Complex Ginzburg-Landau equations in high dimensions and codimension two area minimizing currents

Transcription

1 J Eur Math Soc 2, c Springer-Verlag & EMS 2000 Erratum Fang-Hua Lin Tristan Rivière Complex Ginzburg-Landau equations in high dimensions and codimension two area minimizing currents J Eur Math Soc 1, ) In our previous paper [2] Complex Ginzburg-Landau equations in high dimensions and codimension two area minimizing currents, we studied the asymptotic behavior of energy minimizing solutions of the Ginzburg- Landau equations But the η-compactness Lemma was for arbitrary solutions which may not be energy minimizing We found there is a gap in this version of the proof of the η-compactness Lemma Lemma II7) This η-compactness Lemma as well as asymptotic behavior of arbitrary solutions are treated in our forthcoming paper [3] Here we shall simply present our earlier proof of the η-compactness for energy minimizing solutions All the statements in the rest of the paper [2] are not affected by this modification The arguments from II60) to II68) have to be modified in the following way Starting from II60) we say: In fact we will be mainly interested in p such that W 1,p ) H 1 2 ), that is p 2 1 n We will now extend the map ũ into a map w in all of B t by using as less energy as possible We will decompose into a union of disjoint cubes having edges of length,where will be taken very small to be fixed later) We will first extend ũ between and Asitisprovedin[1] it is possible to choose this union of cubes such that, if C k denotes the corresponding k-skeleton for this union of cubes for 2 k n 1), we FH Lin: Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY , USA T Rivière: Ecole Normale Supèrieure de Cachan, Centre de Mathèmatiques et de leurs Applications, Unitè associeè au CNRS URA-1611, 61, avenue du prèsident Wilson, F Cachan Cedex, France

2 88 Fang-Hua Lin, Tristan Rivière have in the same time ) 1 n 1 k K 2 k n 1 C k r1 2 ) ) 2 1 n 1 k r1 K 2 C k 2 k n 1 C k ũ d ũ p K ) 1 n 1 k ũ d ũ p 2 k n 1 1) Where K is a constant depending only on n not on nor on First of all, will be chosen such that ũ 1/2inC 2 Combining II47) of [2] and 1) we get r1 2 K C 2 ) 1 n 3 η ũ 2 C r 1 Thus, this is the case if η/ n 3 is sufficiently small independantly of ) Denote by Ck the k-skeleton homothetic to C k, contained in,forthe homothetie of rate t and let D t k+1 be the k + 1-skeleton in the interior of B t \ B t having C k Ck as boundary we will construct the extension w of ũ in B t \ B t on D k by induction on k Fork = 3wetake wx) =ũ t x ) in D 3 x We clearly have D 3 w 2 w d w p K D 3 C 2 K ) 1 n 4 r1 ) ) n 4 r1 ) K 2 2 D 3 ) 1 n 4 ũ d ũ p 2)

3 Ginzburg-Landau equations 89 Before to construct the extension w of ũ in the interior of B t \ B t we construct the extension w of ũ in,ontheck by induction on k On each cell of the 3-skeleton ) C3,extendw radially from the boundary of the cell, where wx) =ũ t x, to the center of the cell One verifies that we have x ) 1 n 4 w 2 C w 2 C C3 C2 r1 C 3 C 3 ) ) n 4 r1 C w d w p C ) 1 n 4 ũ d ũ p 2 w 1 2 a e in C 3 Repeating these extensions on Ck until k = n 1 we obtain that w 2 C r1 2 ) 2 r1 C 2 w d w p C ũ d ũ p 3) w 1 2 a e in Now we construct w in D k by induction on kwehave D 4 = D 3 C 3 C 3 Here also, in each cell, we choose w to be the radial extension, relative to the center of the cell, from it s value on the boundary We establish in the same way inequalities like 2), replacing D 3 by D 4 and n 4byn 5 Repeating this procedure until k = n, we get in particular w 2 C D n =B t \B t r1 2 ) r1 C 2 B t \B t 4)

4 90 Fang-Hua Lin, Tristan Rivière Finally we extend w in B t in the following way We have w 1/2 in a e and w W 1,2 ) Thus w w d w w L2 ) and since [ w d w d w ] = 0, w and HdR 1 ) = 0, there exists φ W 1,2 ) such that w w d w w = dφ and φ = 0 Thus w = w exp iφ + iφ 0,whereφ 0 [0, 2π), and from II68) of [2] and 3) we deduce that ) 1 φ W 1,p ) C q η γ Cq η γ 2 for some fixed 0 <γ<1, where p = 2 1 By Sobolev embedding we n have ) 1 φ 1 C η γ 2 2 H 2 γ + C η 2 Bt ) Let φ be the harmonic extension of φ in B t,wehave φ 2 C φ 1 B H 2 C ηγ Bt ) t + C η γ 5) We take w/ w =exp iφ + iφ 0 in B t For the modulus w of w in B t we choose w = ω,whereω is the solution of the following problem ) ω + ω = 1 in B t r 1 6) ω = w in In [4] we proved that min Bt ω min Bt w 1/2and 1 ω 2 ω 2 r1 + B t 7) C ) ) r1 1 + ) ) 1 ũ 2 2 2

5 Ginzburg-Landau equations 91 Combining 4), 5), 7) and the minimality of ũ in B t we get r1 1 + ) ũ 2 2 C η γ + B t + C + Cη γ Choose = Cη n recall that we had already chosen Cη n 3 ) Using II44) of [2] and going back to the usual scale we get, for β = inf 1 n 3,γ), E tr1 tr 1 ) E n 2 Cηβ r 1 r n 2 + η β 1 where t 1/2 Thus, because of the monotonicity formula we have II68) of [2] and the proof can be ended like in [2] References 1 F Bethuel: The approximation problem for Sobolev maps between two manifolds Acta Math 167, ) 2 FH Lin, T Rivière: Complex Ginzburg-Landau equations in high dimensions and codimension two area minimizing currents JEMS 1, ) 3 FH Lin, T Rivière: Complex Ginzburg-Landau Equations in High dimensions and minimal submanifolds II In preparation 4 T Rivière: Line vortices in the U1)-Higgs model COCV )