By N. M. Ivochkina, A. P. Oskolkov (auth.), O. A. Ladyzhenskaya (eds.)
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Additional resources for Boundary Value Problems of Mathematical Physics and Related Aspects of Function Theory
N). о is selected so that hk" 1-\1( lк .. ~ .... ,n), we oЬtain n .!. .!.. k (к=-\, ... , n). \m+i, ... mн = ~~ ... rt. the inequality From this, with the aid of inequality (9'), we oЬtain (20) For entire functions of finite degree, for G=Е " , and for trigonometric polynomials, taken over а period, inequalities (19) and (20), with the indicated values of the constants С, were first oЬtained Ьу S. М. Nikol'skii  Ьу another method. ~, ... , Ьу а number of other authors. ·, G' i v; G) PtL·,G'; v·, G).
L- \ . Let the solution u. ) . ;. are functions of the corresponding classes P(t·,G'; v; 6-) . of the form (1 О) (for certain particular types of domains G, the functions simple algebraic polynomials). , Ье an arbltrary function of the form given in Eq. 4 , ... n ). L>-rlщ)-. ,. The possibllity of oьtaining an estimate for Et,t. ,,w;(G), (31) is based on the inequality (31) and theorems giving an estimate for \U. i. "wt. G)' An estimate for Ru. ~ ll,i (G) may Ье oьtained Ьу using а theorem of I.
LQ')' and this, the limiting form of Eq. (52) will Ье \ ( 4'"' \j1 and r in L, lQ'), converges to ct> uniformly in от·. Thanks to ~~ • У~ U ' Щ. ) U:ч т U;. \1""< - cr 1q> dx. т, whence, in view of the sufficient arЬitrariness of ф and the fact that u belongs to w~·~·а~ l ) ' it follows that t1 satisfies Eqs. (3). Thus, we have proved the following theorem: Theorem 1. )• and U"'IUw:·\Qт). The norm of the solution 'U, namely, nunw'·'lQT•)• is also determined Ьу these same quantities. " and the boundaries 51 of the domain nt in Е~ must have uniformly bounded "norms" in е for all t Е: [ОТ) and must satisfy conditions 1) and 2), given on page 42.
Boundary Value Problems of Mathematical Physics and Related Aspects of Function Theory by N. M. Ivochkina, A. P. Oskolkov (auth.), O. A. Ladyzhenskaya (eds.)