Y . L . Xin and Yihu Yang, Regularity ofp-harmonic maps

Si - i \P 4 /

51 - 1 \p 4 /

as 8 --► 0. On the other hand, we have the right hand side of (3.3)

= - J (\аф\^\Уи^ + \афГ'7и^'7\аф\)*1

Si - i

w>e P

f (-^\dф\^'^\Viuw)\^ + -^ViuwУ■V\dф\^']*l

1____ , „ , ... 1_

r . 2

0^w^c\^ P^

Since Vw = 0 a.e. on {x|w(x) = 0}, it is easy to see that /3(8) -♦ 0 and

A ( ß ) - ^ - f (\аф\Р\7и\^ + -7и^'7\аф\П*1 as e-► 0. s»-i\ P J

Thus , (3.3) holds true for all и e C^(S^~^). In particular, taking w = 1 we obtain (3.4) J {±^lV\dф\h'-A\dф\^'^')*lè-^^^ I \dф\^>*l.

Si - 1 \P /4 51-1

Let us now give an upper bound of the left hand side of (3.4) by using Lemma 2.3. In the following discussion, we will restrict ourself to 5+"- as a domain manifold. A direct computation shows

- Л\аф\^ = {р-^1)\афГ'\7\аф\\'^\афГ'А\аф\. P

By using (2.4) and the curvature conditions on 5+"^ and N we have

( / ? - - 1 ) |#Г2|Р|#||2 + |#|^-М|#|~Р,^<т((^),|#Г-^ф,0

^ ( p - - 2 ) \dфГ'\F\dфf^\dфГ' ( \Vdф\' + (l^2)\dф\'^'^^^

where Ä'> 0 is the upper bound of the sectional curvature of the target manifold. Hence,

\dфГ'A\dф\ ~ К^ЫФ1 ЫфГ'ф^е,}

^\dфГЧ\f^dф\' - \V\dфf ) '^\dфГ'i ( l^2 ) \dф\'^^f^