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Sobolev spaces. With applications to elliptic partial differential equations. Transl. from the Russian by T. O. Shaposhnikova. 2nd revised and augmented ed. (English) Zbl 1217.46002
Grundlehren der Mathematischen Wissenschaften 342. Berlin: Springer (ISBN 978-3-642-15563-5/hbk; 978-3-642-15564-2/ebook). xxviii, 866 p. (2011).
The material contained in this volume is an expanded and revised version of [V. G. Maz’ya, “Sobolev spaces” (Berlin etc.: Springer-Verlag) (1985; Zbl 0692.46023)]. This new edition of the book 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.
The contents of the volume is divided into 18 chapters, as follows: 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 $$L_1^1(\Omega)$$; 6. Integrability of Functions in the Space $$L_p^1(\Omega)$$; 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 $${\mathbb R}^n$$ 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 $$L^l_p(\Omega)$$ into Other Function Spaces; 16. Embedding $$L^l_p(\Omega,\nu)\subset W_r^m(\Omega)$$; 17. Approximation in Weighted Sobolev Spaces; 18. Spectrum of the Schrödinger Operator and the Dirichlet Laplacian.
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.

##### MSC:
 46-02 Research exposition (monographs, survey articles) pertaining to functional analysis 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
##### Keywords:
Sobolev space; Sobolev embeddings; Poincaré inequality; density
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