Sobolev spaces play an outstanding role in modern analysis, in particular, in the theory of partial differential equations and its applications in mathematical physics. They form an indispensable tool in approximation theory, spectral theory, differential geometry etc. The theory of these spaces is of interest in itself being a beautiful domain of mathematics. The present volume includes basics on Sobolev spaces, approximation and extension theorems, embedding and compactness theorems, their relations with isoperimetric and isocapacitary inequalities, capacities with applications to spectral theory of elliptic differential operators as well as pointwise inequalities for derivatives. The selection of topics is mainly influenced by the author's involvement in their study, a considerable part of the text is a report on his work in the field. Part of this volume ¿rst appeared in German as three booklets of Teubner-Texte zur Mathematik (1979, 1980). In the Springer volume "Sobolev Spaces", published in English in 1985, the material was expanded and revised. The present 2nd edition is enhanced by many recent results and it includes new applications to linear and nonlinear partial differential equations. New historical comments, five new chapters and a signi¿cantly augmented list of references aim to create a broader and modern view of the area.
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In July 2009 the Senior Whitehead Prize of the London Mathematical Society was awarded to Professor Maz'ya. He has also received the Celcius Gold Medal from the Swedish Royal Society in Uppsala (2004), the Verdaguer Prize from the Academie de France (2003), and the Humboldt Research Prize (1999) in Germany. In 2002 he was elected to the Royal Swedish Academy of Sciences and in 2001 he became Corresponding Fellow of the Scottish National Academy. He was an invited speaker at the International Congress of Mathematicians (2002) and on the occasion of his 70th birthday (2008) two international conferences in Rome and Stockholm were organized. In 2009 five volumes dedicated to him were published in USA, Italy and Germany.
Introduction.- 1 .Basic Properties of Sobolev Spaces.- 2 .Inequalities for Functions Vanishing at the Boundary.- 3.Conductor and Capacitary Inequalities with Applications to Sobolev-type Embeddings.- 4.Generalizations for Functions on Manifolds and Topological Spaces.- 5.Integrability of Functions in the Space L 1/1(O).- 6.Integrability of Functions in the Space L 1/p (O).- 7.Continuity and Boundedness of Functions in Sobolev Spaces.- 8.Localization Moduli of Sobolev Embeddings for General Domains.- 9.Space of Functions of Bounded Variation.- 10.Certain Function Spaces, Capacities and Potentials.- 11 Capacitary and Trace Inequalities for Functions in Rn with Derivatives of an Arbitrary Order.-12.Pointwise Interpolation Inequalities for Derivatives and Potentials.- 13.A Variant of Capacity.- 14.-Integral Inequality for Functions on a Cube.- 15.Embedding of the Space L l/p(O) into Other Function Spaces.- 16.Embedding L l/p(O) ¿ W m/r(O).-17.Approximation in Weighted Sobolev Spaces.-18.Spectrum of the Schrödinger operator and the Dirichlet Laplacian.- References.- List of Symbols.- Subject Index.- Author Index.
From the reviews of the second edition:
"The present augmented and revised edition has five new chapters, new historical comments, and a significantly extended list of references. ... It is well written and clearly comes up to the authors aim of creating a broader and modern view of the area. It can be warmly recommended to anybody working in the field of partial differential equations." (G. Teschl, Monatshefte fur Mathematik, Vol. 166 (1), April, 2012)
"This is a revised and enlarged edition of a book first published in English in 1985. ... The book is largely an account of the author's own work on the subject. ... Maz'ya's massive book will ... continue to be a fundamental reference for those who work in the field." (Fernando Q. Gouvea, The Mathematical Association of America, June, 2011)
"This new edition of the book is enhanced by many recent results and it includes new applications to linear and nonlinear partial differential equations. ... This comprehensive volume is very well written and well structured. It will certainly serve as a valuable reference work for graduate students and researchers working in related fields." (Teodora-Liliana Radulescu, Zentralblatt MATH, Vol. 1217, 2011)
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