) that there exists su h that H λ lnε ψ ′ ε if ≥ | |, the order parameter has vorti es. However, we are not able to λ c al ulate the riti al value for whi h H λ lnε as ε 0. C1 ∼ c| | → This is due to the te hni al di(cid:30) ulty arising from the very rapid os illations of the N maximal super ondu ting density in , see Se tion 2 for more details on erning this point. d 0 Remark 1.3. Noti e that if one formally makes → in (1.6), one would obtain N 0 that the jump a ross tends to . This agrees with experimental and theoreti al predi tions that the Josephson e(cid:27)e t would be absent in jun tions made up of very thin normal materials, see [9℄. Remark 1.4. Let us introdu e the ve tors εn(x) ( iA )ψ (R ℓ)eiθ Xε− = · ∇ψ−(Rε ℓε)eiθ − , ε (cid:18) − (cid:0) (cid:1) (cid:19) εn(x) ( iA )ψ(cid:0) (R+ℓ)ei(cid:1)θ X+ = · ∇− ε ε , θ [0,2π[, ε ψε (R+ℓ)eiθ ∀ ∈ (cid:18) (cid:0) (cid:1) (cid:19) (cid:0) (cid:1) and the matrix exp(2√ad) 1 1 2√a − Ma,d = exp(2√ad)+1 .   0 1   Then, (1.6) and (1.7) an be written in the equivalent form lim X+ M X =0, (1.8) ε 0 ε − a,d ε− L2(S1) → (cid:13) (cid:13) (cid:13) (cid:13) whi h justi(cid:28)es the boundary ondition postulated by deGennes in [12℄. For thi ker rings, we have a result analogous to that of Theorem 1.1. Ω = D(0,1) S N (ψ ,A ) ε ε Theorem 1.5. Let , and as in (1.1), (1.2), and be a ℓ=ℓ(ε) minimizer of (1.3). Assume in addition that satis(cid:28)es, ε ℓ(ε) (ε 0), c>0, ε ]0,1], ℓ(ε) cεln lnε . (1.9) ≪ → ∃ ∀ ∈ ≤ | | a>0 λ>0 Then,given ,thereexists su hthat,iftheappliedmagneti (cid:28)eldsatis(cid:28)es 2√aℓ(ε) H λexp lnε , (1.10) ≤ − ε | | (cid:18) (cid:19) we have n(x) ( iA )ψ lim ε · ∇− ε ε 2√a =0, (1.11) ε→0(cid:13) (cid:20) ψε (cid:21)N − (cid:13)L2(S1) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) lim [ψ ] =0. (cid:13) (1.12) ε 0 ε N L2(S1) → x (cid:13) (cid:13) n(x)= x R2 (cid:13)0 (cid:13) Here, x for all ∈ \{ }, is the unit outward normal ve tor of any dis R2 | | in . 4 AYMANKACHMAR Noti ethatthe resultofTheorem1.5agreeswiththatofTheorem1.1whenone d + takes formally → ∞ in (1.6). WementionalsothattheresultofTheorem1.5isvalidformagneti (cid:28)eldsslightly mu h lower than that of Theorem 1.1. Te hni ally, this is due to the fa t that we N an not ex lude the energy of vorti es in (see Se tion 2). But heuristi ally, the reason is that the maximal super ondu ting density (i.e. the positive minimizer of H = 0 (1.3) for ) in the regime of Theorem 1.5 is small inside the jun tion, hen e the pri e of a vortex be omes for magneti (cid:28)elds less than that of Theorem 1.1. However, we were not able in this ase to prove that the riti al (cid:28)eld for vortex exp 2√aℓ(ε) lnε nu leationhastheorderof − ε | |, thoughtheoreti alpredi tionsin (cid:16) (cid:17)N the physi s literature say that vorti es in would be present for magneti (cid:28)elds mu h below than that of a bulk super ondu tor, see [10℄. ℓ(ε) = c ε lnε The result of Theorem 1.5 is still valid up to lengths ∗ | |, where c ]0,1[ ∗ ∈ is su(cid:30) iently small (this an be he ked through minor modi(cid:28) ations of the argument). However, sin e the orresponding magneti (cid:28)eld will be small H 1 1 ≪ , we do not fo us on this last regime . Asaby-produ toftheanalysisthatweshall arryout,wegetaresult on erning N the onservationof the urrent a ross . The urrentis de(cid:28)ned as the ve tor (cid:28)eld j = iψ ,( iA )ψ = (iψ ,(∂ iA1)ψ ),(iψ ,(∂ iA2)ψ ) , (1.13) ε ε ∇− ε ε ε 2− ε ε ε 2− ε ε (, ) (cid:0) (cid:1) (cid:0) C R2 (cid:1) where · · denotes the s alar produ t in when identi(cid:28)ed with . The exa t result on erning the urrent is the following. Theorem1.6. InbothregimesofTheorems1.1&1.5,the ir ulationofthe urrent is almost onserved a ross the jun tion: τ(x) j τ(x) j = ε1/2 lnε as ε 0, ε ε (1.14) (cid:12)(cid:12)Z|x|=R+ℓ · −Z|x|=R−ℓ · (cid:12)(cid:12) O(cid:16) | |(cid:17) → (cid:12) (cid:12) τ(x)(cid:12)= x⊥ x R2 0 (cid:12) where (cid:12) x for all ∈ \{ }. (cid:12) | | Finally, we ommentonsomepastwork on erningnon-homogeneoussuper ondu tors and pinning models. Unlike to our situation, pinning models previously onsidered orrespond to a term in the Ginzburg-Landau fun tional having the (a ψ 2)2 a form −| | , with being a smoothfun tion. The (cid:28)rst analyti workprobably a=a ε ε goesba kto [1℄, where , assumedalwayspositive,maydepend on withthe lnε a C lnε ε restri tionthatit annotos illatequi kerthan | |(|∇ |≤ | |). Compared with oursituation, t0he disu ontinuityuoftheC oe(cid:30) ients leadsto an order∂pSarameter os illating between and ε with |∇ ε|≥ ε in a thin neighborhood of . Later, a in [6℄, the authors deal with the ase when has isolated zeros, and prove that a 1 vorti es appear (cid:28)rst at the zeros of for magneti (cid:28)elds having order . In [4℄, a the authors allow the fun tion to have negative values, but the hypothesis of its smoothness for es the order parameter to be small on the boundary of the normal side (hen e, the surfa e super ondu ting sheath in the normal side is absent). Moreover,the situationin [4℄ ismorerelatedto the aseofdomainswithholesand relies on the analysis arried out by the same authors in [3℄. More re ently, the author of the present paper showed in [18℄ that pinning of vorti es is observed for a magneti (cid:28)elds near the (cid:28)rst riti al (cid:28)eld when the fun tion is a step fun tion. Let us also mention the very re ent work of Alama, Bronsard and Sandier in [5℄ 1 H ≪ 1 It is more likely that the regime is treated, without additional restri tions on the H H =0 de ayof ,bybifur ation arguments. Inthepresentpaper, wetreat indetailthe ase . JUMP CONDITIONS FOR JOSEPHSON JUNCTIONS 5 where a layered super ondu ting model has been investigated. There, the expression of the riti al (cid:28)eld above whi h vorti es are dete ted depends strongly on the thi kness of the normal regions separating the super ondu ting layers. Organization of the paper. Se tion 2 is devoted to a des ription of the main points of the argument. Se tion 3 is devoted to a preliminary analysis of the minimizers of (1.3). Se tion 4 is devoted to an analysis of an auxiliary variational problem (this is the variational problem (2.5) des ribing the Meissner state). Se tion 5 is devoted to prove a lower bound of the fun tional (1.3). In Se tion 6, we exhibit a vortex-less regime and we a hieve the proofs of Theo- rems 1.1, 1.5 and 1.6. Finally, inAppendixA,weproveauniquenessresult on erningthesolutionofthe anoni al equation without magneti (cid:28)eld (this is Eq. (2.1)), and in Appendix B, we dis uss the di(cid:30) ulSty1 behind the estimate of the energy of a on(cid:28)guration with vorti es on the ir le R. 2. Sket h of proof R2 Canoni al Equation in for the ase without magneti (cid:28)eld. Ageneralte hniqueto ta kleasymptoti problems,(su essfullyappliedby Hel(cid:27)er & o-authorsforlinearproblems( f. e.g. [16℄),andLu-Panfornon-linearproblems ( f. e.g. [22℄)), is to understand the anoni al situation. So we onsider the ase H =0 without magneti (cid:28)eld, , and work with the parti ular domains, N =R ] ℓ,ℓ[, S =R2 N. × − \ ℓ(ε)=dε When , this leads to the equations: ∆u=(1 u2)u in R R [ d,d] , −∆u+au−= in R ] ×d{,d[\, − } (2.1)  − × − ∂ u(,t )=∂ u(,t ), u(,t )=u(,t ), t= d,  x2 · ± x2 · ∓ · ± · ∓ ±  Sin e Eq. (2.1) arises as a limiting equation of a riti al point of (G-L), we fo us on solutions in the lass = u H2 (R2) L (R2) : u 0 . C { ∈ loc ∩ ∞ ≥ } Using the argument of [19, Se tion 4℄, Eq. (2.1) admits in C a unique solution R2 (x ,x ) U(x ) 1 2 2 ∋ 7→ given by β(a,d)exp(√2 x ) 1 2 | | − if x d, (2.2) U(x2)= β(a,d)exp(√2|x2|)+1 | 2|≥  A(a,d)[exp(√a x2 )+exp( √a x2 )] if x2 <d. | | − | | | | β(a,d) > exp( √2d) A(a,d) > 0 where the onstants − and are determined expli - itly, but the only important remark is that U (d) U ( d) exp(2√ad) 1 ′ = ′ − =2√a − , U(d) −U( d) exp(2√ad)+1 − hen e the onditions of deGennes are veri(cid:28)ed exp(2√ad) 1 U′(t) = 1 2√aexp(2√ad)−+1 U′(t) . (2.3) U(t)   U(t) (cid:18) (cid:19)t=d 0 1 (cid:18) (cid:19)t= d −   6 AYMANKACHMAR The ase without magneti (cid:28)eld in a bounded domain. H = 0 Now we return to minimizing (1.3) when . We prove in [19, Theorem 1.1℄ (u ,0) ε that (1.3) has, up to a gauge transformation, a unique minimizer where u H2(Ω) 0<u <1 Ω ε u ε ε ε ∈ is a real-valued fun tion, in (for small enough), and solves the equation 1 a ∆u = (1 u2)u 1 u 1 . − ε ε2 − ε ε S − ε2 ε N ε 0 Then, by a blow-up argument, we generalize (2.3) asymptoti ally as → , see Theorem 3.2. The ase with magneti (cid:28)eld: A vortex-less regime. (ψ,A) H > 0 Now we return to minimizers of (1.3) when . Following an idea of Lassoued-Mirones u[21℄, we introdu e a normalized density ψ ϕ= . u ε ϕ 1 Then, | |≤ and we are led to work with the fun tional (see Theorem 3.5): (ϕ,A)= u2 ( iA)ϕ2dx Fε,H ε| ∇− | ZΩ 1 + u4(1 ϕ2)2dx+ curlA H 2dx. 2ε2 ε −| | | − | ZS ZΩ (B(a ,r )) i i i Then, following [25, 27℄, we onstru t a family of disjoint balls (see r lnε 10 Proposition 5.2) su h that i i ≤| |− and x Ω :Pϕ(x) 1 lnε−2 B(ai,ri). { ∈ | |≤ −| | }⊂ i [ α ]0,1[ This permits to obtain, for a given number ∈ 2 , a lower bound of the energy (see Theorem 5.4): ε,H(ϕ,A) M0(ε,H)+ (2παmi(ε) lnε 2H)di CH lnε−10, (2.4) F ≥ | |− − | | i X C >0 d ϕ/ϕ ∂B(a ,r ) i i i where is an expli it onstant, is the degree of | | on , m (ε)= min u2(x), i ε x∈B(ai,ri) and M (ε,H)= inf (1,A). 0 ε,H (2.5) A H1(Ω;R2)F ∈ By omparing (2.4) with what is existing in the literature ( .f. [26℄), we suspe t α that the lower bound (2.4) is not optimal in the sense that should be equal to 1 ε,H . This restri tion is a tually due to the parti ular expression of F , where a ϕ N penalization term for | | is absent in the energy of . = H h The in(cid:28)mum in (2.5) is a hieved by a unique ve tor (cid:28)eld A u2ε∇⊥ ε, where h : Ω ]0,1[ ε 7→ satis(cid:28)es a London equation with weight (see (4.4)). Thus, we get the upper bound (ϕ,A) M (ε,H). ε,H 0 F ≤ d s When this upperbound ismat hed with(2.4), wededu e thatall the ′i areequal 0 to provided that H απ infm (ε) lnε . i ≤ i | | ℓ(ε) ε a =R (cid:16) (cid:17) i If ≫ and | | , we have by Theorem 3.4 that 2√aℓ(ε) m (ε) exp . i ≈ − ε (cid:18) (cid:19) JUMP CONDITIONS FOR JOSEPHSON JUNCTIONS 7 S1 Sin e we an not ex lude the possibility of a vortex ball entered on the ir le R, ℓ(ε) ε H we restri t ourselves when ≫ to magneti (cid:28)elds satisfying 2√aℓ(ε) H λexp lnε ≤ − ε | | (cid:18) (cid:19) in order to insure the absen e of vorti es. Now, in the absen e of vorti es we get an energy estimate (see Theorem 6.1) 1 ( iA)ϕ2+ curlA Hh 2 dx+ (1 ϕ2)2dx 1 (ε 0). | ∇− | | − ε| ε2 −| | ≪ → ZΩ ZS (cid:0) L2 (cid:1) ϕ Then, we implement -estimates for the equations of in order to dedu e that [ϕ] 0 in L2(S1), ϕ 1, n(x) ( iA)ϕ 0 in L2(∂N), N → | |→ · ∇− → whi h permits us to dedu e Theorems 1.1 and 1.5, see Se tion 6.1. 3. Preliminary analysis of minimizers 3.1. The ase without applied magneti (cid:28)eld. This se tion is devoted to a summary of the main results obtained in [19℄ whi h deal with minimizers of (1.3) H =0 when the applied magneti (cid:28)eld . A = 0 H = 0 We keep the notation introdu ed in Se tion 1. Upon taking and in (1.3), one is led to introdu e the fun tional 1 a (u):= u2dx+ (1 u2)2dx+ u2dx, (3.1) Gε |∇ | 2ε2 − ε2 ZΩ ZS ZN H1(Ω;R) de(cid:28)ned for fun tions in . We introdu e C (ε)= inf (u). 0 ε (3.2) u H1(Ω;R)G ∈ The next theorem is a summary of Theorem 1.1 in [19℄. a >0 d>0 ε ε ]0,ε [ 0 0 Theorem 3.1. Given aHnd1(Ω;R), there existsu suC h2(tSh)at fCor2(aNll) ∈ , ε the fun tional (3.1) admits in a minimizer ∈ ∪ su h that 0<u <1 in Ω. ε Ω,N S Furthermore, with our hoi e of the domains and in (1.1) and (1.2), the u ε fun tion is radial. H =0 (u ,0) ε If , minimizers of (1.3) are gauge equivalent to the state . ε 0 Let us just mention why we fo us only on the regime → in the statement u 0 of Theorem 3.1. Noti e that ≡ is a riti al point of the fun tional (3.1), so we would like to ex lude the possibility that this riti al point is stable. To this end, we de(cid:28)ne the following eigenvalue : 1 λ (a,d,ε)=inf φ2 φ2 dx 1 |∇ | − ε2| | (cid:26)ZS(cid:18) (cid:19) a + φ2+ φ2 dx : φ H1(Ω), φ =1 . |∇ | ε2| | ∈ k kL2(Ω) ZN(cid:16) (cid:17) (cid:27) λ (a,d,ε) < 0 λ (a,d,ε) 1 1 Then when , the orresponding eigenfun tion of provides u 0 us with a test on(cid:28)guration whose energy is below that of ≡ . On the other λ (a,m,ε) ε 0 1 hand, this last ondition of the sign of is easily veri(cid:28)ed when → , thanks in parti ular to the min-max prin iple. N Letusre allthenotationofthejumpa ross introdu edin(1.4). Usingablow- up argumentand a result on erning uniqueness of riti alpoints of the fun tional 8 AYMANKACHMAR (3.1) in the model ase of the entire plane, we establish now Theorem 1.1 in the H =0 ase when . u ε Theorem3.2. Let bethepositiveminimizerof(3.1)introdu edinTheorem3.1. ℓ(ε)=dε Then, if , we have n(x) u exp(2√ad) 1 lim ε ·∇ ε 2√a − =0, (3.3) ε→0(cid:13) (cid:20) uε (cid:21)N − exp(2√ad)+1(cid:13)L∞(S1) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) lim [u ] =0, (cid:13) (3.4) ε 0 ε N L∞(S1) → (cid:13) (cid:13) (cid:13) (cid:13) lim sup u (R ℓ)eiθ A(a,d) =0. ε (3.5) ε→0 θ∈[0,2π[(cid:12) ± − (cid:12)! n(x)= x x R2(cid:12)(cid:12) (cid:0)0 A(a(cid:1),d)e>0 (cid:12)(cid:12) 2 Here, x for all ∈ \{ }, and is an expli it onstant . | | ℓ(ε) ε On the other hand, if ≫ (this overs the regime (1.9)), then we have, n(x) u lim ε ·∇ ε 2√a =0, (3.6) ε→0(cid:13) (cid:20) uε (cid:21)N − (cid:13)L∞(S1) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) lim sup u (R ℓ)eiθ A(a) =0, ε (3.7) ε→0 θ∈[0,2π[(cid:12) (cid:0) ± (cid:1)− (cid:12)! A(a)>0 (cid:12) (cid:12) where is an expli it onstant. ℓ(ε)=dε d>0 (r,θ) Proof. Letustreatthe asewhen , . Let bepolar oordinates, 0<r <1 π <θ <π , − , and set t=R r, s=Rθ. − s [ Rπ,Rπ[ 0 Given ∈ − , we de(cid:28)ne the res aled fun tion, R 1 1 R R R uε(s,t)=uε (R εt)eiε(s−s0)/R , − <t< − , π <s s0 <π . − ε ε − ε − ε (cid:16) (cid:17) u ε Tehe equation of be omes: ∆ u =(1 u2)u , 0<t< 1 R, s s <πR, − ε ε − ε ε −ε | − 0| ε e  −∆εueε+amuεe=e0, R−ε1 <t<0, |s−s0|<πRε, ∂u ∂u ε(e,t )= e ε(,t ), u (,t )=u(,t ) t= d, ε  ∂et · ± ∂et · ∓ · ± · ∓ for ± where ∆ = 1 ε t −2∂e2+∂2 e ε ∂ . ε − R s t − (R εt) t (cid:18) (cid:19) − Now, by ellipti estimates, the fun tion uε onvergesto a fun tion u in Wl2o,c∞(R2). u Furthermore, solves (2.1) in C, and by [19, Lemma 5.2℄, there exist onstants k ,c >0 u(0,k )>c u(s,t)= 0 0 0 0 e su hthat . Thus,we on ludebyTheoremA.1that U(t) U , where is given in (2.3), and therefore, by oming ba k to the initial s ale, t k 0,1,2 , limεk u (s,t) U =0, ε ∀ ∈{ } ε→0 (cid:13)(cid:13) − (cid:18)ε(cid:19)(cid:13)(cid:13)Wk,∞({|s−s0|≤πRε,|t|≤(1−R)ε}) (cid:13) (cid:13) s [ πR,πR[ and the onvergen e is uni(cid:13)form with repe t t(cid:13)o 0 ∈ − . This yields (3.3)- (3.5). e 2 A(a,d) Theexpressionof isgivenexpli itlyintheAppendix. JUMP CONDITIONS FOR JOSEPHSON JUNCTIONS 9 ℓ(ε) ε (cid:3) The statements on erning the ase when ≫ follows from [19, (5.20)℄. u ε We shall need the following remarkable properties of , that distinguish the di(cid:27)erent regimes onsidered in this paper. ℓ = dε Lemma 3.3. With the notations and hypotheses of Theorem 3.2, if , there c(a,d)>0 exists an expli it onstant su h that u (x)>c(a,d), x Ω. ε ∀ ∈ ℓ(ε) u ε For the ase when satis(cid:28)es (1.9), the asymptoti behavior of be omes 0 N ompletely di(cid:27)erent in the sense that it is lose to inside . In order to be pre ise we introdu e the fun tion : βexp(√2t) 1 V(t)= − (t 0), V(t)=A exp(√at) (t<0), (3.8) βexp(√2t)+1 ≥ S together with the `signed distan e' to the boundary of , t (x)=dist(x,∂S) (x S), t (x)= dist(x,∂S) (x D(0,1) S). S S (3.9) ∈ − ∈ \ β A Here the onstants and are given by : √2+√a+2 √2+√a+2 √a β = , A= − . (3.10) √a √2+√a+2+√a ℓ(ε) Theorem 3.4. Assume that satis(cid:28)es (1.9). Then, we have t (x) S u V =o(1) (ε 0), ε (3.11) − ε → (cid:13) (cid:18) (cid:19)(cid:13)L∞(Ω) (cid:13) (cid:13) (cid:13)t V (cid:13) where the fun tions (cid:13)S and have bee(cid:13)n introdu ed in (3.8)-(3.9). ε ]0,1] ε g(ε) ]0,1[ 0 Moreover, there exist a positive onstant and a fun tion ∋ 7→ ∈ g(ε) 1 ε ]0,ε ] 0 su h that ≪ and for all ∈ , one has the estimate (3.12) √at (x) √at (x) S S (A g(ε))exp u (x) (A+g(ε))exp , x N. ε − ε ≤ ≤ ε ∀ ∈ (cid:18) (cid:19) (cid:18) (cid:19) Proof. The asymptoti behaviorin (3.11)hasbeen obtained in [19℄. We haveonly N to prove the improved estimate in , i.e. (3.12). Let us show how one an obtain the lower bound. Let us introdu e the fun tion : δt (x) S v (x)=Cexp , ε ε (cid:18) (cid:19) C δ where and are positive onstants to be spe i(cid:28)ed later. t S Let us re all that by de(cid:28)nition, the fun tion is written as R ℓ(ε) x if R ℓ(ε) x R, tS(x)= x− R −ℓ(|ε)| if R−< x ≤R|+|≤ℓ(ε), (cid:26) | |− − | |≤ R ]0,1[ S N where the onstant ∈ has been introdu ed for de(cid:28)ning and , see (1.1) and (1.2). t N S Consequently, the fun tion is smooth in ea h of the following two annuli of : N = x N : R ℓ(ε)< x <R , N = x N : R< x <R+ℓ(ε) . + − { ∈ − | | } { ∈ | | } One then he ks easily that a δ2 a ε ∆(u v )+ (u v ) = 1 x 1 v − ε− ε ε2 ε− ε ε2 − δ2 ± δ| |− ε δ2 h a εi 1 v in N . ≥ ε2 − δ2 − δ(R ℓ(ε)) ε ± (cid:20) − (cid:21) 10 AYMANKACHMAR ε 0 It is a result of the asymptoti formula (3.11) that there exist a onstant and a ]0,ε ] ε f(ε) ]0,1[ f(ε) 1 ε 0 0 fun tion ∋ 7→ ∈ su h that ≪ as → and u A f(ε), ε ]0,ε ]. ε|∂N − ≤ ∈ 0 (cid:12) (cid:12) Therefore, gathering all th(cid:12)e above re(cid:12)marks, we get for ε2 ε δ = a+ , C =A 2f(ε) s 4(R ℓ(ε))2 − 2(R ℓ(ε)) − − − ε ]0,ε ] 0 and when ∈ , a ∆(u v )+ (u v ) 0 in N − ε− ε ε2 ε− ε ≥ ± (3.13) ( uε(x) vε(x)>0 on ∂N. − u x = R ε Two ases may o ur regarding the gradient of on the ir le | | , either u (R) 0 u (R)>0 ′ε ≤ or ′ε . u (R) 0 If ′ε ≤ , then we get in addition to (3.13) ∂ (u v )>0 on (∂N ) N. ε ε + ∂ν − ∩ N+ ν Here,were allthat N± denotetheunit outwardnormalve torsofthe boundaries N of ±. Therefore, the appli ation of the strong maximum prin iple yields that u v 0 in N . ε ε + − ≥ This last lower bound when ombined with (3.13) yields a ∆(u v )+ (u v ) 0 in N − ε− ε ε2 ε− ε ≥ − ( uε(x) vε(x)>0 on ∂N . − − u v 0 N ε ε Hen e by the strong maximum prin iple, − ≥ in −. Therefore, we dedu e that u v 0 in N, ε ε − ≥ u ε whi h is nothing but the lower bound we wish to prove for the fun tion . The u (R)>0 N N same argument holds when ′ε , but by hanging the roles of + and −. (cid:3) Theproofoftheupperboundfollowsthesamelinesabove,soweomitthedetails. 3.2. The ase with magneti (cid:28)eld. This se tion is devoted to a preliminary H =0 analysisoftheminimizersof(1.3)when 6 . Themainpointthatweshallshow C (ε) 0 ishowtoextra tthesingularterm ( f. (3.2))fromtheenergyofaminimizer. Noti e that the existen e of minimizers is standard starting from a minimizing sequen e ( f. e.g. [15℄). A standard hoi e of gauge permits one to assume that the magneti potential satis(cid:28)es divA=0 in Ω, ν A=0 on ∂Ω, (3.14) · ν ∂Ω where is the outward unit normal ve tor of . Ω N With this hoi e of gauge, one(ψis,Aab)le to pCro1v(Ωe;(Cw)henCth1(eΩb;oRu2n)daries of and are smooth) that a minimizer is in × . One has also the following regularity ( f. [19, Appendix A℄), ψ C2(S;C) C2(N;C), A C2(S;R2) C2(N;R2). ∈ ∪ ∈ ∪ The next lemma is inspired from the work of Lassoued-Mirones u( f. [21℄).