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Grundlehren der mathematischen Wissenschaften

Sobolev Spaces

with Applications to Elliptic Partial Differential Equations

Authors: Maz'ya, Vladimir

  • New, expanded and revised edition of Sobolev Spaces, originally published in the Springer Series in Soviet Mathematics (1985) Enhanced by many recent results Includes new applications to linear and nonlinear partial differential equations New historical comments, five new chapters and the significantly augmented list of references create a broader, modern view of the field

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About this book

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 first 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 significantly augmented list of references aim to create a broader and modern view of the area.

About the authors

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.

Reviews

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 Rădulescu, Zentralblatt MATH, Vol. 1217, 2011)


Table of contents (18 chapters)

  • Basic Properties of Sobolev Spaces

    Maz’ya, Vladimir

    Pages 1-121

  • Inequalities for Functions Vanishing at the Boundary

    Maz’ya, Vladimir

    Pages 123-229

  • Conductor and Capacitary Inequalities with Applications to Sobolev-Type Embeddings

    Maz’ya, Vladimir

    Pages 231-253

  • Generalizations for Functions on Manifolds and Topological Spaces

    Maz’ya, Vladimir

    Pages 255-286

  • Integrability of Functions in the Space $L^{1}_{1}(\varOmega )$

    Maz’ya, Vladimir

    Pages 287-321

Buy this book

eBook $139.00
price for USA (gross)
  • ISBN 978-3-642-15564-2
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover $179.00
price for USA
  • ISBN 978-3-642-15563-5
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Rent the ebook  
  • Rental duration: 1 or 6 month
  • low-cost access
  • online reader with highlighting and note-making option
  • can be used across all devices
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Bibliographic Information

Bibliographic Information
Book Title
Sobolev Spaces
Book Subtitle
with Applications to Elliptic Partial Differential Equations
Authors
Translated by
Shaposhnikova, T.O.
Series Title
Grundlehren der mathematischen Wissenschaften
Series Volume
342
Copyright
2011
Publisher
Springer-Verlag Berlin Heidelberg
Copyright Holder
Springer-Verlag Berlin Heidelberg
eBook ISBN
978-3-642-15564-2
DOI
10.1007/978-3-642-15564-2
Hardcover ISBN
978-3-642-15563-5
Series ISSN
0072-7830
Edition Number
2
Number of Pages
XXVIII, 866
Additional Information
Originally published under Vladimir G. Maz'ja in Springer Series of Soviet Mathematics
Topics