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By Fabrice Bethuel, Haïm Brezis, Frédéric Hélein (auth.)

ISBN-10: 0817637230

ISBN-13: 9780817637231

ISBN-10: 1461202876

ISBN-13: 9781461202875

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In n on each 8wj , j = 1,2, ... , n onoG tPi = 0 and 211' -211' o if w is the disc centered at 11'i , if w is the disc centered at ni, otherwise . (ii) The function l1i associated to the ion [Pi, Di] is the solution of Al1i = 0 l1i = Const. l1i =0 and n, on each 8wj , j = 1,2, ... ,n, onoG, if w is the disc centered at Pi, otherwise. (iii) The function 8i associated to qi is the solution of in A8i = 0 and in n 8i = Const. on each 8i = 0 onoG { OBi _ 0 JOWj all So that, in fact, Bi == o. OWj, j = 1,2, ...

Moreover E~ = LrV~212 where ~2 is the solution 0/ (20). 3. SI- valued harmonic maps with prescribed isolated singularities. The canonical harmonic map As previously let G be a smooth, bounded and simply connected domain in R 2 • Fix n points al, a2, ... ,an in G, and n integers dl, d2' ... ,dn in Z. Set d = E~=1 di and let 9 be a map from aG into 8 1, such that deg(g, aG) = d. We propose to describe the class C of all smooth harmonic maps u from n = G\ {al, ... , an} into 81, such that: (i) deg(u,aj) = d j (where deg(u,aj) denotes the degree of u restricted to any small circle centered at aj), (ii) u is continuous up to aG and u = 9 on aGo Let ~o be the solution of in G, (22) onlJG.

Finally, we have, by (10), Sup v :$ Inf v + Sup v - Inf v + X n n 8G lJG lJG lJG lJG :$ Inf v + Sup v - Inf v + X =Supv+X. L :$ 0 by (1). Thus Sup v :$ X. L G) - X. L G) • Hence This proves (8). We now prove (9). Pi( 8wi). Pil2 ~ 211" log(ll/ p) - 211"A. 2. Pi be the solution of (6). Assume di > O. Pi > 0 811 - (12) ~ on uW•• . Proof. pt In {iVIt't1 2 + In ( 81t'i It't J/)w; 811 2 kiVIt'tl2 + 211" di It't(8Wi) ~ 1/'Vlt'tI • Thus It't = 0 in = n and hence It'i :$ 0 in n. We now turn to the proof of (12).

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Ginzburg-Landau Vortices by Fabrice Bethuel, Haïm Brezis, Frédéric Hélein (auth.)

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Categories: Mathematical Physics