Boundary Value Problems of Mathematical Physics and Related by N. M. Ivochkina, A. P. Oskolkov (auth.), O. A. Ladyzhenskaya PDF

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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Example text

I (р> P•Pi .. jЦ)>'S, ~i •t,'S~ c,=Co"'st ... & 'S;. i <Р>• "1 t JA"" Со м ~о. Ч (р> н~, (\~J tp)). tp>- as 1р1--, and all the functions we have introduced will Ье assumed to Ье continuously differentiahle functions of their arguments. ) Ье а thrice differentiahle solutionof Eq. >"M, Weoьtainnowthe estimate m_gд:. )\. For this we introduce in place of u. X. ,. ,. )I 'М. *The results given here appeared in [1] in aЬbreviated 47 form. 48 О. А. LADYZHENSКAYA AND N. N. URAL'TSEVA function v Ьу means of the equation: u..

VA '11- '41;. t) are, in а definite sense, small, even for small Т, if Ч', "'•, and J only possess а certain smoothness. Appro,qmate solutions will Ье found Ьу the method of Rothe, and their convergence to the solution of proЬlei'n (З) will Ье estaЬlished with the aid of two а priori estimates. LADYZHENSКAYA 11. Let us divide QT Ьу the planes t =tк. KAt • кh into layers, am let us denote Ьу n .. the cross section of Q" made Ьу the plane t•tк, and Ьу Sк the boundary n,.. , ... ;. {К) t (4) dl11 in which utK)!!

In the variaЬle Xk(K=I, ... ,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 [6] Ьу another method. ~, ... , Ьу а number of other authors.

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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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