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

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**Example text**

On 8G (resp. , gi) 8F =/ 8T [Note that F (resp. Gi ) is well defined as a single-valued function since J8G / = 0 (resp. J8B(0,"p ) gi = 0)]. Let '1'* be the solution of in G, on8G. 4 to v = W~ - \11* we see that L 8B(0. Sup n ,p) i=l since IlgiIlLoo(8B(0"p» $ (Gi - '1'*) - Inf (Gi 8B(0••p) - '1'*) = O(p) C. , D. Gilbarg and N. , 8'11 (66) 8Xl { 8'11 8X2 8\11* = 8X2 8'11* = - 8X l in G in G 26 1. Energy estimates for SI-valued maps so that '11 satisfies (67) 6'11=0 { 8'11 = 8'11* = 8v lh f in G on8G.

Renormalized energy 25 Hence, we have in {lp, on 8G, on 8B(ai, p), i = 1,2, ... , n, (62) i = 1,2, ... , n, where F (resp. Gi ) is a primitive with respect to arc length of / (resp. on 8G (resp. , gi) 8F =/ 8T [Note that F (resp. Gi ) is well defined as a single-valued function since J8G / = 0 (resp. J8B(0,"p ) gi = 0)]. Let '1'* be the solution of in G, on8G. 4 to v = W~ - \11* we see that L 8B(0. Sup n ,p) i=l since IlgiIlLoo(8B(0"p» $ (Gi - '1'*) - Inf (Gi 8B(0••p) - '1'*) = O(p) C. , D. Gilbarg and N.

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).