Existence and regularity results for some variational problems related to harmonic maps

Nonlinear Analysis 52 (2003) 355 – 356

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Erratum

Existence and regularity results for some variational problems related to harmonic maps (Nonlinear Analysis 47 (2001) 1703–1714
Atsushi Tachikawa Department of Mathematics, Faculty of Science and Technology, Science University of Tokyo, Noda, Chiba 278-8510, Japan Received 16 November 2001; accepted 22 January 2002

In the proof of Theorem 2.4 of the above-mentioned paper p. 1708, the author gave the following estimate:


∗ ∗  |u| d 6 Ef (w) + 0 {|u|2 + −=(2 −) } d e(u) d 6 Ef (w) + 0





6 c3 (Ef (w);
; g; ; ; 0 ) + c4 (
; g; h; 0 )








e(u) d:

However, the last inequality is not correct. In the last term c4 depends also on uL∞ . Therefore the remaining part of the proof is not valid. We must replace the condition on  by a stronger one and the second half of the proof must be changed. In Theorem 2.4 the condition on  must be replaced by the following one:   2(m − 1) :  ∈ 0; (m − 2) From the 6fth line of p. 1710, the proof of Theorem 2.4 should be changed as follows: Now, let us estimate the right-hand side of (2.17). Though we proceed as if we are assuming that m = 3 or 4, the following proof is valid for m = 2 as well if one replace 2∗ by a su8ciently large constant.




PII of the original article : S0362-546X(01)00303-0 E-mail address: [email protected] (A. Tachikawa).

0362-546X/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 3 6 2 - 5 4 6 X ( 0 2 ) 0 0 0 9 7 - 4

356

A. Tachikawa / Nonlinear Analysis 52 (2003) 355 – 356

Since we are assuming (2.2), the minimality of u implies that


∗ ∗ e(u) d 6 Ef (w) + 0 |u| d 6 Ef (w) + 0 {|u|2 + −=(2 −) } d








6 Ef (w) + c3 (
; 0 )−=(2

∗

−)

+ c4 (
; g; h; 0 )R02

∗

−2





e(u) d:

Here, we used Young’s inequality and the PoincarCe inequality. By choosing  = ∗ 1=2c4 R02 −2 , we get the following a priori estimate:
∗ ∗ (2.18) |Du|2 d x 6 c5 (g; h; ; 0 ;
; Ef (w))(1 + R0(2 −2)=(2 −) ):


Moreover, using the PoincarCe inequality we get from (2.18) that 
1=2∗ ∗ ∗ (2∗ −2)=2∗ 2 uL2∗ 6 R0 |u| d x 6 K0 (1 + R(2 −2)=(2 −) );

(2.19)

∗

where K0 = c51=2 . It is nothing to see that K0 satis6es lim

0 ; Ef (w)→0

K0 (g; h; ; 0 ;
; Ef (w)) = 0:

(2.20)

On the other hand, since we are assuming that m 6 4, (2.19) implies that uL4 6 ∗ ∗ c6 (
; m) · K0 (1 + R(2 −2)=(2 −) ). Moreover, by condition (2.5) with  ∈ [0; 4=(m − 2)), we see that 2∗ m ui fi (u; e(u))Lq 6 c7 (b2 ; q;
)uL2∗ for q = ¿ :  2 Consequently, if m 6 4 and (2.5) holds for some  ∈ [0; 4=(m − 2)), then we obtain from (2.17) and (2.19) u2L∞ (
) 6 c4 {c6 (1 + c7 )K0 (1 + R0(2

∗

−2)=(2∗ −)

) + wL2q } + w2L∞ (
) :

(2.21)

w2L∞ (
)

Now, from (2.20) and (2.21), we can see that if b2 ; 0 ; Ef (w) and are su8ciently small we have (2.9). When we can take R0 = +∞ in the condition C(q0 ; R0 ), for any given b0 ; b1 ; b2 ; b3 and f, we can proceed as in the above proof choosing R0 su8ciently large so that R20 is greater than the right-hand side of (2.21). Such a choice is possible since 2∗ − 2 ¡2 (2.22) 2∗ −  by the assumption on . For the case m = 2, for any  we can proceed as in the above proof by replacing 2∗ by a su8ciently large constant so that (2.22) holds.