AJT0n t 3ft n

We do this so that there shall be upper bounds Plt P2 and P3 to the constants Pv P2 and P8in lemmas 7.4.1-7.4.3 for the surface Ji?(t) f|

Lemma 7.4.6

If conditions (1) and (2) of lemma 7.4.4 hold and if (4) there is some Qs such that

(by lemma 7.4.1, this implies condition (3)), then there exist positive constants (depending on W, e, a, <2X and Qs) such that

||K, jT(i) n ^+||4+a < Ps>a{||K,^(0)n ^+||4+a+||F,^(t)||3+0}- (7-30)

From lemma 7.4.4 one has an inequality for ||K, M?(t) n r To obtain an inequality for ||K, 3i?(t)n one forms the 'energy' tensor S"6 for the first derivatives KTJtc and proceeds as before. The divergence S^^ can now be evaluated by differentiating equations (7.20):

= AadKIjMFp(}iececcJOcJP + (terms quadratic in KTj,

KTJic and K1Jlcd with coefficients involving Acd,

With the possible exceptions of BcPIQJ[d and CPIQJid, these coefficients are all bounded on in the case a = 0. When integrated over the surface Jf(t') n the term in (7.31) involving BcPIQJfa is

There is some Qt such that for all t', (7.32) is less than or equal to

Qi f IDBIIDKI ID2K| dcr

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