Renormalized Energy with Vortices Pinning Effect

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1 Renormalized Energy with Vortices Pinning Effect Shijin Ding Department of Mathematics South China Normal University Guangzhou, Guangdong 5063, China Abstract. This paper is a successor of the previous paper in the Journal of Partial Differential Equations []. We derive in this paper the renormalized energy to further determine the locations of vortices in some case for the variational problem related to the superconducting thin films having variable thickness. Key words: Renormalized energy, Superconductivity, Vortices, Pinning mechanism. Classification:35J55, 35Q40. Introduction Three-dimensional thin films of superconducting material, say ( δa(x), δa(x)), are modeled as two-dimensional objects by Q. Du and M.D. Gunzburger in []. The reduced model is derived in [] (see (.) in the following). It is believed from numerical computation that vortices are pinned near the relatively thin regions of the sample. Scaling the physical parameters in the model derived in [], we considered in [] the following functional G ε (u, A) = a(x){ A u + ε ( u ) + da }, (.) where is a bounded smooth domain in R, a(x) is smooth such that 0 < a 0 a(x) a 0, a, D a a 0 in with constant a 0 > 0, u H (, R ) and A, the vector potential, is a real valued -form: A = A dx + A dx, A u = u iau. Since a Dirichlet-type condition is not consistent with the gauge invariance, we proceed instead as follows to have a well-posed minimization problem (see [3] or This work is supported by the Natural Science Foundation of China (No ).

2 []). Let d > 0 be an integer and g : R be a smooth function. Consider the space ( see [3] or []) V = {(u, A) H (, R ) H (, R ) : u = on, deg(u, ) = d > 0, J τ = g on } where τ denotes the unit tangent vector to such that (n, τ) is a direct, n denotes the exterior normal to, and J = (iu, A u) where (a, b) = (ab+ab) for complex numbers a, b. In [], we chose the gauge transformation as follows ([]): div(a(x)a) = 0 in, A ν = 0 on. From [], we know, there is a minimizer (u ε, A ε ) of (.) in V such that div(a(x)a ε ) = 0 in, A ε ν = 0 on. Let m = min a(x) and a (m) = {x a(x) = m}. or or Case I. a (m) : N = Carda (m) d, I(i) N = Carda (m) < d, I(ii) Case II. a (m) : N = Card(a (m) ) d, II(i) N = Card(a (m) ) < d. II(ii) We obtained in [] the following results. Theorem. Suppose that I(i) (or I(ii), or II(i), or II(ii)) holds. Let (u εn, A εn ) be any sequence of minimizers of (.). There are a subsequence, (u εn, A εn ), points a,, a N0 (N 0 < ) and u, A smooth except at a,, a N0 such that div(a(x)a ) = 0 in, (.) A ν = 0 on, and N 0 u εn u strongly in Hloc( \ {a i }) and in W,p (), p <. (.3) i= Set h εn = curla εn, h = curla. We have N 0 h εn h strongly in Hloc( \ {a i }) and in W,p (), p <. (.4) i=

3 Theorem. Let u, h, a,, a N0 be as in Theorem.. We have (i) a,, a N0 a (m), if I(i) (or I(ii), or II(i)) holds; (ii) Let d i = deg (u, a i ), then a,, a N0 a (m), if II(ii) holds (.5) d i =, N 0 = d, if N d, (.6) N 0 d i, d i = d, if N < d. (.7) (iii) u and h solves, respectively, the following equations i= (a(x) u ) = a(x)u u, in \ N 0 u =, a.e., i= {a i } (.8) ( (a(x)h a(x) ) ) + h = π N 0 (ah ) ν = ag,. i= d i δ ai, in, (.9) Theorem.3 Let Z ε = {x u ε (x) = 0}. Then for any δ > 0, there exists ε 0 > 0 such that for 0 < ε ε 0, Z ε O δ (a (m)) = {p dist (p, a (m)) < δ}. (.0) A natural question is that if N=Carda (m) d, how can we further determine the locations of the vortices a, a,, a d in a (m)? This paper is to answer this question. The purpose of this paper is to show that the vortices can be further determined by minimizing a renormalized energy defined on (a (m) ) d. Preliminaries In order to derive the renormalized enery, we need some preliminary lemmas to describe the energy expressions for the energy-minimiing harmonic maps with weight a(x) satisfying either I(i) or II(ii). Definition.. Let be a bounded smooth, simply connected open domain in R. Fix n points b, b,, b n in a (m) and n integers d, d,, d n. Set d = n j= d j and let g be a map from onto S such that deg(g, ) = d. 3

4 Let Φ 0 be the solution of div( a(x) Φ 0) = n j= πd j δ bj, in, a(x) nφ 0 = g τ g, on, Φ 0 = 0. (.) where n and τ are the normal and tangential vector on such that (n, τ) is a direct. The unique harmonic map u 0 : \ {b, b,, b n } S such that (). deg(u 0, b j ) = d j ; (). u 0 is continuous up to and u 0 = g which is associated to Φ 0, namely, u 0 x u 0 = a(x) x Φ 0, in \ {b, b,, b n }, u 0 x u 0 = a(x) (.) x Φ 0, in \ {b, b,, b n }. The function u 0 is called the canonical harmonic map with weight associated to the configuration (g, b j, d) where d = (d, d,, d n ). First of all, we can prove the following Lemma. Let u 0 be a canonical harmonic map associated to (g, b j, d) with b j a (m), then as ρ 0, a(x) u 0 = mπ ρ n d j log j= ρ + w(g, b j, a(x), d) + O(ρ ), (.3) where ρ = \ n j=b(b j, ρ) and w = w(g, b j, a(x), d) is an energy defined as follows w = w(g, b j, a(x), d) = n π a(b j )d j d k log b j b k + n Φ 0 (g g τ ) π d j R 0 (b j ), (.4) j k j= where Φ 0 is defined by (.) and n R 0 (x) = Φ 0 (x) a(b j )d j log x b j. j= Proof. Since u 0 =, we have a(x) u 0 = a(x)( u 0 x u 0 + u 0 x u 0 ) ρ ρ = ρ a(x) Φ 0 n = a(x) Φ 0 n Φ 0 B(b j,ρ) a(x) Φ 0 n Φ 0. (.5) 4 j=

5 Using (.)-(.) and define n R 0 (x) = Φ 0 (x) a(b j )d j log x b j j= which is a smooth harmonic function on. Set S j (x) = Φ 0 (x) a(b j )d j log x b j. (.6) Note that S j (x) is a smooth function and near b j it satisfies Thus we have B(b j,ρ) a(x) Φ 0 n Φ 0 = div( a(x) S j(x)) = a(b j )d j a(x) log x b j, S j (x) = Φ 0 (x) a(b j )d j log ρ, on B(b j, ρ), n S j (x) = n Φ 0 a(b j ) d j ρ, on B(b j, ρ), (.7) n S j (b j ) = R 0 (b j ) + a(b j )d j log a k a j. B(b j,ρ) j k a(x) ( ns j + a(b j ) d j ρ )(S j + a(b j )d j log ρ) and we get from a simple computation that B(b j,ρ) a(x) Φ 0 n Φ 0 = B(b j,ρ) a(x) S j + πd j S j (b j ) + πd j log ρ + o(). (.8) Combining (.5) and (.8), we are led to n a(x) u 0 = π d j log n ρ j= ρ + w j= B(b j,ρ) a(x) S j + o() (.9) which yields the conclusion. Q.E.D It follows from [4] that the canonical harmonic map can be expressed as and u 0 (z) = n j= z b j z b j eiφ div(a(x) φ) = a(x) ( θ + + θ n ), in, where φ = φ 0 on and φ 0 : R is such that and g(z) = n j= z b j z b j eiφ 0 θ k = ( (y β k), x α k z b k z b k ) where z = x + iy, b k = α k + iβ k. 5

6 Lemma. Let G be a bounde simply connected smooth domain in R and ω j (j =,,, n)be n sub-domain in G such that ω j G and ω j ω k = (j k). Denote = G \ n ω j. Define j= E = {ψ H (, S ) : ψ τ ψ = g, on G, deg(ψ, ω j ) = d j }, where g is a given function as above. If v E such that E(v) = a(x) v = E = inf a(x) ψ, (.0) ψ E then E = E(v) = where Φ solves the following problem div( a(x) Φ ) = 0, in, a(x) Φ, (.) Φ = const. = c j, on ω j, j =,, n, Φ ω = πd j a(x) n j, j =,, n, Φ a(x) n = g, on G. Proof. Since v takes its value in S we have and hence v x v x = 0, v = v v x + v v x, (v v x ) + ( v v x ) = 0. x x Setting D = ( v v x + a(x) Φ x, v v x + a(x) Φ x ), we have divd = 0, in ; ω j D n = 0, j =,, n. Then there exists a function H on such that D = ( H x, H x ), that is v v x = H x Φ a(x) x, v v x = H x + Φ a(x) x, (.) and a(x) v = a(x) Φ + a(x) H +( H x Φ x H x Φ x ). 6

7 Noting that ( H x Φ x H x Φ x ) = = = G = = = 0, G div(h Φ, H Φ ) x x H Φ n τ H Φ ω j τ H Φ τ H Φ τ G G j= Φ ( (v v τ ) + a(x) Φ n ) we have a(x) v = a(x) Φ + a(x) H a(x) Φ, for any v E. On the other hand, the problem has a solution v : S if and only if and v v x = F, v v x = F, F x = F x Γ j F τ πz, F = (F, F ), for each connected component Γ j of where Z denotes the set of integers. Hence it follows from the definition of Φ that the problem v v x = Φ a(x) x, v v x = Φ a(x) x, admits a solution v : S and we have a(x) v = a(x) Φ. 7

8 Moreover, we may see that v v τ = g on G and πdeg(v, ω j ) = v v τ = ω j ω j a(x) Φ n = πd j which means that v E and E is attained by v and E = a(x) Φ. The lemma is proved. Q.E.D Now we consider a particular case that ω j = B(b j, ρ) and further consider the limit as δ 0. Denote δ = \ n j= where is now a simply connected domain. Define E δ = { ψ H ( δ, S ) : ψ ψ τ = g, on, deg(ψ, B(b j, δ)) = d j }. We deduce from the above lemma that the minimum ν δ = inf a(x) ψ (.3) ψ E δ δ is attained by a function in E δ and ν δ = δ a(x) Φ δ where Φ δ is determined by the above lemma with ω j replaced by B(b j, δ). Denote the minimizer of problem (.3) by v δ and v 0 is the canonical harmonic map. Similarly to [3], it is not difficult to prove the following three lemmas Lemma.3 As δ 0, v δ converges to v 0 in C k (K) on every compact subset K of ( \ {b,, b n }). Lemma.4 As δ 0, a(x) v δ = π δ where w is defined as above. n a(b j )d j log j= δ + w + O(δ) Lemma.5 Let u δ and u be the minimizers of the minimization problems inf a(x) ψ (.4) ψ E δ δ and respectively, where inf ψ E a(x) ψ δ (.5) E δ = { ψ H ( δ ) : ψ ψ τ = g + a(x) nξ δ, deg(ψ, B(b j, δ)) = d j }, E = { ψ H ( δ ) : ψ ψ τ = g + a(x) nξ, deg(ψ, B(b j, δ)) = d j }. If ξ δ converges to ξ in W, () as δ 0, then when δ 0 we have a(x) u δ = δ a(x) u + o(). δ (.6) 8

9 3 Main result and its proof Now we are in a position to derive the renormalized energy for our problem. Our main result of this paper is Theorem 3. in this section. Consider d distinct points b,, b d in a (m) and δ > 0. We choose δ sufficiently small such that B(b j, δ), j {,, d}, (3.) B(b j, δ) B(b k, δ) =, for j k. Denote δ = \ d j=b(b j, δ) and consider the space W δ defined by W δ = { (u, A) H ( δ, S ) H (, R ) such that (iu, τ (( ia)u)) = g on, deg(u, B(b j, δ)) = } and the functional (note that now, ( u ) = 0) E δ (u, A) = a(x) ( ia)u + δ We define a minimization problem Note that the condition is equivalent to a(x) curla. µ δ (b) = inf (u,a) W δ E δ (u, A). (3.) (iu, τ (( ia)u)) = g, on u u τ = g + A τ, on. Lemma 3. µ δ (b) is achieved for δ sufficiently small so that (3.) holds and moreover, µ δ (b) mπd log δ + C (3.3) where C depends only on the configuration (b j ) and δ 0 here δ δ 0 such that (3.) holds. Proof. The proof is similar to that of Lemma. in [] by choosing suitable comparison element in W δ. Lemma 3. Suppose that δ is small enough so that (3.) holds. For the minimizers (u δ, A δ ) of problem (3.), we have a(x) curla δ C, (3.4) a(x) ( ia δ )u δ mπ log C, (3.5) δ δ where C depends only on (b j ) and δ 0. 9

10 Proof. In the proof we omit the subscript δ for simplicity. Let (u, A) be a pair of minimizer of (3.) satisfying the gauge condition We know that div(a(x)a) = 0, in, A n = 0, on. (3.6) a(x)a = curlξ = ( ξ x, ξ x ) (3.7) for some scalar function ξ solving ξ a(x) ξ = a(x)h = a(x)curla, in, a(x) ξ = 0, on. We have a(x) ( ia)u = δ a(x) curla = δ (3.8) δ a(x) u + a(x) ξ + (i u curlξ, u),(3.9) δ a(x) ξ a(x) a(x) ξ (3.0) here we have used the relations u(x) = on δ, a(x) A = a(x) ξ and We have a(x)[iau u iau u ] = (i u curlξ, u). (iu curlξ, u) δ = ξcurl(iu, u) + δ = j= j= ξ(iu, u τ ) ξ(iu, u τ ) (3.) where we have used the fact that u = e iφ for some local function φ, so curl(iu, u) = curl φ = 0 locally. To estimate the right hand side of (3.), we proceed as follows. From deg(u, B(b j, δ)) = and u(x) = on δ, we may write u = e i(θ j+η), on, U j \ B(b j, δ) in some neighbourhood U j of b j. Here η is some real single-valued function on U j \ B(b j, δ). We now have, for x U j \ B(b j, δ), u = i( θ + η)e i(θ+η) which implies η = θ ie i(θ+η) u 0

11 and finally η ( u + On the boundary of B(b j, δ), we have (iu, u τ ) = δ + u τ. We deduce by integrating by parts along B(b j, δ), ξ(iu, u τ ) = π ξ j + where ξ j = ξ, η j = B(b j, δ) ). (3.) x b j ξ τ (η η j ) (3.3) η. B(b j, δ) For any ξ solving (3.8), by Sobolev imbedding and elliptic estimates, we have for any 0 < α < ξ C 0,α C α ξ L. (3.4) From this we have B(b j,δ) ξ x j = ξe j n = (ξ ξ(b j))e j n, B(b j,δ) B(b j,δ) ξ x j πc α δ α ξ L. By standard trace theorem, we obtain ξ xj (3.5) ξ xj Cδ ξ. (3.6) B(b j, δ) B(b j,δ) Combining (3.5)-(3.6), we have ξ C(δ + δ α ) ξ. (3.7) Similarly we also have η η j Cδ η Cδ( u + π log ). (3.8) δ δ Combining (3.3), (3.7) and (3.8), we conclude ξ τ (η η j ) Cδα ξ L ()( u L ( δ ) + (log δ ) ). (3.9) This finishes the estimate for (3.). For the first term on the right hand side of (3.9), we have, by Lemma., a(x) u mπd log C. (3.0) δ δ

12 Combining (3.), (3.3), (3.5), (3.0) and Lemma 3., we obtain ( δ a(x) ξ a(x) a(x) ξ + a(x) ξ ) where +4π ξ(b j ) + R(δ) C (3.) j= R(δ) = [( ξ j ξ(b j )) + ξ τ (η η j )]. (3.) j= Combining (3.9) and (3.4), we verify that R(δ) C ξ L ()(δ α u L ( δ ) + δ α/ ). (3.3) Using Lemma 3. and the equality u = ( ia)u + iua, we obtain a(x) u a(x)[ ( ia)u + A ] δ δ a(x)[ ( ia)u + δ a (x) ξ ] Going back to (3.3), we deduce that By standard elliptic estimates, we have (3.5), (3.6) and (3.) yield (3.4) and (3.5) then follows. C log δ. (3.4) R(δ) 0, as δ 0. (3.5) ξ(b j ) C ξ L (). (3.6) j= ξ L () C. Lemma 3.3 Let ξ be the solution of ( ( ξ a(x) a(x) ξ a(x) )) + a(x) ( ξ a(x) ξ ) = π d j= δ bj, in, ξ = 0, on, a(x) n( ξ a(x) a(x) ξ ) = g on. where h = ξ a(x) a(x) ξ = curla. Then we have (3.7) ξ δ ξ, in W, () as δ 0. (3.8)

13 Proof. We first remark that problem (3.7) is equivalent to ( (a(x)h a(x) )) + h = π d δ bj, in, j= ξ = 0, on, n (a(x)h ) = g on. (3.9) and these equations come from (.9) and (3.7). On δ, we have, by th fat that (u δ, A δ ) are energy minimizer, ( ( ξ a(x) δ a(x) ξ a(x) δ)) + ( ξ a(x) δ ξ a(x) δ) = π d δ bj, in δ, On the other hand, j= ξ δ = 0, on, a(x) n( ξ δ a(x) a(x) ξ δ) = g on. = a(x) n( ξ δ a(x) a(x) ξ δ) From Lemma 3., we know that (iu δ, τ ( ia δ )u δ ) (3.30) = π + o(). (3.3) ξ δ L () + ξ δ L () C and by passing to a subsequence if necessary, we may assert that ξ δ ξ, weakly in W, (). Passing to the limit in (3.30) and using (3.3), we see that ξ verifies the desired equation and hence ξ = ξ. The full sequence therefore converges by uniqueness of the limit. By the fact that (u δ, A δ ) is an energy minimizer, we have a(x) ( ia δ )u δ + curla δ δ a(x) ( ia )u + curla (3.3) δ Expanding the first term on both sides of above inequality as in (3.9), we obtain a(x) curla δ + δ a(x) ξ δ + (i u δ curlξ δ, u δ ) curla + a(x) ξ + (i u curlξ, u ). 3

14 Since ξ δ ξ weakly in W, (), we obtain a(x) curla δ curla + o(), that is by (3.0) a(x) ξ δ ξ + o(). This implies the strong convergence of ξ δ to ξ in L in view of the weak lower semi-continuity (with dx replaced by dν = a(x)dx) and then Lemma 3.3 follows. Lemma 3.4 Let where Then we have µ δ = inf a(x) ψ, ψ G δ δ G δ = { ψ H ( δ, S ), deg(ψ, B(b j, δ)) =, ψ τ = i(g + a(x) nξ )ψ, on }. δ a(x) u δ µ δ 0, as δ 0. (3.33) Proof. Since τ u δ = i(g + a(x) nξ δ )u δ, on we see that u δ E δ (see Lemma.5) and then a(x) u δ inf δ ψ E δ On the other hand we have from Lemma.5 that µ δ δ = µ δ + o(). δ a(x) ψ := µ δ δ. This implies a(x) u δ µ δ + o(). δ Recalling that R(δ) 0 as δ, we deduce E δ (u δ, A δ ) = a(x) u δ + ( δ a(x) ξ + a(x) ξ a(x) a(x) ξ ) + π ξ (b j ) + o().(3.34) j= Hence, in view of (3.34), we have uncoupled u and A in the functional E δ (u, A). So in order to minimize E δ (u, A), it suffices to minimize µ δ, and this yields a(x) u δ µ δ + o(). δ The Lemma is proved. Q.E.D 4

15 Lemma 3.5 Let < p <, the map u δ remains bounded in W,p and u δ converges to u strongly in W,p, and in C k loc( \ d j={b j }), where u is defined by where φ solves with boundary condition u = d j= z b j z b j eiφ (3.35) div(a(x) φ ) = a(x) ( θ + + θ d ), x, (3.36) τ φ = (i d j= z b j z b j, τ( z b j z b j )) + a(x) nξ + g, (3.37) where θ k = ( y β k z b j, x α k z b j ) with z = x + iy, b k = α k + iβ k, k =,, d. Proof. Since u δ takes its value in S, we have div(iu δ, curlu δ ) = 0, in δ. Let Φ be the solution of div( a(x) Φ) = π d δ bj, j= a(x) nφ = g + a(x) nξ δ. (3.38) We have and div((iu δ, curlu δ ) + Φ) = 0, a(x) = = 0. ((iu δ, curlu δ ) + a(x) Φ) n (iu δ, τ u δ ) + a(x) nφ Hence there is some H δ H ( δ, R) such that (iu δ, curlu δ ) = a(x) Φ + curlh δ, that is u δ u δx = H δ x u δ u δx = H δ x + 5 a(x) a(x) Φ x, Φ x,

16 We have from this equality that a(x) u δ = δ δ a(x) Φ + a(x) H δ + ( H δ Φ H δ Φ ). δ x x x x Note that ( H δ Φ H δ Φ ) δ x x x x = Φ H δ τ + = = o(), j= Φ H δ τ Φ( u δ τ u δ + Φ a(x) n ) where we have used the fact ξ δ ξ in W, () and Φ( u δ τ u δ + Φ a(x) n ) = 0. Therefore a(x) u δ = δ δ a(x) Φ + a(x) H δ + o(). δ It follows from Lemma. that δ a(x) Φ is the minimum defined in Lemma. with g replaced by g + a(x) nξ δ. We then have from Lemma. that From Lemma 3.4, we deduce that Hence δ a(x) Φ µ δ 0, as δ 0. (3.39) δ H δ 0. (iu δ, curlu δ ) + a(x) Φ L ( δ ) 0, as δ 0, and the convergence in C k loc( \ d j={b j }) of Lemma 3.5 follows from Lemma 3.3 and the convergence in W,p follows from the fact that Φ d j x b j, near b j. 6

17 Lemma 3.6 We have for any configuration (b j ) ( \ a (m)) d, µ δ (b) = W (b) + mπd log δ + o(), as δ 0 (3.40) where W is defined by the renormalized energy W (b) = π a(b j ) log b j b k π R(b j ) j k j= +π ξ (b j ) + Φ(g + j= a(x) nξ ) + a(x) ( ξ a(x) a(x) ξ ) n ξ (3.4) where ξ is the solution of the following problem ( ( ξ a(x) a(x) ξ a(x) )) and R is given by + a(x) ( ξ a(x) ξ ) = π d j= δ bj, in, ξ = 0, on, a(x) n( ξ a(x) a(x) ξ ) = g on. (3.4) R(x) = Φ(x) m log x b j. (3.43) j= In addition, Φ is the solution of div( a(x) Φ) = π d j= δ bj, in, a(x) nφ = g + a(x) nξ, on Proof. We deduce from Lemma 3.3-Lemma 3.5 and (3.34) that µ δ (b) = δ a(x) u δ + ( a(x) ξ + a(x) ξ a(x) a(x) ξ ) + π (3.44) ξ (b j ) + o(). (3.45) j= Moreover, multiplying (3.4) by ξ, and then integrating over, we have π ξ (b j ) = j= a(x) ( ξ a(x) a(x) ξ ) n ξ a(x) ξ a(x) a(x) ξ 7 a(x) ξ. (3.46)

18 Lemma.5 and Lemma.4 imply a(x) u δ = mπd log + w(b, d, a(x), g) + O(δ) (3.47) δ δ where w(b, d, a(x), g) = mπ log b j b k π R(b j ) + j k j= Here Φ is the solution of (3.38) and R denotes R(x) = Φ(x) m log x b j. j= Combining (3.45)-(3.47), we obtain the desired result. Finally we prove the main result of this paper: a(x) Φ nφ. Theorem 3. Let a j (j =,, d) be as in Theorem. with N = Card( a (m)) d. Then the configuration a = (a j ) minimizes the renormalized energy W (b) in ( a (m)) d where W (b) is given by (3.4). Proof. For b 0 a (m), we define an auxiliary minimization problem I(ε, δ) = inf a(x) v + v Q B(b 0,δ) ε a(x)( v ), Q = {v H (B(b 0, δ), R ), v B(b0,δ) = e iθ }. Let b = (b j ) be a configuration of d points in a (m). comparison function v ε,δ in H () as follows: (). v ε,δ = u δ on δ = \ d j=b(b j, δ); We construct a (). On B(b j, δ) \ B(b j, δ ), v ε,δ is a harmonic map from B(b j, δ) \ B(b j, δ ) into S with boundary condition v ε,δ B(bj,δ) = u δ and v ε,δ B(bj, δ ) = z b j z b j ; (3). On B(b j, δ ), v ε,δ is the minimizer of I(ε, δ ). In above (u δ, A δ ) is a minimizer of problem (3.). Then we have B(b j,δ) a(x) v ε,δ + ε a(x)( v ε,δ ) I(ε, δ ) + C(δ) (3.48) where C(δ) 0 as δ 0. It follows that G ε (u ε, A ε ) G ε (v ε,δ, A ε ) E ε (v δ, A ) + di(ε, δ ) + C(δ) µ δ (b) + di(ε, δ ) + C(δ) W (b) + mπd log δ + di(ε, δ ) + C(δ). (3.49) 8

19 On the other hand, it follows fromthe definition of the functional G ε (u, A) that G ε (u ε, A ε ) = a(x)[ ( ia ε )u ε + curla ε + ε ( u ε ) ] = a(x) ( ia ε )u ε + curla ε + δ 4ε ( u ε ) ] δ + j=[ a(x) ( ia ε )u ε + ( u B(a j,δ) 4ε ε ) ] B(a j,δ) a(x) ( ia ε )u ε + curla ε δ + j=[ a(x) u ε + ( u B(a j,δ) 4ε ε ) ] + O(δ) (3.50) B(a j,δ) where we have used the fact u ε and A ε L C and δ = \ d j=b(a j, δ). Arguing as in the proof of Lemma VIII. of [5] and using Lemma., we have G ε (u ε, A ε ) W (a) + mπd log δ + j=[ a(x) u ε B(a j,δ) + ( u 4ε ε ) ] + O(δ) (3.5) B(a j,δ) where a = (a j ). The summation term on the right hand side of (3.5) can be dealt with by the same arguments as in [4]. That is, we have j=[ We finally have B(a j,δ) a(x) u ε + ( u 4ε ε ) ] di(ε, δ ). (3.5) B(a j,δ) G ε (ψ ε, A ε ) di(ε, δ ) + W (a) + mπd log δ + C (δ) (3.53) where C (δ) 0 as δ 0. Combining (3.49)-(3.53), we obtain, by letting δ 0, The theorem is proved. W (a) W (b), b = (b j ) ( a (m)) d. 9

20 References [] S.Ding, Z.Liu, Pinning of vortices for a variational problem related to the superconducting thin films having variable thickness, J. Partial Diff. Eqns., 0 (997) [] Q. Du and M.D. Gunzburger, A model for superconducting thin films having variable thickness, Physica D, 69 (993) 5 3. [3] F. Bethuel, T. Riviére, A variational problem related to superconductivity, Annal, IHP, analyse non Linéaire, (995), [4] A. Beaulieu, R. Hadiji, Asymptotics for minimizers of a class of Ginzburglandau equations with weight, C. R. Acad. Sci. Paris, Ser I, 30 (995) [5] F. Bethuel, H. Brezis and F. Hélein, Ginzburg-Landau Vortices, Birkhäuser, 994. [6] S. Ding, Z. Liu, W. Yu, Pinning of vortices for the Ginzburg-Landau functional with variable coefficient, Appl. Math.-JCU, B (997)