Sobolev Spaces

1. Front Matter

   Pages I-XXVIII

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2. Basic Properties of Sobolev Spaces

   • Vladimir Maz’ya

   Pages 1-121

3. Inequalities for Functions Vanishing at the Boundary

   • Vladimir Maz’ya

   Pages 123-229

4. Conductor and Capacitary Inequalities with Applications to Sobolev-Type Embeddings

   • Vladimir Maz’ya

   Pages 231-253

5. Generalizations for Functions on Manifolds and Topological Spaces

   • Vladimir Maz’ya

   Pages 255-286

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

   • Vladimir Maz’ya

   Pages 287-321

7. Integrability of Functions in the Space \(L^{1}_{p}(\varOmega )\)

   • Vladimir Maz’ya

   Pages 323-404

8. Continuity and Boundedness of Functions in Sobolev Spaces

   • Vladimir Maz’ya

   Pages 405-434

9. Localization Moduli of Sobolev Embeddings for General Domains

   • Vladimir Maz’ya

   Pages 435-458

10. Space of Functions of Bounded Variation

    • Vladimir Maz’ya

    Pages 459-509

11. Certain Function Spaces, Capacities, and Potentials

    • Vladimir Maz’ya

    Pages 511-548

12. Capacitary and Trace Inequalities for Functions in ℝⁿ with Derivatives of an Arbitrary Order

    • Vladimir Maz’ya

    Pages 549-609

13. Pointwise Interpolation Inequalities for Derivatives and Potentials

    • Vladimir Maz’ya

    Pages 611-655

14. A Variant of Capacity

    • Vladimir Maz’ya

    Pages 657-668

15. Integral Inequality for Functions on a Cube

    • Vladimir Maz’ya

    Pages 669-692

16. Embedding of the Space \(\mathaccent"7017{L}^{l}_{p}(\varOmega)\) into Other Function Spaces

    • Vladimir Maz’ya

    Pages 693-735

17. Embedding \(\mathaccent "7017{L}^{l}_{p}(\varOmega, \nu) \subset W^{m}_{r}(\varOmega)\)

    • Vladimir Maz’ya

    Pages 737-753

18. Approximation in Weighted Sobolev Spaces

    • Vladimir Maz’ya

    Pages 755-768

19. Spectrum of the Schrödinger Operator and the Dirichlet Laplacian

    • Vladimir Maz’ya

    Pages 769-801

20. Back Matter

    Pages 803-866

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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.

• 46E35, 42B37, 26D10
• Sobolev spaces
• general domains
• integral inequalities
• isoperimetric and isocapacitary inequalities
• spectrum of differential operators

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)

• Department of Mathematics, Linköping University, Linköping, Sweden

  Vladimir Maz'ya

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.

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