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 frst 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 signifcantly augmented list of references aim to create a broader and modern view of the area.
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 für 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. Gouvêa, 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)
