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This commemorative volume honors the memory of S. L. Sobolev by presenting eighteen papers reflecting the area of Sobolev's main contributions: applications of functional analysis to differential equations. The papers examine various problems in the theory of partial differential equations (linear and nonlinear) and the theory of differentiable functions of several real variables. Applications to problems of mathematical physics and approximate methods of conformal mapping are also treated. Foreign (e.g., Leray, Lions, Fichera) as well as domestic (e.g., Besov, Oleinik, Il'in) authors are represented. ... Read more

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This updated monograph has been compiled on the basis of lectures given by Academician Sobolev on the theory of cubature formulas. The concept of generalized functions is considered and the classical problem of approximate integration of functions of several independent variables is discussed. The work reveals the diverse relationships between theoretical approximate integration and problems in such areas as mathematical analysis, functional analysis, computational mathematics and number theory.
S.L. Sobolev was, before his death in 1988, an Academician at the USSR Academy of Sciences and President of the National Committee of Soviet Mathematics. ... Read more

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Kto kontroliruet umy, tot pravit mirom. Trevozhnye sobytiia, razvivaiuschiesia po skhozhemu stsenariiu v Rossii i SShA, napolnili izvestnuiu istinu konkretnym i predel'no zloveschim soderzhaniem. Mnogie vazhnye persony, zanimaiuschie krupnye gosudarstvennye posty, stanoviatsia ob'ektom izoschrennykh manipuliatsii. Pautina, sotkannaia dlia oblechennykh vlast'iu muzhei, nevesoma, nevidima glazu, no chrezvychaino prochna. Chelovechestvu iavleny sovershenno nevidannye dosele tekhnologii podavleniia i unichtozheniia lichnosti. Kazhdomu, kto pytaetsia sorvat' zloveschie plany Avaddona, grozit neotvratimaia gibel'.. Chego zhe dobivaetsia Sverkhorganizatsiia, za kotoroi ugadyvaetsia chudovischnyi, poistine necheloveskii razum? Otkuda vzialis' vse eti kazhuschiesia fantasticheskimi tekhnologii? I est' li esche shans dlia spaseniia? Na eti voprosy nezavisimo drug ot druga pytaiutsia otvetit' analitik rossiiskikh spetssluzhb Romantsev i amerikanskaia zhurnalistka Kolkhauer.. .. ... Read more

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The spaces of functions with derivatives in Lp, called the Sobolev spaces, play an important role in modern analysis. During the last decades, these spaces have been intensively studied and by now many problems associated with them have been solved. However, the theory of these function classes for domains with nonsmooth boundaries is still in an unsatisfactory state.

In this book, which partially fills this gap, certain aspects of the theory of Sobolev spaces for domains with singularities are studied. We mainly focus on the so-called imbedding theorems, extension theorems and trace theorems that have numerous applications to partial differential equations. Some of such applications are given.

Much attention is also paid to counter examples showing, in particular, the difference between Sobolev spaces of the first and higher orders. A considerable part of the monographis devoted to Sobolev classes for parameter dependent domains and domains with cusps, which are the simplest non-Lipschitz domains frequently used in applications.

This book will be interesting not only to specialists in analysis but also to postgraduate students. ... Read more