 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

Example text

G: S • .!! S 't. ~ Р· Ву inequality (16"), р. t&) where h is an arЬitrary positive number, satisfying the inequalities: o'h' Н, where (2) Н is а *We make the assumption p~i for simplicity. Of the estimates introduced below, the fundamentalones can Ье oьtained, with some complication in the arguments, even for р: i. V. 28 Р. IL'IN parameter appearing in the definition of the class А ( 1-1), to which the domain G- Ьelongs; Wh (:х:) is an n -dimensional ball of radius h' with center at х' and с~ and с~ are constants, independent of and 'L.

N.. , lf, Ф, ЩК·1), and Sк. This is proved in the manner employed in (1] for а Navier-Stokes system, linearized according to Stokes (see Chap. 6, §11, and Chap. З, §5). The presence of linear terms of lower order entails no dШiculties. From §5, Chap. З of , we shall need an estimate of the form (6) (estimate of V. А. ) of tn .. ), which vanishes on Sк. о ( n .. ). The constant ~~ in inequality (7) depends on the "norm" of S. in с' . The assumption we make concerning the smoothness of the boundaries St , t Е: t О, Т 1, consists in taking the constant ~~ in inequality (7) to Ье common for all the St, t «> [ О,Т ].

Then inequality 50 О. А. LADYZHENSKAYA AND N. N. URAL'TSEVA (13) yields whence where Thus,'L having oьtained an upper bound for 1J'~a. ,. ('Х) • То 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.