Asymptotics for solutions of the Ginzburg-Landau equations with Dirichlet boundary conditions

(1)

(2) (3)

ASYMPTOTICS FOR SOLUTIONS OF THE GINZBURG-LANDAU EQUATIONS WITH

.pIRTCHLE',l'.B9:UN1?ARY .CONDITIONS

JONGMIN HAN

ABSTRACT. In this paper we study some asymptotics for solutions of the Ginzburg-Landau equations with Dirichlet boundary con- ditions. We consider the solutions (Ue,Ae) which minimize the Ginzburg-Landau energy functional Ee(u,A). We show that the solutions (ue, A e) converge to some (u., A.) in various norms as the coupling parameterto-To.

1. Introduction

Let n be a smooth bounded simply connected domain in R^2. We consider two smooth functions

satisfying luol = 1 on an, ^divAo=·0 and deg(g,an) ^{= 0 where 9 =}

UoIan· Here deg(g,an) is the winding number of9 consideredasa map from an ^{into SI.} Consider the following Ginzburg-Landau equations with Dirichlet boundary conditioris, denoted by (P),

2 1 ₂

D_Au+"2(1-!u! )u=O inn

€

2 '/,( - - )

curIA+2"uDAu-uDA U =0 inn u=9 , A = Ao ^onan

where ^€ > 0, (DA)j = aj - iAj and curl²A = -~A+'\7(divA). Here u : 0. ~ C is the complex .order parameter and A : n ~ R² is the magnetic vector potential.

Received April 3, 1998.

1991Mathematics Subject Classification: 35B40, 35J65.

Key words and phrases: Ginzburg-Landau equations, asymptotic behavior.

in n

on an,

The Ginzburg-Landau model has been proposed in 1950 to give phenomenological descriptions on superconductivity at low tempera- ture. Physically lul represents the density of the superconducting elec- tron pairs, so called Cooper pairs, in the superconducting material.

When lul = 0,the material remains in normal conducting state. When

lul = 1, the material remains in superconducting state.

For physical reasoning, it is natural that the Ginzburg-Landau equa- tions be solved under a Neumann boundary condition (See [4]). But one can consider boundary value problems with the boundary data simply imposed on the gauge potential itself. In other words we can consider directly the influence of the external gauge potential instead of the external magnetic field to the superconductor. Such a problem has arisen in [7] and some existence results have been established. In this paper we shall consider the Ginzburg-Landau equations constraint to Dirichlet boundary condition.

The Ginzburg-Landau equations are the Euler-Lagrange equations of the following energy functional

(4) E€(u, A) = ~ ~^{IDAUl2+IFAI2+} 2~2^(1-luI2)2.

Here FA = a1A2 -~Al is the magnetic field. The functionalE€(u,A) has an important property; it is gauge invariant. This means that

(u,A) ---t(ueix,A+ 'VX).

for any smooth functions X : n ^---t R. In Neumann boundary value·

problem, it is possible to fix the gauge as divA = 0 (see [4]). In that context, we shall consider the problem (P) under the condition divA = O.

For simplicity of analysis one may consider the case that the gauge field vanishes identically, i.e., A =O. In this simpler form the equations (P) are reduced to

(5) { ~u+ _~(1 -luI²)u = 0 u=g

and the functional E€(u, A) is reduced to

E€(u, A) = ~

L

The equation (5) has many interesting properties and has been widely studied in recent years (See [6]). One of remarkable results was ob- tained by Bethuel, Brezis and Helein [1,2], a very detailed asymptotic characterization in the limit ^{€ - 4}°for the solutions to (5) minimizing E€(u,A). The Brouwer degreea~deg(9,00) plays a cruCial role iuthe asymptotic analysis for the solutionsu€ corresponding to (5k

When d = 0, u€ converges to a harmonic map which minimizes

over the space H~(O) = {u E W¹,2(0,Sl) : u = 9 on aO}. When d=1= 0, the analysis is more complicated because H~(O)= 0. If0 is a starshaped regionu€ converges the canonical mapu*defined as follows;

there are exactlydpoints ab ... ,adsuch that

is a harmonic map. The location ofaI, ... ,ad can be shown to be the set which minimizes the renormallzed energy (See [2] for the definition of the renormalized energy).

In this paper we are concerned with the solutions (u€, A€) of (P) which minimize the functional E€ (^U,A) in an appropriate function space and consider a generalization of the works [1], the case d = 0, when the gauge field is nontrivial. In section 2 we state our main results; (u€, A€) converges to some(u*, A*) in various norms. The ex- istence and uniqueness of(u*, A*) is proved. In section 3 we show that it converges in C1,0(0). In section 4 we show convergences in higher derivatives and compute convergence speeds.

REMARK. After finishing his work, the author found out that the asymptotic analysis of solutions was established in more general case degg=1= °with Neumann type boundary conditions by F. Bethuel and T. Riviere [3].

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2. Statements of the main Theorem

We define function spaces by

H ={u E W¹,2(n,C):ulan ⁼ g},

V = ^{{A E W1},2(n,R2) : Alan =^A_{o ,}^divA=^D},

Ho= W5,2(n,C), Vo= {A E W5,2(n,R²) : divA=D}, X =H x V, Xo =^Ho x VO.

In what follows we shall often write Wk,p(n) instead of Wk,p(n,C) or Wk,p(n,R²⁾ if there is no risk of confusion. We observe that for BEVo,

II^Blltro =

L

L

DEFINITION 1. We say that (u,A) is a weaksolution to (P) ifv = u - UoE Ho andB = A - Ao E VO ^satisfy

f ^DAo+B(UO+v)DAo+BW+ 1₂(Iuo+vl² -l)(uo+v)w = 0

in ^E

L

for all (w,K) E Xo.

THEOREM 2. Thereexists a minimizer(u€,A€) ofE€ overX which is a weaksolution to (P) in X. Moreover, this solution is smooth up to boundary.

Proof. Existence of a weak solution was shown in [7]. We rewrite (1) and (2) as

~u^{= 2iA .V'u}+IAI²u - 1₂(1 - lul²)u

E

Z - -

~A= 2(UD,4U - uDAu)

Since Wl,2(n, C) c LP(n,C) and W¹,2(n, R 2) c LP(n,R 2) for all P2: 1, standard elliptic arguments show that u andA are smooth. D

Let HI = {uEH: lul = I}. Since deg(g,an) ^{= 0,HI} is nonempty.

We consider the minimization problem of the following functional

over the spaceHI- Itis well known (see [1]) that J has aminimizer in HI which is the unique smooth solution of

(8)

{

6.u+ulV'ul² = 0

lul =¹

u=g

inn inn

on n.

Next we consider the minimization problem of the following functional

over the space Xl = HI X V.

THEOREM3. The variational equations of the functional I (u, A) are given by

(9) (10)

6.u+ulV'ul^{2 =} 0

f i ----

cur A+"2(uDAu - uDAu) = 0 u=g, A=Ao

inn

inn

onan.

These equations admit a unique smooth solution (u*, A*) which mini- mizes I (u, A) over Xl and u* isthe unique solution of (8).

Proof Let (u, A) be a critical point of the functional lover Xl.

For given v E Ho and B E Vo, if we let w(t) = (u +tv)/Iu + tvl, then for all sufficiently small Itl, w(t) E HI so that w(O) = u and w'(O) =(v - u²v)/2._ Differentiating lul² =1, we obtain

(11) { ^uV'u_21V'u1⁺²₊^u"Vuu6.u⁼+⁰ u6.u= O.

Leta(t) = I(w(t), A +tB). Then

2a'(0) = ~Re[V(v - u 2v)V U+iUV(v - u2v). A+i(v - u 2v)Vu. A + 2iuVu . B)+2A . B +_2FAFB

= ~ Re[-~u(v ^{- u 2v)+}2iVu· A(v - u 2v))

+2B· [Re(iuVu)+A - ~A)

= ( !v(-~u+u 2~u - 2iu2Vu . A - 2iVu· A)

in ²

1 .

+"2v(-~u+u2_~u+2iu2Vu . A +2iVu . A)

+2B· [Re(iUVu)+A - ~A]

=0.

Hence the Euler-Lagrange equations are {

_~u+u2~u+2iu2Vu·A+ 2iVu· A = O.

-~A+A +Re(iuVu) = o.

By means of (11) and the fact that lul = 1, these equations are equiva- lent to (9) and (10). Since (9)isequal to (8), (9) has a unique solution.

Since (10) is linearin A,it has a unique solution.

Next we show the existence of minimizer of I(u, A) over Xl following [7). Let (un, An) be a minimizing sequence of lover Xl, and let us write Un = Uo +Vn and An ⁼ Ao+B n for (vn, Bn ) E Xo. Then

2I(Un,An)2: ~^IFAo+FBnl22: ~llFBnl2 ^-

L

Thus _{IIBnll~o ~} 4I(un , An)+2I1AolI~. Since luo+vnl = 1, 2I(un, An)

2:

L

= ~^{IVuo +Vvnl2 +2Re[i(uo+vn)V(uo+Vn) . (Ao+B n)]}

+IAo+B n^l2

1uo+vn^l2

2:

L

L

Hence (V_{n ,}Bn ) is uniformly bounded.inXoso that passing to a subse- quence, if necessary, we see that there is (v*, B*) E Xo

{

:

for all I

Let (u*, A*) = (uo+V*, A o+^B*). On the other hand, we observe that 2I(un ,An) =

k

where

A(v, B) =ll'Vu^ol2 +2'Vuo . 'Vv

+2Re[i(uo+^v)'V(uo+^{v) . (Ao}+B)]

+IAo+BI21uo+vI²+/FAD1²+2FAo ' FB.

Since_{A(vn, B n)}^--+ A(v*, B*), from the weakly lower semicontinuity of norms

r ^{12 2}

2I(un, An) ~Jf}^l'Vv* + ^{/FB.I )}+A(v*,B*) = 2I(v*,B*), which implies that (u*,A*) is a minimizer of lover Xl' 0

We are now in a position to state our main results. We establish

THOEREM 4. Let _(ue,Ae) beany'solutions of(l) and (2) which are minimizers ofEe(u, A) over the spaceX. Then we have

(i) and

for all 0< ^Q:< 1.

(ii) We have

lIue - u.llu"'co) :::; CE² and IIAe - A.IILooco) :::; CE³/2 .

(iii) We have

Ilu^{e -} u*ilctocCo) :::; CE² and IIA^{e -} A.llc1kocCO) :::; CE² for every nonnegative integer k.

Proof Theorems 11, 12 and 14. o

We will prove this Theorem in the next two sections following [lJ.

3. Convergence in C¹,a(f2)

From now on let us denote by (u€,AeJ the solutions to (P) which are minimizers ofE€(u,A) over the space X.

PROPOSITION 5. (u€,A€) ---+ (u*,A*) strongly in X.

Proof. Letv€ =u€ - Uo and B€ =A€ - Ao. ^Since (u*,A*) EX,

Asinthe proof of Theorem 3, we have

and { 1 22 1 ( 4 1

C* ;::: 1n2€2 (l-lu€l) ;::: 4€21nlu€1 - 2€21f21.

Thus (B€) is uniformly bounded inVo and (u€) is uniformly bounded inL4(f2,C). On the other hand,

C* ;:::kIDAO+B.(UO+v€)1²

= kl'VUO+'Vv€1² +2Re[iU€'V(uo +v€) . (Aa+B€)]

+IAo+B€1²Iu€1²

;::: k ~1'Vv€12- C kl'Vuol2+lu€1⁴+IAol⁴+IB€1⁴.

Hence(v€) isuniformly boundedinHo. Now there exists a subsequence (u€n' A€J = (v€n +uo, B€n +Aa) and (iL, A) E X so that

{

(u€n,A€n) ~ (fi,A) weaklyin X

(u€n' A€J^---+ (fi,A) strongly inLP for all 1:Sp<00.

Since

we get lul = 1and hence (u,A) E Xl. We also find as in the proof of Theorem 3that

liD

which implies that(iL,.A)is a rnininriierofloverXl. By theuniquenes~

of minimizer we have(u,A) = (u*,A*).

We observe by (12) that

(13)

klDAEn Uen - DA.U*1²+IFAEn - FA.l²

=kiD AEn Uen¹²+IDA. U*¹²+IFAEJ 2+IFA.l² - 2Re(DAEn Uen . DA. U*) - 2FAEn FA.

~

2l

Since(uen , Aen )converges weakly to(u*, A*) inX, the right hand side of (13) goes to zero. Thus (uen , AeJ converges strongly to (u*, A*) in X. The convergence of whole sequence follows from the uniqueness of

(u*, A*). 0

LEMMA 6. (i) luel ~ ¹ inn .

(ii) (A_{e )} is uniformly bounded in W2,2(n,R2 ) .

(iii) IIVueIlLoo(f2) ~Go/E , Co is independent of^E.

(iv) IU_eI^{-+ 1} uniformly onn.

Proof. (i) Ifwe letWe = (1 -lue^I2 )/E2

, then using (1) we find that

Then the maximum principle implies that We ~ O.

(ii) Since .6.Ae E L²(n) by (7) and (12), standard elliptic estimates give

IIAeIlW 2,2(f2) ~ C(n)(IIAe Il L 2(f2) +II.6.Ae ll£2(f2) +IIA^{ollw 2,2(f2)}

~ G(n, C*,IIAollw2,2(f2)'

(iii) Let

whereq€ is a solution

Then by (6) v€ satisfies

tlq€ =0 in0 q€ =g on 80.

tlv€ = 2iA€ . \7u€+ IA€1²u€ - 2(1-lu€11 ²)u€=h€ in0

€

v€ = 0on 8n.

Since lu€j ::;1 and(A€) is uniformly boundedinLOO(n,R2)by Sobolev imbedding, we observe that

and

It follows from (i), (ii) and Lemma A.2 that

and thus

This achieves the desired estimate

(iv) We first show that lu€1 ^-+ 1 uniformly on every compact subset of O. Let K cc O. Assume the contrary. Then there would be a

sequenceEn _~0 andXn E K so that 1-!u€n(xn)1² ~ ^Q for some fixed

Q >O. Since(ufn,Afn )~ (u.,A.) inX, by (12)

Let ~n = QEn/4Co < dist(K,oO) for all sufficiently small En. Here Co is the constant in (iii). IfIx - xnl ~ ~n, then by (iii)

Now

a contradiction.

Since0 is smooth, there is a constant C depending only on 0 such that

for all x E 0 and for all sufficiently small ~ > O. Using this fact, we can show that for any x E on, ^IUfI^{- ?} 1 uniformly on 0.nB(x,~) for some ~> 0 as above, and the proof is completed. 0

In the rest of this paper we assume by Lemma 6 that lu€1 ~ 1/2.

LEMMA 7. Let!; = Lj,k l(uf)xjXkI^2• Then

whereC is independent ofE.

Proof. For simplicity, we drop the subscriptE. By direct calculation,

1

Ux₃6ux "_, =UX"₃[2iA .Vu+^{IAI2u - -2}_E ^{(1 - lul2)u}}X,"

=2iux j(VAj .Vu) +^2iAj (Vux j •Vu) + 2uAj (VAj .Vu) +IAI²1Vul²

1 1

+ 2(IVu121u12 +u2Vu· Vu) - 2(1-luI2)IVuI2.

E €

Hence by (1)

6(IVuI2)=2/2+ 2Re[2iux j(VAj •Vu)

+ 2iAj(Vux j •Vu) +^2uAj (VAj . Vu)]

1 2D²u

+21A121Vu12+ 2luVu+uVul2+ _A_IVuI2.

E U

Since 1/2 ::; lul ::; 1 and A ^E Loo(0, R²) by Sobolev embedding theo- rem, we have

2f2 ::; -2Re[2iux j(VAj . Vu) +2iAj (Vux j •Vu) +2uAj(VAj .Vu)}

+ 6(jVuj2) + 21D~u11Vu12

lul

::; 6(IVuj2) + C( IVAIIVul²+ flAllVul+ lulIAI IVAIIVul ) + C( IAllVul + IAI²1ul+ 16ul ) IVul2

::; f2 + 6(IVuI2) +C(l+ IVAI² + \VuI^{4) •}

This gives the desired inequality. o

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PROPOSITION 8. Ue is uniformly boundedin Wl~~(O,C).

Proof. Given 8>0, there is a number R > 0 so that

h(X,R)nnlV1J.,!2 < ~

for eachx E o. Sinceu€ ---7 u* in H, for all sufficiently small ^€

for each x E O. Let K cc O. Fix a point x E K and choose r <

min{R, dist(K,aO)}. Let ( be a smooth function with support in B(x,r) satisfying 0::; (::; 1 and (=1 on B(x,r/2). Furthermore, we may assume that for each h E £1(0)

I In h1V(12

1 ' I Inh(t1()^I < C^r In Ihl·

Multiplying (15) by (2, we obtain

1n{;(2::;Cr(1+InIVueI2t1(2+lIVA€12(2+ InIVu€1⁴⁽²⁾

(17) ::;C^r(1+ In 21Vu€^12(t1( +

In

::;Cr (l+ ^{Inlvu€14(2).}

SinceW1,1(0) isembedded in£2(0),we see that for each hEW1,1(0)

Let h = IVu€12(. Then

In IVu€1⁴(2 ::; Cr(1n IVu€12( + IVu€12IV(1 +IVu€lf€()2

::; C^{r (1}+ In IVu€lf€cf·

Consequently, by (16), (17), (18) and Holder inequality we obtain

In^{;(2 ~ ^Cr(l ^+(j

In

Therefore if(j is sufficiently small, we conclude that

[ /: ~ ^{[ /;(2} ~Cr.

i^B(xQ,r/2) in

Henceu€ is uniformly bounded inWi~;(n,C).

LEMMA 9. For all€> 0,

o

[ 18u€/2 _~ C.

ian 8v

Here C is independent of€ and v is the outward unit normal vector field onan.

Proof. LetU=(U_{1 ,}U2 ) be a smooth vector field onn^{so thatU=1/}

onan. We note that by the uniform boundedness ofu€ inH, 2Re

l

2l

l

=

2l

On the other hand, by (6), (12) and the fact that lu€/ = 1 onan

2Re

l

= 2ReIn2iA€ . V'u€(U . V'u€)+IA€1 2u€(U . V'u€)

1 2

- 2u€(1 -lu€1 )(U· V'u€)

€

= 0(1) - 1₂ [(1 - lu€/2)(u€ \7u€ . U+u€V'u€ . U)

€ in

=0(1) +2~2 InU . \7(1 -lu€1²)2

= 0(1) - - ; [(1 -lu€12)2divU 2€ in

=0(1).

Hence

in

Ifwe set ^T =vl.. = (-V2,V1), the tan~entvector to an,then IVu€1² =

1~12+_{l~j2 on}art sinceu€ = 9 onon, wecondudethat

knl~::12⁼knl~12+0(1)=O(1).

PROPOSITION 10. u€ is uniformly boundedinW2,2(n,C).

o

mU mU

on {X2 = O}nau,

Proof. In view of Proposition 8, it is enough to compute uniform estimates ofu€ in W²,2(n,C) near the boundary ofn.

Let ^XQ E 00. From the smoothness of an, changing coordinates if necessary, we can find a smooth function h : R ^---t R with h(O) =

XQ such that for some r > 0, nn B(xo,r) = {x ^E B(xo,r)lx2 >

h(xl)}. Furthermore, there is r' > 0 so that the image ofU = {y ^E B(O, r')IY2 >O} liesinnnB(xQ, r) diffeomorphically under the corre- spondence (XI,X2) ^f-7 (Xl,X2 - h(xl)) = (yI,Y2) ^E U.

Let ii.€ (YI, Y2) = Ufo(YI, Y2 + h(Y1)) and rL(YI, Y2) = A€ (Yb Y2 +

h(Yl)). It is easy to check that if (u€,A€) is a weak solution of (P), then (ii.o A€) is a weak solution of .

(19)

{

L~€^: 2iib~ ~ii.€ ~ IA€~2~€ ^- _~(1 -1ii.(12)ii.^fo LA€ - 2(u€DA.u€ - u€DA.u€)

(ii.€,A€) = (ii.Q,Ao) where

L ='"LJi,j 8Yj aij 8Yi '8 ( 8 ) all = 1 , al2 =a2l = - h',a22 =1+h'2,

b= _(bI,~), bl =AI, ~ =A2 - ^h'Ab

iJ^{A. =} V - iA€,

V = (ayl , ay2 ), V= (ay~,(1- h')8Y2).

Let1; ⁼ 2:_j^,k!(ii.€)yjYk\2. We want to get an inequality of type (15) for

!? Since the right hand side of (19-2) belongs toW²,2(U,R^2), byellip- tic estimates we have IIAEIlw2,2(U,R2) ~ C and hence IIA€ 11Loo(U,R2) ~

C. For simplicity we drop the subscriptEand write u instead. ofu^and

so on. By direct calculations, we see that

Differentiating the first equation of (19), we find that (21)

2Re[uyk L(uyk )]

= 2Re[-uyk((aij)YkuY.)Yi +uyk (2ib· "Vu+^IAI2u - 2(1-lu I1 2)u )Yk]

E

= -2Re[uyk ((aij)Yk uY.)Yj]+2Re[2i(bj)YkUYjUYk +2ibjuYiYkUYk]

+21A121"Vu12+4Re[uAj"VAj · "Vu] +^21u"VU1 +^u"Vul2

E

2 .

+-(Lu - 2ib· "Vu -IAI2u)I"VuI2.

u

Since IIAliL=(o) ~C and ILul ~ CC!+l"Vu!), by (20) and (21) L(lVuI2)2: 20f2 - CI"Vul(I"Vul+J) - C(IVbll"VuI2+flbll"Vu!)

- ClulIAII"VAII"Vul- C(lbll"Vul+ IAI21ul+ ILu!)lV'uI2 2: Of2 - C(l+I"VAI2+l"VuI⁴).

Here eis the ellipticity constant ofL. Thus

Choosing a test function (" with support inB(O,r') satisfying 0~ ("~ 1 and ("=1 on B(O,r'/2), we are led to

Ifthe last term of the righthand side is uniformly bounded, then

Using the same arguments used in the proof of Proposition 8, we con- clude that

whichprove::: the propositiun.

Itremains to show that (22)

First, we observe by integration by parts that

r^{L(I'VuI2)(2 =} r^{l'Vul 2L(2+2} r a121'VuI 2((2)Yl

Ju Ju J{Y2=O}

+ r ^{(a12)Yl!'VuI2(2+ [ a221'VuI2((2)Y2}

J{Y2=O} J{Y2=O}

- [ a22(I'VuI2 )Y2(2.

J{Y2=O}

Invoking Proposition 5 and Lemma 9, we see that all the integrals on RRS are uniformly bounded except for the last term. On the other hand, it follows from integration by parts that

r a22(I'VuI 2)Y2(2

J{Y2=O}

=2Re r a22uYluYlY2(2+a22u Y2 u Y2Y2(2 J{Y2=O}

=2Re [ - [ (a22(2)Yl BYl u Y2 - [ a22BYlYl u y2 (2

J{Y2=O} hY2=O}

+ r a22u Y2 Uy2y2 (2 ] . J{Y2=O}

The first two terms are uniformly bounded by Lemma 9. To estimate the last term, we note that on {Y2 =O}, by (19)

a22 u Y2Y2 = - (anUyJYl - (a21 u Y2)Yl

- (a12UyJY2 - (a22)Y2uY2 +2ib·'Vu+ JAI²u.

Now using Lemma 9 and integrating by parts, we find that Re

1

. {Y2=O}

= Re

1

- (a22)Y2/UY212(2 +2i(b·V'U)U_{y2 (2} +IA1²uuy2 (2

= 0(1) - Re

1

=0(1) - 2Re

1

{Y2=O}

= 0(1)+

1

= 0(1).

Thus (22) is proved. o

THEOREM 11. Let (uE,AE) ^{be any} solutions of (1) and (2) which are minimizers ofEE(U,A) over the space X. Then we have

and

forall 0 < a < l.

Proof We note that by Proposition 10, (V'uE ) is uniformly bounded in LP(n,C2) for all p > 2 and so is (~AE) inLP(n,R2) by (7). This verifies the assertion for AE • Let

Then, since lu€1 2::1/2, (14) reads (23)

Multiplying (23) byW~-1 and using Holder inequality, we find that

Inw~::; ²Inw~-1IDA<U(12::; Cpllw(II~;;)o)"

Consequentiy, by (6) we conclude that (u() is uniformly bounded in W²,p(n,C) for allp > 2, which completes the proof. 0

4. Convergences in higher derivatives

Since 9 : an ^{-> S1} is of degree 0, there is a smooth function <Po :

an ^{-> R so that9 = ei<Po.} We also denote by<Poits harmonic extension inn. It is easily seen (BBH1) that u* =^ei<po.

THEOREM 12. Wehave (i) Ilu( - u*IILoo(o) ::;C€2, (ii) IIA( - A*IILoo(o) ::; CE³/2

•

Proof. We investigate that by the maximum principle (see Theorem 3.7 [5]), (23) yields

IIw(!ILoo ::;C.

Hence if we let ^u( = p(ei<p< withp( = lu(I(this is well defined because lu(I2:: ! ), ^then

(24)

Take the imaginary part of(6) to obtain (25)

(26)

Multiplying p( on both sides of (25) and using the fact that Ll<po =0 and divA( =^0, we find that

{

-Ll(<p( - <Po) = div((p~ - 1)(\7<p( - A()) in n

<Pt - <Po =⁰ on an.

Now elliptic estimates (see Chapter 8 [5]) applied to (26) yield (27) 1I<p( - <poIIL=CO) :::; CII(p~ - 1)(\7<p( - A()IILoc(o)

:::; C1lp~ - 1I1L=(0) :::; CE²•

Hence

lu€ - u*I ~ ¹(P€ - 1)eicp<1 +leicp<- eicpoI

~ Ip€ -11+I'P€ - 'Pol _~ C€2, and (i) is proved.

To prove (ii) we now apply Lemma A.2 to (6) and (9) to obtain IIVu€ - Vu*1I1oo(Q) ~ Cllu€ - u*IILOO(Q) ~ C€2.

Since we have

(2~ .

IVP€! _~ IVu€ - Vu*1 +I(P€ - l)V'P€!+I'P€ - 'Po!IV'P€!+lV'P€- Vcpo!

~ C€+C€2+IV'P€ - V'Pol.

Next, we rewrite (25) as

2 2

(29) ~('P€ - 'Po) = --Vp€· V'P€+-A€· Vp€.

P€ P€

Since 'P€ - 'Po = 0 on ao, by Lemma A.2, (27) and (28) we have IIV'P€ - V'Pollloo(Q) ~ CIIVP€11Loo(Q) II'P€ - 'PoIILoo(Q)

~ C€2(C€+IIV'P€ - V'PoIILOO(Q».

Thus we obtain

(30) IIV'P€ - V'POHLOO(Q) ~ Cf.^3/2• On the other hand, equations (7) and (10) read

{ ~A€ ⁼ p;A€+Re._(iu€Vu€)

(31) ~A* =A* +Re(w*Vu*).

Since by (24) and (30)

1Re(iu€Vu€) - Re(iu*Vu*)1 ~ ^{I(p; -} l)V'P€1+IV'P€ - V'Pol ~^C€3/2, it follows from Lemma A.3 that

o

LEMMA 13. We have (i) IIV<PeIlCzkocCO) ~ C, (ii) IIAellC_Zoc^k CO) ~ C, ,..., 111 - Pe 11 .~ ,....

(ill) ii~^iic{'occn) ~ ^L-,-

Proof We use inductions on k. The case k = 0 follows from Theo- rem 11 and (24). We take real and imaginary parts of (6) and (7) to obtain

(32) (33) (34)

2 2 1 2

f::,.Pe = pe IV<PeI - 2peA .V<Pe +IAcl Pe - 2"(1 - Pe)Pe

€

2 2

f::,.<Pe = - - V pe . V <Pe+ - Ae . V Pe

Pe Pe

f::,.Ae = p;Ae - p;V<Pe.

It comes from (32) that (f::,.Pe) is uniformly bounded in Cl~c(n) and hence

11PeIlwk+2,PCO) :SCp Vp< 00

Zoc and

(35)

Similar arguments applied to (33) deduce

!I<Pe!lwk+2,Pcn) _~ Cp and !IAeIl_wk+2,Pcn) _~ Cp Vp < 00.

loc loc

Therefore (36)

Now (f::,.<Pc) is uniformly bounded in Wl:~l,p(n)so that

In particular, (37)