By N. M. Ivochkina, A. P. Oskolkov (auth.), O. A. Ladyzhenskaya (eds.)

ISBN-10: 1475746660

ISBN-13: 9781475746662

ISBN-10: 1475746687

ISBN-13: 9781475746686

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**Extra info for Boundary Value Problems of Mathematical Physics and Related Aspects of Function Theory**

**Sample text**

Uи .. ,~. n. n. , 5•• • (ЗО) Thanks to relations (30) and (26), we have 1 ~} IAtK-0-Pк'\A\1<-1) At n. i:>U(K) and, hence, we obtain from Eq. Upt>u~ 1 ,n. u\1 1,a,_,"'"[ I'U:tl2.. l,_, 0 к_,). , (31) ~ C 1 C 1 1~uU 1,n. l\,nк) Let us replace the first term in inequality (22) Ьу the smaller quantity in inequality (32) and then sum the resulting inequalities with respect to к from К•2. , take account of condition (5), (8), and the estimate (15), and choose all the t. so small that the coefficients for i р ~ u U~ 11 and 'р l!.

Is а function of class ;г (2 m+l ; С5; s·~; (28) G), then, on the basis of inequalities (15), (27), and {8), we shall have l ~j. "' ) \\ 'С Е г. )t-s-~·i)' t"' LG\A. m [G (~"') 1-(t-s-~.. ~) , where с~ is independent of m. Convergence of the series (28) follows from inequality (29) and the convergence of the series (23 ). \ 111... tm 11~·"'... -f я~.

Complete the proof, we need to show the existence of а solution 'fl'lr) of Eq. (12), having the properties stated earlier. Let us introduce, instead of 'f(11), а new function '( l 'U') Ьу means of the equation ч' ( 11)= е 'ltv>. Substituting into Eq. (12), we oьtain for m~ -7 11 ' 2. ~ +ее \~'\. (14) Let us find а solution of this equation for which ~ 'L О. have Let us consider ~ as the argument, and S =- '( as the unknown function. Then, from Eq. (14), we d3 -sd~ For "'-=-"5 е-~, we main where rJ- >о oьtain, from Eq.

### Boundary Value Problems of Mathematical Physics and Related Aspects of Function Theory by N. M. Ivochkina, A. P. Oskolkov (auth.), O. A. Ladyzhenskaya (eds.)

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