REGULARITY OF THE MINIMIZER FOR THE D-WAVE GINZBURG-LANDAU ENERGY

Transcription

1 METHODS AND APPLICATIONS OF ANALYSIS. c 2003 International Press Vol. 0, No., pp , March REGULARITY OF THE MINIMIZER FOR THE D-WAVE GINZBURG-LANDAU ENERGY TAI-CHIA LIN AND LIHE WANG Abstract. We study the minimizer of the d-wave Ginzburg-Landau energy in a specific class of functions. We show that the minimizer having distinct degree-one vortices is Holder continuous. Away from vortex cores, the minimizer converges uniformly to a canonical harmonic map. For a single vortex in the vortex core, we obtain the C 2 -norm estimate of the fourfold symmetric vortex solution. Furthermore, we prove the convergence of the fourfold symmetric vortex solution under different scales of δ.. Introduction. In this paper, we investigate the minimizer of the d-wave Ginzburg-Landau energy E ɛ,δ (u) = 2 u ɛ 2 ( u 2 ) δ x y u 2 dx dy, (.) defined on a class of functions V g = {u = u(x, y) : u H g (; C), x y u = h L 2 () in distribution sense} (.2) with the norm defined by u 2 = u 2 H + x y u 2 L 2. Hereafter, is a bounded smooth domain in R 2, g : S is a smooth map with degree d N, 0<ɛ,δ are small parameters, and H g (; C) ={u H (; C) : u = g}. The d-wave Ginzburg-Landau energy describes high-temperature superconductors. From [2], [7] and [8], we learned the d-wave Ginzburg-Landau energy without the magnetic field given by G(u) = R 2 u ( u 2 ) β ( 2 x y) 2 u 2 dx dy, (.3) 2 where β is a positive constant. Rotating the coordinates by 45, we may obtain G(u) = R 2 u ( u 2 ) 2 +2β x y u 2 dx dy, (.4) 2 Hereafter, we assume that u and all the derivatives of u decay fast as (x, y). Such an assumption is consistent with the results in [9] and [9]. Then we may transform (.4) into (.) up to some constants. For the minimization of (.), we may use the standard direct method to obtain the energy minimizer u δ ɛ of E ɛ,δ ( ) over the function class V g.herewehaveusedthe fact that both H () and L 2 () are reflexive Banach spaces. The minimizer u δ ɛ is a weak solution of u + ɛ 2 ( u 2 ) u δ 2 x 2 y u =0 on, (.5) Received October 5, 2002; accepted for publication June 9, Department of Mathematics, National Chung-Cheng University, Minghsiung, Chia-Yi 62, Taiwan (tclin@math.ccu.edu.tw). Department of Mathematics, University of Iowa, Iowa City, IA , USA (lwang@math.uiowa.edu). 8

2 82 T.-C. LIN AND L. WANG The highest derivative term is δ x 2 y 2 which is a degenerate elliptic operator. Such a term has a small divisor δ and may lose derivatives by the standard bootstrap argument for (.5). This may cause the main difficulty to get the regularity of u δ ɛ. Until now, there is not any regularity theorem for (.5). Now we state a general regularity theorem of u δ ɛ as follows: Theorem I. Suppose δ = δ ɛ is a positive constant which may depend on ɛ. Then there exists a minimizer u δ ɛ of (.) over V g such that u δ ɛ C 2 ( ) = O(ɛ +δ). (.6) For (.5), we may rescale the spatial variables by ɛ, and obtain u +( u 2 ) u δɛ 2 2 x 2 y u =0 on ɛ, (.7) where ɛ {(x, y) :(ɛx,ɛy) }. From physical literature (cf. [2] and [7]), the coefficient δɛ 2 of x 2 y 2 u is positive and bounded i.e. 0 <δ= O(ɛ 2 )asɛ 0+. Hereafter, we only consider such a quantity for δ. By the same argument of [6], we have Theorem A. Suppose 0 <δ= O(ɛ 2 ) as ɛ 0+. Then there exists a minimizer u ɛ of (.) over V g such that (i) u ɛ has d degree-one vortices in, (ii) E ɛ,δ (u ɛ )=πd log ɛ + O() as ɛ 0+, (iii) u ɛ converges to u (up to a subsequence) strongly in L 2 () and weakly in Hloc (\{a,,a d }), (iv) (a,,a d ) d is a global minimizer of the renormalized energy W g in [3]. Here u is a canonical harmonic map defined by u (z) = d j= z a j z a j eih(z), z, (.8) and h is a real-valued harmonic function. For the product in (.8), we have used the fact that R 2 is equivalent to C. Actually, the vortices of u ɛ may arbitrarily tend to a j s (up to a subsequence) as ɛ goes to zero. Away from the vortex cores B ρ (a j ) s, we obtain a uniform convergence of u ɛ as follows: Theorem II. Suppose 0 <δ= O(ɛ 2 ) as ɛ 0+. Then for ρ>0, the minimizer u ɛ converges to u (up to a subsequence) uniformly on \ d j= B ρ (a j ) as ɛ goes to zero, where u and (a,,a d ) d are defined in Theorem A. Hereafter, B ρ (a j ) is the disk in R 2 with radius ρ and center at a j. To estimate u ɛ in the vortex cores B ρ (a j ) s, we may simplify the minimization problem by setting = B (0), where B (0) is the unit disk in R 2 with center at the origin. Moreover, we consider a modified minimization problem given by Minimize E ɛ,δ over W 0 = V g0 W, (.9)

3 REGULARITY OF THE d-wave MINIMIZER 83 where g 0 e iθ on and W = {u = a +4k (r) e i (+4k) θ on,a +4k (r) R, r [0, ],k Z}. k Z Hereafter, (r, θ) denotes the polar coordinates in R 2. The function space W provides fourfold symmetry for the minimizer. Actually, fourfold symmetry is a characteristic of vortex states in d-wave superconductors (cf. [5], [7], [9]). Please note that the function space W 0 is a subspace of V g0. Hence we cannot assure that the energy minimizer u ɛ of E ɛ,δ on W 0 is a weak solution of (.5). In [], we prove that u ɛ is a weak solution of (.5) and we obtain the H -norm estimate as follows: Theorem B. Assume that 0 < δ = O(ɛ N ), where N is a positive constant independent of ɛ. Then there exists a minimizer u ɛ of (.9) such that u ɛ is a weak solution of (.5) and u ɛ = u ɛ 0 + v ɛ, (.0) where u ɛ 0 f 0 ( r ɛ ) eiθ is the unique energy minimizer of E ɛ,0 over H g 0 () (cf. [?]), v ɛ W H 0 () satisfies v ɛ L2 () = O( δɛ ), v ɛ H () = O( δɛ 2 ). (.) Hereafter, u ɛ is called the fourfold symmetric vortex solution of (.5). From [6], [0], [2], one may know qualitative theorems of u ɛ 0. Then Theorem I implies that v ɛ C 2 ( ) = O(/ɛ). (.2) The upper bound of (.2) may tend to infinity as ɛ goes to zero. By the fourfold symmetry of u ɛ, we may improve the estimate (.2) by Theorem III. Assume that 0 <δ= O(ɛ N ) as ɛ goes to zero, where N 4 is a constant independent of ɛ. Then v ɛ C 2 ( ) = O( δɛ 2 ), (.3) Theorem III is essential to prove the stability of the fourfold symmetric vortex solution u ɛ. Actually, H -norm estimate (cf. Theorem B) cannot assure the stability of u ɛ. We may consider the associated quadratic form given by Q ɛ (w) = for w V 0, where 2 w 2 2ɛ 2 ( u ɛ 2 ) w 2 + ɛ 2 (u ɛ w) δ w xy 2 dx dy, (.4) V 0 = {u = u(x, y) : u H 0 (; C), x y u = h L 2 () in distribution sense}, d 2 and Q ɛ (w) = 2 dt E 2 ɛ,δ (u ɛ + tw) t=0 is the associated second variational form. In Corollary I, we will use (.3) to prove Q ɛ (w) > 0forw V 0, w L 2 0, provided

4 84 T.-C. LIN AND L. WANG the parameter δ is sufficiently small. Therefore the fourfold symmetric vortex solution u ɛ is stable if the parameter δ is sufficiently small. From (.3), it is remarkable that if N>4, then v ɛ C 2 0asɛ goes to zero. ( ) Theorem III gives us the C 2 -norm estimate of the perturbation term v ɛ in. Can we have C α -norm, 2 <α<, estimate of v ɛ? To answer this question, we state another result for the estimate of v ɛ in C α -norm, 2 <α<, as follows: Theorem IV. Assume that 0 <δ= O(ɛ N ) as ɛ goes to zero, where N 6 is a constant independent of ɛ. Suppose lim inf η 0+ { η x η 2 xv 2 ɛ 2 + yv 2 ɛ 2 dx dy = O(ɛ 6 ), (.5) 2 +y 2 } for 0 <ɛ<ɛ 0,whereɛ 0 is a positive constant. Then for 2 <α<, v ɛ Cα ( ) = O( δɛ 3 ). (.6) Theorem IV implies that the C α -norm, 2 <α<, estimate of v ɛ may depend on the behavior of xv 2 ɛ and yv 2 ɛ near the boundary. It is remarkable that the boundary condition of u ɛ is { uɛ = g 0 on, (.7) x y u ɛ =0 on, Hence the boundary condition of v ɛ is { vɛ =0 on, x y v ɛ = x y u ɛ 0 on. (.8) As ɛ goes to zero, x y v ɛ = x y u ɛ 0 x y e iθ 0. Thus 2 v ɛ may not tend to zero on, and it is possible that 2 xv ɛ and 2 yv ɛ may have boundary layer on. Therefore (.5) is necessary to Theorem IV. To understand more on the structure of a single vortex, we may rescale the spatial variables by ɛ i.e. we set ũ ɛ (x, y) =u ɛ (ɛx,ɛy), for (x, y) ɛ. Then ũ ɛ is a weak solution of (.7). For the convergence of ũ ɛ,wehave Theorem V. Let ũ ɛ (x, y) =u ɛ (ɛx,ɛy), for (x, y) ɛ. Then ũ ɛ converges weakly (up to a subsequence) to ũ in Hloc (R2 ). Furthermore, ũ is a weak solution of u +( u 2 )u =0 on R 2, if δ = o(ɛ 2 ), (.9) u +( u 2 )u λ x 2 y 2 u =0 on R 2, if δ = λɛ 2, (.20) where λ is a positive constant independent of ɛ. In particular, if δ = O(ɛ N ),N > 4, then ũ = f(r) e iθ,wheref is the unique solution of f + r f r 2 f +( f 2 ) f =0, r >0, (.2) f(0) = 0,f(+ ) =.

5 REGULARITY OF THE d-wave MINIMIZER 85 The equation (.9) has been investigated to find solutions with vortex structures (cf. [3], [4], [6], [2]). However, the uniqueness of (.9) with lim u(x, (x,y) y) eiθ = 0 has not yet been proved. Hence it is still open that ũ = f(r) e iθ if 0 < δ = O(ɛ N ), 2 <N 4. In the rest of this paper, we will prove Theorem I and introduce a general regularity theorem in Section 2. In Section 3, we will prove Theorem II. In Section 4 and 5, we will complete the proof of Theorem III, IV and V, respectively. Acknowledgement. The research of the first author is partially supported by a research Grant from NSC of Taiwan. The research of the second author is partially supported by NSF Grant General Regularity Theorem. In this section, we will provide a proof of Theorem I. To prove Theorem I, we need a crucial Lemma given by Lemma I. Assume is a bounded smooth domain in R 2.Letu V g satisfy x y u 2 A, (2.) u 2 H () B, (2.2) where A, B are positive constant and V g is defined in (.2). Then u is of C 2 ( ) and u C 2 ( ) C A + B, (2.3) where C is a positive constant depending only on. It is remarkable that Lemma I is a general regularity theorem for functions satisfying (2.) and (2.2). Now we prove Lemma I as follows. From extension theorem (cf. []), we may extend the function u on a cube Q =[a, b] [α, β] such that Q x y u 2 C 0 A, (2.4) u 2 H (Q) C 0 B, (2.5) where C 0 is a positive constant depending on, a<b; α<βare constants. By (2.5) and Fubini Theorem, there exists x 0 [a, b] such that u(x 0, ) H ([α,β]) B 0, (2.6) where B 0 =2C 0 B/ b a. Hence by Sobolev embedding, u(x 0, ) C 2 ([α, β]). Fix x [a, b] arbitrarily. Without loss of generality, we may assume u is smooth on Q.

6 86 T.-C. LIN AND L. WANG Then β u 2 (x, y)+u 2 y(x, y) dy α 2 =2 +2 β α β α β +2 α β α u 2 (x 0,y) dy +2 u 2 y(x 0,y) dy +2 β α β u 2 (x 0,y)+u 2 y(x 0,y) dy +2 x x 0 β α u xy (t, y) dt 2 dy x u(x, y) u(x 0,y) 2 dy u y (x, y) u y (x 0,y) 2 dy (by Triangle inequality) β α x x 0 u x (t, y) dt 2 dy 2B 0 +2 x x 0 u x (t, y) 2 + u xy (t, y) 2 dt dy (by Holder inequality) α x 0 2[B 0 + b a C 0 (A + B)] ( (2.4), (2.5)), where u y = y u and u xy = x y u. Hence u(x, ) 2 H ([α,β]) C (A + B), for x [a, b], where C is a positive constant depending only on and b a. Thus by Sobolev embedding, u(x, ) C 2 C 2 A + B, (2.7) ([α,β]) for x [a, b], where C 2 is a positive constant depending only on, b a and β α. Similarly, we may obtain u(,y) C 2 C 3 A + B, (2.8) ([a,b]) for y [α, β], where C 3 is a positive constant depending only on, b a and β α. Therefore by (2.7) and (2.8), we may complete the proof of Lemma I. Now we want to prove Theorem I. From the standard direct method, it is easy to obtain a minimizer u δ ɛ V g of (.). Let u 0 ɛ be a minimizer of the energy functional E ɛ,0 (u) = 2 u 2 + 4ɛ 2 ( u 2 ) 2 on H g (). From [3], we learned the quantitative properties of u 0 ɛ. Then it is easy to check that Hence E ɛ,0 (u 0 ɛ)+ 2 δ E ɛ,0 (u 0 ɛ)=πd log + O(), (2.9) ɛ x y u 0 ɛ 2 = O(ɛ 2 ). (2.0) x y u δ ɛ 2 E ɛ,0 (u δ ɛ)+ 2 δ = E ɛ,δ (u δ ɛ) E ɛ,δ (u 0 ɛ) = E ɛ,0 (u 0 ɛ)+ 2 δ x y u δ ɛ 2 x y u 0 ɛ 2 = E ɛ,0 (u 0 ɛ)+δo(ɛ 2 ) ( (2.0)).

7 REGULARITY OF THE d-wave MINIMIZER 87 Hence Moreover, u δ 2 ɛ 2 E ɛ,δ (u δ ɛ) E ɛ,δ (u 0 ɛ) x y u δ ɛ 2 = O(ɛ 2 ). (2.) = πd log ɛ + O(δɛ 2 ) ( (2.9), (2.0)). Thus u δ ɛ 2 H () = O(log ɛ + δɛ 2 ). (2.2) Therefore by (2.), (2.2) and Lemma I, we may complete the proof of Theorem I. 3. Minimizer with Multiple Vortices. In this section, we assume 0 <δ= O(ɛ 2 )asɛ goes to zero. From Theorem A in Section, we obtain a minimizer u ɛ having d degree-one vortices near a j,j =,,d. By Theorem I, the minimizer u ɛ is of C 2 ( ) and u ɛ C 2 = O(ɛ ). Such an upper bound is unbounded as ɛ tends ( ) to zero, and cannot assure any strong convergence of u ɛ. The main purpose of this section is to prove Theorem II and get a bounded estimate for the C 2 -norm of u ɛ away from the vortex cores. NowwewanttoproveTheoremII.Letρ>0beasmall constant and ρ \ d j= B ρ (a j ), where a j s are defined in Theorem A. We may define a comparison map v ɛ given by { uɛ in B v ɛ = ρ (a j ),j =,,d, w ɛ in ρ, (3.) where w ɛ is the minimizer of the energy functional E ɛ,0 (w; ρ )= ρ 2 w 2 + 4ɛ 2 ( w 2 ) 2 over the function class H g ( ρ). Here the boundary condition g is defined by { g on, g = on B ρ (a j ),j =,,d. u ɛ From Theorem A and [3], we may obtain quantitative properties of w ɛ. Now we define the following energy functionals: E ɛ,δ (u ɛ ; d j= B ρ (a j )) d = 2 u ɛ ɛ 2 ( u ɛ 2 ) δ x y u ɛ 2, (3.2) j= B ρ(a j) E ɛ,δ (w ɛ ; ρ )= ρ 2 w ɛ ɛ 2 ( w ɛ 2 ) δ x y w ɛ 2. (3.3)

8 88 T.-C. LIN AND L. WANG Then E ɛ,δ (u ɛ ; d j= B ρ (a j )) + E ɛ,0 (w ɛ ; ρ )+ 2 δ x y u ɛ 2 ρ E ɛ,δ (u ɛ ; d j= B ρ (a j )) + E ɛ,0 (u ɛ ; ρ )+ 2 δ x y u ɛ 2 ρ = E ɛ,δ (u ɛ ) E ɛ,δ (v ɛ ) = E ɛ,δ (u ɛ ; d j= B ρ (a j )) + E ɛ,δ (w ɛ ; ρ ) = E ɛ,δ (u ɛ ; d j= B ρ (a j )) + E ɛ,0 (w ɛ ; ρ )+ 2 δ x y w ɛ 2, ρ i.e. ρ x y u ɛ 2 x y w ɛ 2. (3.4) ρ Similarly, i.e. E ɛ,δ (u ɛ ; d j= B ρ (a j )) + u ɛ 2 2 ρ E ɛ,δ (u ɛ ; d j= B ρ (a j )) + E ɛ,0 (u ɛ ; ρ )+ 2 δ = E ɛ,δ (u ɛ ) E ɛ,δ (v ɛ ) = E ɛ,δ (u ɛ ; d j= B ρ (a j )) + E ɛ,δ (w ɛ ; ρ ), ρ x y u ɛ 2 u ɛ 2 E ɛ,δ (w ɛ ; ρ ). (3.5) 2 ρ Hence by (3.4), (3.5) and [3], we obtain ρ x y u ɛ 2 K 0, (3.6) ρ u ɛ 2 K 0, (3.7) where K 0 is a positive constant depending on ρ. Thus (3.6), (3.7) and Lemma I imply that u ɛ C 2 ( ρ) K, (3.8) where K is a positive constant depending on ρ. Therefore by (3.8) and Arzela- Ascoli Theorem, we may complete the proof of Theorem II. Note that (3.8) provides a bounded C 2 -norm estimate of u ɛ on ρ. 4. Estimate of a Single Vortex. In this section, we assume = B (0) is a unit disk with center at the origin, and 0 <δ= O(ɛ N )asɛ goes to zero, where N 4

9 REGULARITY OF THE d-wave MINIMIZER 89 is a constant independent of ɛ. From Theorem B and using energy comparison, we have E ɛ,0 (u ɛ 0)+ 2 δ x y u ɛ 2 E ɛ,0 (u ɛ )+ 2 δ x y u ɛ 2 = E ɛ,δ (u ɛ ) E ɛ,δ (u ɛ 0) = E ɛ,0 (u ɛ 0)+ 2 δ x y u ɛ 0 2, i.e. x y u ɛ 2 x y u ɛ 0 2. (4.) Then by (4.), (.0) and Holder inequality, we obtain x y v ɛ 2 = O(ɛ 2 ). (4.2) Here we have used some properties of u ɛ 0 (cf. [6], [2]). From (.), (4.2) and extension theorem (cf. []), we may extend v ɛ to a cube Q =[, ] [, ] such that v ɛ 2 H (Q ) = O( δɛ 2 ), (4.3) Q x y v ɛ 2 = O(ɛ 2 ). (4.4) Here we have used the fact that δ = O(ɛ N ),N 4asɛ goes to zero. Now we want to prove Theorem III. Without loss of generality, we may assume v ɛ is smooth on Q and satisfies x v ɛ 2 δɛ 2, (4.5) Q y v ɛ 2 δɛ 2, (4.6) Q Q x y v ɛ 2 ɛ 2. (4.7) By (4.6) and Fubini Theorem, there exists x 0 [, ] such that y v ɛ (x 0,y) 2 dy 2 δɛ 2. (4.8) From Theorem B, u ɛ is of fourfold symmetry on i.e. v ɛ is of fourfold symmetry on

10 90 T.-C. LIN AND L. WANG Q. Hence we may set x 0 [0, ]. Moreover, 2 +2 y v ɛ (x, y) 2 dy x y v ɛ (x, y) y v ɛ (x 0,y) 2 dy y v ɛ (x 0,y) 2 dy (by triangle inequality) =2 x y v ɛ (s, y) ds 2 dy +2 y v ɛ (x 0,y) 2 dy x 0 2 x x 0 x y v ɛ 2 +2 y v ɛ (x 0,y) 2 dy (by Holder inequality) Q 6 δɛ 2 ( (4.7), (4.8)), for x x 0 δ, i.e. y v ɛ (x, y) 2 dy 6 δɛ 2 for x I [x 0 δ, x 0 + δ] [0, ]. (4.9) Let Q 2 =([, ]\I ) [, ]. Then (4.6) implies that Q 2 y v ɛ 2 dx dy δɛ 2. (4.0) Hence by Fubini Theorem and the fourfold symmetry of v ɛ, there exists x [0, ]\I such that As for (4.9), we obtain y v ɛ (x,y) 2 dy 2 δɛ 2. (4.) y v ɛ (x, y) 2 dy 6 δɛ 2 for x I 2 [x δ, x + δ] [0, ]\I. (4.2) By induction, there exist x k [0, ]\ k j= I j such that y v ɛ (x k,y) 2 dy 2 δɛ 2, (4.3) y v ɛ (x, y) 2 dy 6 δɛ 2, for x I k+, (4.4) for k =0,, 2, N, where I k+ [x k δ, x k + δ] [0, ]\ k j= I j. Here N = O(/ δ) is a positive integer such that N k= I k =[0, ]. (4.5)

11 REGULARITY OF THE d-wave MINIMIZER 9 Note that the interiors of I k s are disjoint to each other. Similarly, by (4.5) and the same argument as (4.3)-(4.5), we may obtain y l [0, ]\ l k= J k such that for l =0,, 2, M, where x v ɛ (x, y l ) 2 dx 2 δɛ 2, (4.6) x v ɛ (x, y) 2 dx 6 δɛ 2, for y J l+, (4.7) J l+ [y l δ, y l + δ] [0, ]\ l k= J k. Here M = O(/ δ) is a positive integer such that M l= J l =[0, ]. (4.8) Note that the interiors of J l s are disjoint to each other. Sobolev embedding, From (4.4), (4.5) and v ɛ (x, ) C 2 ([,]) C 0 δɛ 2, (4.9) for x [0, ], where C 0 is a positive constant independent of x, y, ɛ and δ. Similarly, by (4.7), (4.8) and Sobolev embedding, v ɛ (,y) C 2 ([,]) C 0 δɛ 2, (4.20) for y [0, ]. Thus (4.9) and (4.20) imply that v ɛ C 2 ([0,] [0,]) C δɛ 2, (4.2) where C is a positive constant independent of ɛ and δ. Therefore by (4.2) and the fourfold symmetry of v ɛ, we may complete the proof of Theorem III. Now we want to prove Theorem IV. From (.5) and (.0), we have where v ɛ δ 2 x 2 y v ɛ = δ 2 x 2 y u ɛ 0 + N ɛ (v ɛ ), (4.22) N ɛ (v ɛ )= ɛ 2 [ ( u ɛ 0 2 )v ɛ +2(u ɛ 0 v ɛ )u ɛ 0 +2(u ɛ 0 v ɛ )v ɛ + v ɛ 2 u ɛ 0 + v ɛ 2 v ɛ ]. (4.23) By (.) and (4.23), it is easy to check that N ɛ (v ɛ ) L 2 () = O( δɛ 3 ). (4.24) Here we have used the assumption that 0 <δ= O(ɛ N ), N 6asɛ goes to zero. Hence by (4.22), (4.24) and [3], we obtain v ɛ δ 2 x 2 y v ɛ L 2 () = O( δɛ 3 ). (4.25)

12 92 T.-C. LIN AND L. WANG Let P C0 () be a real-valued test function defined later. Then using integration by parts, we have P v ɛ ( v ɛ δ x 2 y 2 v ɛ )= P v ɛ 2 δ P v ɛ x 2 y 2 v ɛ = P v ɛ 2 δ P xv 2 ɛ x 2 y 2 v ɛ δ P yv 2 ɛ x 2 y 2 v ɛ = P v ɛ 2 + δ P x 2 y v ɛ yp xv 2 ɛ 2 + δ P y 2 x v ɛ xp yv 2 ɛ 2, i.e. P v ɛ ( v ɛ δ x 2 y 2 v ɛ ) = Now we define the test function P by P v ɛ 2 + δ P ( x 2 y v ɛ 2 + y 2 x v ɛ 2 ) 2 δ 2η 2 yp 2 xv ɛ xp 2 yv ɛ 2. (4.26) P (x, y) =e x 2 y 2, for (x, y), (4.27) where η>0 is a small parameter. Then it is easy to check that xp, 2 yp 2 0 for x 2 + y 2 η, (4.28) xp, 2 yp 2 α 0 η 2 for η x 2 + y 2, (4.29) where α 0 > 0 is a universal constant independent of η. Hence (4.26), (4.28) and (4.29) imply that P v ɛ ( v ɛ δ x 2 y 2 v ɛ ) P v ɛ 2 2 α 0δη 2 xv 2 ɛ 2 + yv 2 ɛ 2. From (4.24), (4.27) and Holder inequality, we have η x 2 +y 2 P v ɛ ( v ɛ δ 2 x 2 y v ɛ ) P v ɛ L 2 () v ɛ δ 2 x 2 y v ɛ L 2 () (4.30) i.e. P v ɛ ( v ɛ δ x 2 y 2 v ɛ ) 2 2 P v ɛ 2 L 2 () + 2 v ɛ δ x 2 y 2 v ɛ 2 L 2 () P v ɛ 2 dx dy + O(δɛ 6 ) ( (4.24), (4.27)), 2 P v ɛ 2 dx dy + O(δɛ 6 ). (4.3) Hence (4.30) and (4.3) imply P v ɛ 2 α 0 δη 2 xv 2 ɛ 2 + yv 2 ɛ 2 dx dy + O(δɛ 6 ). (4.32) η x 2 +y 2

13 REGULARITY OF THE d-wave MINIMIZER 93 Hence by (.5), (4.32) and Fatou s Lemma, v ɛ 2 = O(δɛ 6 ). (4.33) Therefore by (4.33) and standard theorems for W 2,p estimate and Sobolev embedding, we may complete the proof of Theorem IV. Remark. From [], the minimizer u ɛ of (.9) is a weak solution of (.5). However, u ɛ may not be a global minimizer of E ɛ,δ on V g0. We will prove that u ɛ is a local minimizer of E ɛ,δ on V g0 if δ is sufficiently small. From Theorem B and Theorem III, (.4) becomes Q ɛ (w) = 2 w 2 2ɛ 2 ( u0 ɛ 2 ) w 2 + ɛ 2 (u0 ɛ w) δ w xy 2 dx dy + ɛ 2 O( v ɛ L + v ɛ 2 L ) w 2 L2 ( (.0)) 2 w 2 2ɛ 2 ( u0 ɛ 2 ) w 2 + ɛ 2 (u0 ɛ w) 2 dx dy + O( δɛ 4 ) w 2 L2, ( (.3)) i.e. for w V 0, where Q 0 ɛ(w) By (4.34) and [7], we have provided Q ɛ (w) Q 0 ɛ(w)+o( δɛ 4 ) w 2 L 2, (4.34) 2 w 2 2ɛ 2 ( u0 ɛ 2 ) w 2 + ɛ 2 (u0 ɛ w) 2 dx dy. Q ɛ (w) > 0 for w V 0, w L 2 0, (4.35) 0 <δ= o(λ 2,ɛɛ 8 ), 0 <λ,ɛ = o(), (4.36) where o() is a small quantity which tends to zero as ɛ goes to zero. Actually, λ,ɛ is the minimization of Q 0 ɛ(w) forw H0 (), w L 2 = (cf. [], [7], [8]). Therefore we have Corollary I. Under the same assumptions as Theorem III and (4.36), the fourfold symmetric vortex solution u ɛ is stable. 5. Proof of Theorem V. From Theorem A (ii), we obtain e ɛ (u ɛ )=πlog + O(), (5.) ɛ where e ɛ is the energy density of E ɛ,0 and is defined by e ɛ (u) 2 u 2 + 4ɛ 2 ( u 2 ) 2.

14 94 T.-C. LIN AND L. WANG We let Then û ɛ satisfies and û ɛ = e ɛ (û ɛ ) { uɛ if u ɛ, u ɛ / u ɛ if u ɛ >. û ɛ in, (5.2) e ɛ (u ɛ ) π log ɛ + M 0, (5.3) where M 0 is a positive constant independent of ɛ. By (5.2), (5.3) and the proof of Proposition. in [5], û ɛ has only one essential zero a ɛ in and deg( ûɛ û, B)=, ɛ where B = B ɛ α(a ɛ )andα (0, ). Without loss of generality, we may assume that a ɛ = 0. Now we want to prove Lemma II. For each R 0 >, there exists a positive constant C depending only on R 0 such that e ɛ (û ɛ ) π log ɛ C(R 0), as ɛ 0+. (5.4) \B ɛr0 (0) Proof of Lemma II. Fix R 0 > arbitrarily. Let ɛ = ɛr 0. Then (5.3) implies e ɛ (û ɛ ) e ɛ (û ɛ ) π log ɛ + M 0, (5.5) where M 0 = M 0 + π log R 0. Again, by (5.2), (5.5) and the proof of Proposition. in [5], û ɛ has only one essential zero a ɛ in and deg( ûɛ û ɛ, B ) =, where B = B ɛ α(a ɛ ) and α (0, ). Moreover, by the same argument of Lemma 2.2 in [4] and [5], we obtain e ɛ (û ɛ ) π α log ɛ M, (5.6) \B and e ɛ (û ɛ ) π ( α) log ɛ + M, (5.7) B where M is a positive constant independent of ɛ. Now we claim that e ɛ (û ɛ ) π ( α) log K, (5.8) ɛ B \B ɛ(0) where K is a positive constant independent of ɛ. Without loss of generality, we may assume that B = B θ0 (0),θ = ɛ α 0. Then (5.7) implies that e ɛ (û ɛ ) π log θ 0 ɛ + M. (5.9) B θ0 (0)

15 REGULARITY OF THE d-wave MINIMIZER 95 Moreover, we may rescale the spatial variable and rewrite (5.7) as e ɛ (û ɛ ) π ( α) log ɛ + M, (5.0) B (0) where ɛ = ɛ α. By (5.0) and Fubini theorem (cf. [4]), there exists θ ( ɛ 2 α, ɛ α ) such that θ 0 θ e ɛ (û ɛ ) C( α, M ), and that B θ0 θ (0) deg( ûɛ û ɛ, B θ 0θ (0)) =. Hence by the same argument of Lemma 2.2 in [4] and [5], we have B θ0 (0)\B θ0 θ (0) e ɛ (û ɛ ) π log θ M 2, (5.) and B θ0 θ (0) e ɛ (û ɛ ) π log θ 0 θ ɛ + M + M 2, (5.2) where M 2 is a positive constant satisfying M 2 C 0 θ 0. Here C 0 is a positive constant independent of ɛ. Thus by induction, we may obtain θ,,θ m ( ɛ 2 α, ɛ α ) such that ɛ = θ 0 θ θ m and e ɛ (û ɛ ) π log M k+, (5.3) θ k B θ0 θ k (0)\B θ0 θ k (0) and B θ0 θ k (0) e ɛ (û ɛ ) π log θ 0 θ k ɛ k+ + j= M j, (5.4) for k =,,m, where M j s are positive constants satisfying M k+ C 0 θ0 k for k+ k 0. Note that M j M + C 0 θ j 0 C, where C is a positive constant j= j= independent of ɛ and k. Therefore by (5.3), we may obtain (5.8). By (5.6) and (5.8), we obtain (5.4) and complete the proof of Lemma II. Here we have used the fact that e ɛ (û ɛ ) e ɛ (û ɛ ) because of R 0 >. \B ɛ(0) \B ɛ(0) From Lemma II, it is obvious that e ɛ (u ɛ ) π log ɛ C(R 0), (5.5) \B ɛr0 (0) Hence (5.) and (5.5) imply that e ɛ (u ɛ ) K(R 0 ), R 0 >, (5.6) B ɛr0 (0)

16 96 T.-C. LIN AND L. WANG where K is a positive constant depending only on R 0. Since ũ ɛ (x, y) =u ɛ (ɛx,ɛy), for (x, y) ɛ, then (5.6) implies that e (ũ ɛ ) K(R 0 ), R 0 >. (5.7) B R0 (0) Thus ũ ɛ H (B R0 (0)) K (R 0 ) for all R 0 >, where K is a positive constant independent of ɛ. Therefore we may obtain that ũ ɛ converges to ũ weakly in H loc (R2 )as ɛ goes to zero(up to a subsequence). Moreover, by (.7), it is obvious that ũ is a weak solution of (.9) if δ = o(ɛ 2 ), and ũ is a weak solution of (.20) if δ = λɛ 2, where λ is a positive constant independent of ɛ. By Theorem III, [6] and [2], we may obtain ũ = f(r) e iθ if δ = O(ɛ N ),N >4, where f satisfies (.2), and we complete the proof of Theorem V. REFERENCES [] R. A. Adams, Sobolev spaces, Academic Press, INC., New York, 975. [2] I. Affleck, M. Franz, M. H. S. Amin, Generalized London free energy for high-t c vortex lattices, Phys. Rev. B, 55, R704 (997). [3] F. Bethuel, H. Brezis, F. Helein, Ginzburg-Landau vortices, Birkhauser, 994. [4] H. Brezis, F. Merle, T. Riviere, Quantization effects for u = u( u 2 ) in R 2,Arch. Rational Mech. Anal., 26:(994), pp [5] A. J. Berlinsky, A. L. Fetter, M. Franz, C. Kallin, P. I. Soininen, Ginzburg-Landau theory of vortices in d-wave superconductors, Phys. Rev. Lett., 75(995), pp [6] X. Chen, C. M. Elliott, T. Qi, Shooting method for vortex solutions of a complex-valued Ginzburg-Landau equation, Proc. Roy. Soc. Edinburgh, 24A(994), pp [7] D. Chang, C.Y. Mu, B. Rosenstein, C. L. Wu, Static and dynamical anisotropy effects in mixed state of d-wave superconductors, Phys. Rev. B, 57:3(998), pp [8] Q. Du, Studies of a Ginzburg-Landau model for d-wave superconductors, SIAM J. Appl. Math., 59:4(999), pp [9] M. Franz, C. Kallin, P. I. Soininen, A. J. Berlinsky, A. L. Fetter, Vortex state in a d-wave superconductor, Phys. Rev. B, 53:9(996), pp [0] P. Hagan, Spiral Waves in Reaction Diffusion Equations, SIAM J. Appl. Math., 42:4(982), pp [] Q. Han, T. C. Lin, Fourfold symmetric vortex solutions of the d-wave Ginzburg-Landau equation, Nonlinearity, 5(2002), pp [2] R. M. Hervé, M. Hervé, Étude qualitative des solutions réelles d une équation différentielle liée àl équation de Ginzburg-Landau, Ann. Inst. Henri Poincaré, :4(994), pp [3] F. H. Lin, Static and moving vortices in Ginzburg-Landau theories, Progress in nonlinear differential equations and their applications, 29(997), Birkhäuser Verlag Basel/ Switzerland. [4] F. H. Lin, Complex Ginzburg-Landau equations and dynamics of vortices, filaments, and codimension-2 submanifolds, Comm. Pure Appl. Math., 5:4(998), pp [5] F. H. Lin, Vortex dynamics for the nonlinear wave equation, Comm. Pure Appl. Math., LII:6(999), pp [6] F. H. Lin, T. C. Lin, Vortex State of d-wave Superconductors in the Ginzburg-Landau Energy, SIAM J. Math. Anal., 32:3(2000), pp [7] T. C. Lin, The stability of the radial solution to the Ginzburg-Landau equation, Comm. in PDE, 22(997), pp [8] P. Mironescu, On the stability of radial solutions of the Ginzburg-Landau equation, J. Funct. Anal., 30:2(995), pp [9] J. H. Xu, Y. Ren, C. S. Ting, Structures of single vortex and vortex lattice in a d-wave superconductor, Phys. Rev. B, 53:6(996), pp. 299.