Quantitative, uniqueness, and vortex degree estimates for solutions of the Ginzburg-Landau equation

Size: px

Start display at page:

Download "Quantitative, uniqueness, and vortex degree estimates for solutions of the Ginzburg-Landau equation"

Oswin Robbins
5 years ago
Views:

1 Electonic Jounal of Diffeential Equations, Vol ), No. 61, pp ISSN: URL: o ftp ejde.math.swt.edu ftp ejde.math.unt.edu login: ftp) Quantitative, uniqueness, and votex degee estimates fo solutions of the Ginzbug-Landau equation Igo Kukavica Abstact In this pape, we povide a shap uppe bound fo the maximal ode of vanishing fo non-minimizing solutions of the Ginzbug-Landau equation u = 1 ɛ 2 1 u 2 )u which impoves ou pevious esult [12]. An application of this esult is a shap uppe bound fo the degee of any votex. We teat Diichlet homogeneous and non-homogeneous) as well as Neumann bounday conditions. 1 Intoduction In this pape, we povide votex degee estimates fo solutions of the Ginzbug- Landau equation u = 1 1 u 2 ) ɛ 2 u. The votices of solutions of this equation wee studied by Bethuel, Bezis, and Hélein in [5]. We ecall that x 0 is a votex if it is an isolated zeo of u, and if the degee of u at x 0 is nonzeo.) They pescibed nonhomogeneous bounday conditions u = g with g: S 1 such that deg g = d>0. If is convex, and if ɛ is sufficiently small, they poved that a minimizing solution u has pecisely d distinct votices of degee 1. This esult has been extended to include all bounded smooth domains by Stuwe [17]. It was futhe shown in [5] that thee exist non-minimizing solutions of the Ginzbug-Landau equation whose votex at the oigin is an abitaily pescibed nonzeo intege. In this pape, we find a shap uppe bound in tems of 1/ɛ fo the degee of votices fo solutions which ae not necessaily minimizing. Chanillo and Kiessling poved in [7, Lemma 6] that if x 0 is a votex of degee d N, then the vanishing ode of u is at least d. Theefoe, we may use unique continuation Mathematics Subject Classifications: 35B05, 35J25, 35J60, 35J65, 35Q35. Key wods: Unique continuation, votices, Ginzbug-Landau equation. c 2000 Southwest Texas State Univesity. Submitted June 23, Published Ocotbe 2,

2 2 Quantitative uniqueness and votex degee EJDE 2000/61 methods to addess this poblem. Using the esult of Chanillo and Kiessling, the pape [12] implies that in the homogeneous Diichlet and peiodic cases the degee of u at any votex x 0 is less than Cɛ 2 whee C depends only on. In Theoem 3.1 below we impove this bound to Cɛ 1 and then show that this bound is best possible. The main tool in the poof is a new logaithmically convex quantity fo the Laplacian opeato. Moe pecisely, fo any α> 1and any hamonic function u, thequantity H)= ux) 2 2 x 2) α B 0) is logaithmically convex, i.e., log H) is a convex function of log. Due to cancellations of tems involving α in 2.10) 2.12) below, and due to a gadient stuctue of the Ginzbug-Landau equation, we can choose an appopiate optimal α which gives ou esult. Inspied by an example in [5], we constuct in Remak 3.3 a solution which shows that ou bound Cɛ 1 can not be impoved upon. Theoem 3.2 contains an estimate concening the bounday condition u = g whee g = 1, while Theoem 3.5 coves the Neumann case. Fo popeties of stationay Ginzbug-Landau equation, cf. [5, 8, 15, 16, 17] and to [1, 2, 3, 4, 9, 10, 12, 13] fo vaious esults on logaithmic convexity and unique continuation. 2 Quantitative uniqueness fo systems In this section, we conside nontivial solutions u of the system u = F u 2) u u =0 2.1) whee u C 2, R D ) C, R D )withd N. We assume that R d,whee d 2, and one of the following: a) is a convex bounded domain; b) is a Dini domain; Dini domains ae bounded domains with the following popety: Aound any point thee is a neighbohood N, such that afte a otation of coodinates N lies below a gaph of a function whose nomal is Dini continuous see [14] fo details); c) is a peiodic cube [0,L] d ; in this case, =. As in [12], we ae mainly inteested in peiodic bounday conditions; the papes [1] and [14] enable us to conside homogeneous Diichlet conditions without much change. As fa as the Ginzbug-Landau equation is concened, the nonhomogeneous bounday conditions u = g with g = 1) and homogeneous Neumann conditions du/dν) = 0 ae moe physically elevant and moe widely studied. Theoem 3.1 addesses the homogeneous Diichlet bounday

3 EJDE 2000/61 Igo Kukavica 3 conditions; the non-homogeneous bounday conditions ae consideed in Theoem 3.2, while Theoem 3.5 coves the Neumann case. Let M =max u 2.On F:[0,M] R, we make the following assumptions: i) F C 1 [0,M] ) and F x) λ, x [0,M] 2.2) fo some λ>0; ii) F 0) = 0; iii) F is convex on [0,M]. Conditions ii) and iii) imply F x) xf x), x [0,M] 2.3) and F x) λx, x [0,M]. 2.4) The following is the main esult of this section. We ecall that the ode of vanishing at x 0 is defined as the lagest intege n N 0 = {0, 1,...} such that 1 B x 0 ) B x 0) u 2 = O 2n ), as 0. Hee and in each subsequent occuence, one needs to eplace B x 0 ) with B x 0 ) in the case of peiodic bounday conditions c).) In paticula, if u does not vanish at x 0, then the ode of vanishing is 0. We also add that u may not have any zeos in. Theoem 2.1 Let x 0. The ode of vanishing of u at x 0 is less than C λ +1) whee C is a constant depending only on. If λ is sufficiently small, and if satisfies a) o b), then thee ae no nontivial solutions of 2.1). In these cases, the bound C λ +1) may be eplaced by C λ. Let x 0 andr>0 be such that B R x 0 ) is stashaped with espect to x 0. Fo an abitay α> 1and>0, denote H x0 ) = ux) 2 2 x x 0 2) α B x 0) whee u 2 = u j u j. We will omit the dependency on x 0 when it is clea fom the context. Lemma 2.2 Let q 1, andlet0< 1 < 2 be such that q 2 R. Then log H x 0 q 1 ) H x0 1 ) log H x 0 q 2 ) H x0 2 ) + q2 1)2 2dλ.

4 4 Quantitative uniqueness and votex degee EJDE 2000/61 PoofofLemma2.2 Without loss of geneality, x 0 = 0. Let 0,R)be abitay. Denoting B = B 0), we get H ) = 2α u 2 2 x 2) α 1 = 2α H) 1 x j u 2 j 2 x 2) α) 2.5) whence, by the divegence theoem, H ) = 2α+d 1 H)+ I) 2.6) α+1) whee I) =2α+1) x j u k j u k 2 x 2) α. 2.7) By the divegence theoem, I) = u k j u k j 2 x 2) α+1) B = j u k j u k 2 x 2) α+1 B + u 2 F u 2 ) 2 x 2) α+1. As in 2.5) above, we get I ) = 2) j u k j u k 2 x 2) α+1 1 x m j u k j u k m 2 x 2) α+1) +2) u 2 F u 2 ) 2 x 2) α. Using the divegence theoem on the second integal leads to I ) = 2)+d j u k j u k 2 x 2) α x m j u k jm u k 2 x 2) α+1 1 x m j u k j u k 2 x 2) α+1 νm dσx) B +2) u 2 F u 2 ) 2 x 2) α

5 EJDE 2000/61 Igo Kukavica 5 whee ν =ν 1,...,ν d ) denotes the outwad unit nomal and whee dσx) stands fo the suface measue on. Applying the divegence theoem in the second integal, this time with espect to the vaiable x j,weget I ) = 2α+d j u k j u k 2 x 2) α+1 2 x m m u k F u 2) u k 2 x 2) α+1 4) + x m m u k x j j u k 2 x 2) α + 2 x m m u k j u k 2 x 2) α+1 νj dσx) B 1 x m j u k j u k 2 x 2) α+1 νm dσx) B +2) u 2 F u 2) 2 x 2) α. 2.8) The second tem on the ight hand side of 2.8) equals 1 x m m F u 2 )) 2 x 2) α+1 = d F u 2) 2 x 2) α+1 2) F u 2) x 2 2 x 2) α ; since the bounday integal vanishes by F 0) = 0. On the othe hand, the sum of the fouth and the fifth tem is 1 u m u m x k ν k 2 x 2) α+1 dσx) ν ν B which is due to the fact k u = u/ ν)ν k esulting fom u = 0. This integal is nonnegative since B is stashaped with espect to 0. Then I ) 2α + d I) 2α + d u 2 F u 2) 2 x 2) α+1 + d F u 2) 2 x 2) α+1 2) F u 2) x 2 2 x 2) α B 4) + x m m u k x j j u k 2 x 2) α B +2) u 2 F u 2) 2 x 2) α. 2.9)

6 6 Quantitative uniqueness and votex degee EJDE 2000/61 The sum of the second, the thid, the fouth, and the sixth tem is 1 Ex) 2 x 2) α whee Ex) = 2α + d) u 2 F u 2) 2 x 2) +df u 2) 2 x 2) 2α +2)F u 2) x 2 +2α+2)F u 2) u ) We get Ex) = 2 2α u 2 F u 2) d u 2 F u 2) +df u 2) fom whee, using iii), +2αF u 2) u 2 +2F u 2) u 2) + x 2 2αF u 2) u 2 + df u 2) u 2 df u 2) 2αF u 2) 2F u 2)) 2.11) Ex) 2 2 d) u 2 F u 2) + df u 2)) 2 x 2 F u 2). 2.12) By 2.2) and 2.4) and using x 2 2,weget which leads to I ) 2α + d 4) I)+ Now, let N) =I)/H). Then N 4) ) 4dλ + H) 2 Ex) 4dλ 2 u 2 x m m u k x j j u k 2 x 2) α 4dλH). x m m u k x j j u k 2 x 2) α u 2 2 x 2) α ) x j u k j u k 2 x 2) α ) 2 whence, by the Cauchy-Schwaz inequality, N ) 4dλ, i.e., N 2 ) N 1 ) 2dλ 2 1 ) ), 0 < 1 2 R. 2.13) Let q 1, and let 0 < 1 2 Rbe such that q 2 R. Dividing 2.6) by H) and integating the esulting equality between 1 and q 1 leads to log Hq 1) H 1 ) = 2α+d)logq+ 1 α+1 = 2α+d)logq+ 1 α+1 q1 1 q2 Nρ) ρ dρ N 1 ρ/ 2 ) 2 ρ dρ.

7 EJDE 2000/61 Igo Kukavica 7 Using 2.13), we get log Hq 1) H 1 ) 2α + d)logq+2 2 1)dλ 2 1 )q2 = log Hq 2) H 2 ) )q2 1)dλ and the lemma follows. Again, let x 0 befixed,anddenote h x0 )= B x 0) ux) q2 Nρ) dρ α+1 2 ρ Let R>0 be such that B R x 0 ) is stashaped with espect to x 0. Lemma 2.3 Let α 0, 0 < 1 <4 2 /3and 4 2 R. Then, with the above assumptions, log h x ) h x0 1 ) log h x ) h x0 2 ) + C α + d2 2 λ ) whee C is a univesal constant. PoofofLemma2.3 Denote h) =h x0 )andh)=h x0 ). If 0 <<ρ, then H) 2α h) 2.14) and h) Hρ) ρ 2 2 ) α 2.15) Theefoe, log h2 1) h 1 ) log H3 1) H 1 ) α log 5 log H3 1) H 1 ) since α 0. By Lemma 2.2, log h2 1) h 1 ) log H4 2) H4 2 /3) + C2 2 dλ. Using 2.14) and 2.15) again, we get ou assetion. In Theoem 3.2 below, we will need an inteio vesion of the above lemma, which we state fo sake of completeness. Lemma 2.4 Let u be a solution of u = F u 2) u,wheef is as above, in B R x 0 ).Letα 0,0< 1 <4 2 /3and 4 2 R Then log h x ) h x0 1 ) log h ) x ) h x0 2 ) + C α + d2 2λ whee h x0 ) = B x 0) ux) 2 and C is a univesal constant.

8 8 Quantitative uniqueness and votex degee EJDE 2000/61 Poof of Lemma 2.4 The poof is the same as that of Lemma 2.3. Next lemma will be used in the ovelapping chain of balls agument. Lemma 2.5 Let α 0. Assume that x 1,x 2 and > 0 ae such that B 20 x 1 ) is stashaped with espect to x 2. IfB x 1 )and B x 2 ) intesect, and if ux) 2 KH x1 ) fo some K 0, then ux) 2 K 3 exp C α + λ )) H x2 ) whee C is a constant which depends only on d and diam). Poof of Lemma 2.5 It is easy to check that H x1 ) H x2 4). Theefoe, u 2 KH x1 ) KH x2 4) 2.16) which, by 2.14), implies H x2 8) 8) 2α h8) 8) 2α u 2 C α KH x2 4) whee C denotes a geneic constant which depends only on d and diam. Lemma 2.2 then implies log H x 2 4) H x2 2) log K + C α + λ ) 2.17) and similaly log H x 2 2) H x2 ) 2logK+C α+ λ ). 2.18) α+1 The inequalities 2.16), 2.17), and 2.18) then give log u 2 H x2 4) 3logK+C α+ λ ) α+1 which gives ou assetion. Poof of Theoem 2.1 In the cases a) and c), we can take R to be abitaily lage. Note that, in the case a), log h x 0 4) h x0 ) =0, x )

9 EJDE 2000/61 Igo Kukavica 9 povided diam. Theefoe, by Lemma 2.3, thee is a numeical constant C such that log h x ) h x0 1 ) C α + d ) diam)2 λ, x 0 fo evey α 0and 1 0, diam ). Choosing α = dλ diam, we get log h x 0 2) h x0 ) C λd diam, x 0 fo 0, diam ), and Theoem 2.1 follows. In the case c), the agument is the same. The only diffeence is that 2.19) is eplaced by log h x 0 4) h x0 ) C, x 0 povided diam, whee C is a constant depending only on d. In this case we theefoe obtain log h x 0 2) h x0 ) C1 + λ diam ), x 0 fo 0, diam ), whee C is a constant which depends only on dimension d. The poof in the case b) involves a standad agument employing ovelapping chain of balls cf. [11, 13]). Below, the symbol C denotes a geneic constant depending only on. Fist, we choose >0an 1,...,x m such that 1) Bx 1,/2),...,Bx m,/2) cove ; 2) fo evey j {1,...,m}, the egion Bx j, 10) is stashaped with espect to x j ; 3) if Bx j, 10) intesects, it is assumed that the vaiation of the nomal ν is sufficiently small cf. [14, p. 444]). We fix α = λ + 1. Thee exists j 0 {1,...,m 0 } such that u 2 1 u 2 m whence B /2 x j0 ) u 2 C α H xj0 ). Fo evey j {1,...,m}, thee exists an ovelapping chain of distinct) balls fom 1) connecting B x j )andb x j0 ). Repeated use of Lemma 2.4 then gives u 2 λ+1 C H xj ), j =1,...,m.

10 10 Quantitative uniqueness and votex degee EJDE 2000/61 Theefoe, H xj 2) λ+1 C H xj ), j =1,...,m. An agument paallel to [14, p. 445] then leads to H x 2ρ) λ+1 C H x ρ) fo evey x and abitay ρ 0,/2). Using 2.14) and 2.15), we get the theoem. 3 The degee of Ginzbug-Landau votices Now, we apply Theoem 2.1 to the Ginzbug-Landau equation u = ) 1 ɛ 1 u 2 u 2 u =0, 3.1) whee u: Cis assumed to be nontivial. The domain R 2 is as in the beginning of Section 2 and ɛ>0. Theoem 3.1 The ode of vanishing of u at x 0 is less than ) 1 C ɛ ) whee C is a constant which depends only on. As it was pointed out in the emak following Theoem 2.1, the above bound 3.2) can be eplaced by C/ɛ if satisfies a) o b). By [7], 3.2) then povides an estimate fo the the degee of u at any votex x 0. Recall that x 0 is a votex if ux 0 ) = 0 and the degee of u at x 0 is nonzeo.) Namely, by [7, Lemma 6] and ou Theoem 3.1, the degee at evey votex is less than 3.2). Poof By the maximum pinciple, we conclude ux) 1fox, i.e., M = 1. Taking F x) = 1 1 ɛ 2x+ 2ɛ 2x2 we easily veify that i) iii) ae satisfied with λ = ɛ 2. Theoem 3.1 then follows fom Theoem 2.1. Next, we pesent a esult concening the nonhomogeneous bounday conditions u = g whee g: S 1 is sufficiently egula, e.g. continuous. We assume that is stashaped. In this case, Bethuel, Bézis, and Hélein poved in [5, Lemma X.1] that 1 u 2 ) 2 C0 ɛ 2 3.3) whee C 0 depends only on g and. Theoem 3.2 The ode of vanishing of u at x 0 is less than C/ɛ whee C depends on, the bounday function g, and the distance fom x 0 to.

11 EJDE 2000/61 Igo Kukavica 11 Poof It is easy to check that if ɛ is sufficiently lage, then u does not vanish. Fo instance, we may use the inequality ux) C/ɛ fom [5] whee C depends on g and.) Let x 0, denote R = distx 0, ) and 0 = R/4. We distinguish two cases. Case 1: ɛ R 2 /C C 0 )wheecis a lage enough numeical constant and C 0 is as in 3.3). In this case, we can use analyticity aguments to show that the ode of vanishing is bounded by a constant depending only on, g, andr cf. [12]). Case 2: ɛ R 2 /C C 0 )wheecis lage enough. Then 3.3) implies u 2 R2 C B R/4 x 0) as can be eadily checked. Since also max u =1,weget u 2 C u 2 B Rx 0) B R/4 x 0) whee C depends on, g,andr. Since R = distx 0, ), we have B R x 0 ) =. Theefoe, by Lemma 2.4, we get log B 2x 0) u 2 B x 0) u 2 C + C α + 2R2 ɛ 2 ) fo all α 0 povided <R/3. Choosing α = R/ɛ, weget log B 2x 0) u 2 B x 0) u 2 C ) R ɛ +1. Note that 1 CR/ɛ due to the fact ɛ R 2 /CC 0. Theefoe, B log 2x 0) u 2 B u 2 x 0) C distx 0, ) ɛ and the statement follows. Remak 3.3 Hee we show by means of an example that Theoem 3.1 is shap. Let = B 1 0). We shall show that thee exists ɛ 0 > 0 with the following popety: Fo evey ɛ 0,ɛ 0 ), thee exists a solution u of 3.1) such that the degee of u at 0 is at least 1/Cɛ. We seek this solution in the fom ux) =f)e idθ,wheex=e iθ,witha suitable fixed intege d. We find f as a global minimize of the functional 1 Φf) = f 2 + d2 f 2 + ) 2ɛ 2 f 2 1) 2 d 0

12 12 Quantitative uniqueness and votex degee EJDE 2000/61 in the space V = { f Hloc 1 0, 1) : } f f, L 2 0, 1),f1) = 0. What emains to be shown is that if d is suitably chosen, then the minimize f is not identically zeo. Choose an abitay g V such that 0 <g)<1fo 0, 1), say g) =1 ). Then if ɛ 0,ɛ 0 ), whee ɛ 0 is sufficiently small, and if d =[1/Cɛ] wheecis lage enough, then Φg) < 1 1 2ɛ 2 d=φ0) 0 and 0 can not be the global minimize. Now, ux) =f)e idθ,wheefand d ae as above, is not identically 0, it satisfies u CB 1 ) C B 1 \{0} ) and solves the Ginzbug-Landau equation fo x 0. But then it can be eadily checked that 0 is a emovable singulaity, and consequently 3.1) holds. It also follows immediately that 0 is an isolated zeo. Indeed, in the opposite case, thee would be a sequence 1, 2,... which conveges to 0 such that ux) =0if x = j fo j N. Theefoe, 0 would be a zeo of infinite ode, which is not possible since u 0. The est of the pape is concened with the Ginzbug-Landau equation u = 1 ɛ 2 1 u 2 ) u 3.4) with homogeneous Neumann bounday conditions du dν =0 3.5) whee is a connected bounded C 3 domain. The teatment is simila to but not completely the same as) the Diichlet case. We need to conside a confomal staightening of the bounday, which, in tun, leads us to conside the following analog of 2.1). Let u = vf u 2) u du dν =0, whee = B + R 0 0) = { x 1,...,x d ) B R0 = B R0 0) : x d > 0 } and = { x1,...,x d ) B R0 : x d =0 }. As in Section 2, we denote M =max u 2 and we make same assumptions on F :[0,M] R as befoe. We assume that v is a nonnegative function such that max x vx) M 0 and max x vx) M1. Lemma 3.4 Let α>0,0< 1 <4 2 /3,and4 2 R 0. Then log h2 1) h 1 ) log h4 ) 2) h 2 ) + C α + d2 2λ whee C depends only on M 0 and M 1, and whee h) = B ux) 2. +

13 EJDE 2000/61 Igo Kukavica 13 Poof The poof is simila to that of Lemma 2.3; we only indicate the main steps and point out the main diffeences. As befoe, we let H) = u 2 2 x 2) α. B + Then 2.6) holds with 2.7) whee B =B + ). Afte a shot computation, we get I ) = 2α+d I) 2α+d u 2 vf u 2) 2 x 2) α+1 B + + d vf u 2) 2 x 2) α+1 B + 2) x 2 vf u 2) 2 x 2) α B x m m vf u 2) 2 x 2) α+1 B + 4) + x j j u k x m m u k 2 x 2) α B + +2) u 2 vf u 2) 2 x 2) α B + in place of 2.9). Fom hee, we get I ) 2α + d I) 4dλ max vh) 4) + x j j u k x m m u k 2 x 2) α B + λmax v ) u 2 2 x 2) α+1 B + whee we also used nonnegativity of v. Instead of N ) 4dλ, which we had befoe, we now conclude N ) dλ max v λ2 max v. The est follows as befoe. Now, we etun to the Ginzbug-Landau equation 3.4) with the Neumann bounday conditions 3.5), whee is a C 3 bounded connected domain in R 2. Theoem 3.5 The ode of vanishing of u at x 0 is less than 3.2) whee C is a constant which depends only on. Poof sketch) The poof of the theoem is analogous to the poof os Theoem 3.1. The main diffeence is that we use a confomal map to staighten

14 14 Quantitative uniqueness and votex degee EJDE 2000/61 the bounday. Namely, let x 0. Then, by the Riemann mapping theoem, thee exists R 0 > 0and 0 >0 and a confomal map f: B 0 x 0 ) B + R 0 0) such that fx 0 ) = 0. The equation 3.4) then tansfes to u = 1 ɛ 2 v 1 u 2) u with v =1/ f 2. The bounday of being C 3 guaantees that v and v ae bounded up to the lowe bounday B + R 0 [6]. The est is then established as in the poof of Theoem 3.1, except that we use Lemmas 3.4 and 2.4 instead of Lemma 2.3. Acknowledgement: The autho thanks Manuel Del Pino fo numeous valuable discussions. The wok was suppoted in pat by the NSF gant DMS while the autho was an Alfed P. Sloan fellow. Refeences [1] V. Adolfsson, L. Escauiaza, and C. Kenig, Convex domains and unique continuation at the bounday, Revista Matemática Ibeoameicana ), [2] S. Agmon, Unicité et convexité dans les poblèmes difféentiels, Séminaie de Mathématiques Supéieues, No. 13 Eté, 1965), Les Pesses de l Univesité demontéal, Monteal, Que.,1966. [3] F.J. Almgen, J., Diichlet s poblem fo multiple valued functions and the egulaity of mass minimizing integal cuents, Minimal Submanifolds and Geodesics, Noth-Holland, Amstedam, edito M. Obata, 1979, pp [4] R. Bummelhuis, Thee-sphees theoem fo second ode elliptic equations, J. d Analyse Mathématique ), [5] F. Bethuel, H. Bézis, and F. Hélein, Ginzbug-Landau votices, Pogess in Nonlinea Diffeential Equations and thei Applications, 13. Bikhäuse Boston, Inc., Boston, MA, [6] S.B. Bell and S.G. Kantz, Smoothness to the bounday of confomal maps, Rocky Mountain J. Math ), [7] S. Chanillo and M.K.-H. Kiessling, Cul-fee Ginzbug-Landau votices, Nonlinea Analysis ), [8] M. Del Pino and P.L. Felme, Local minimizes fo the Ginzbug-Landau enegy, Math.Z ),

15 EJDE 2000/61 Igo Kukavica 15 [9] N. Gaofalo and F.-H. Lin, Monotonicity popeties of vaiational integals, A p weights and unique continuation, Indiana Univ. Math. J ), [10] C. Kenig, Restiction theoems, Caleman estimates, unifom Sobolev inequalities and unique continuation, Lectue Notes in Math., 1384, Spinge, Belin-New Yok, 1990, pp [11] I. Kukavica, Nodal volumes fo eigenfunctions of analytic egula elliptic poblems, J. d Analyse Mathématique ), [12] I. Kukavica, Level sets fo the stationay Ginzbug-Landau equation, Calc. Va ), [13] I. Kukavica, Quantitative uniqueness fo second ode elliptic opeatos, Duke Math. J ), [14] I. Kukavica and K. Nystöm, Unique continuation on the bounday fo Dini domains, Poceedings of AMS ), [15] F.-H. Lin, Solutions of Ginzbug-Landau equations and citical points of the enomalized enegy, Ann. Inst. H. Poincaé Anal. Non Linéaie ), [16] C. Lefte and V. Rădulescu, Minimization poblems and enomalized enegies elated to the Ginzbug-Landau equation, An. Univ. Caiova Se. Mat. Infom ), ). [17] M. Stuwe, On the asymptotic behavio of minimizes of the Ginzbug- Landau model in 2 dimensions, Diffeential Integal Equations ), Igo Kukavica Depatment of Mathematics Univesity of Southen Califonia Los Angeles, CA kukavica@math.usc.edu

Measure Estimates of Nodal Sets of Polyharmonic Functions

Measure Estimates of Nodal Sets of Polyharmonic Functions Chin. Ann. Math. Se. B 39(5), 08, 97 93 DOI: 0.007/s40-08-004-6 Chinese Annals of Mathematics, Seies B c The Editoial Office of CAM and Spinge-Velag Belin Heidelbeg 08 Measue Estimates of Nodal Sets of