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.
Author(s): Vladimir Maz'ya (auth.)
Series: Grundlehren der mathematischen Wissenschaften 342
Edition: 2
Publisher: Springer-Verlag Berlin Heidelberg
Year: 2011
Language: English
Pages: 866
Tags: Analysis
Front Matter....Pages I-XXVIII
Basic Properties of Sobolev Spaces....Pages 1-121
Inequalities for Functions Vanishing at the Boundary....Pages 123-229
Conductor and Capacitary Inequalities with Applications to Sobolev-Type Embeddings....Pages 231-253
Generalizations for Functions on Manifolds and Topological Spaces....Pages 255-286
Integrability of Functions in the Space $L^{1}_{1}(\varOmega )$ ....Pages 287-321
Integrability of Functions in the Space $L^{1}_{p}(\varOmega )$ ....Pages 323-404
Continuity and Boundedness of Functions in Sobolev Spaces....Pages 405-434
Localization Moduli of Sobolev Embeddings for General Domains....Pages 435-458
Space of Functions of Bounded Variation....Pages 459-509
Certain Function Spaces, Capacities, and Potentials....Pages 511-548
Capacitary and Trace Inequalities for Functions in ℝ n with Derivatives of an Arbitrary Order....Pages 549-609
Pointwise Interpolation Inequalities for Derivatives and Potentials....Pages 611-655
A Variant of Capacity....Pages 657-668
Integral Inequality for Functions on a Cube....Pages 669-692
Embedding of the Space $\mathaccent"7017{L}^{l}_{p}(\varOmega)$ into Other Function Spaces....Pages 693-735
Embedding $\mathaccent "7017{L}^{l}_{p}(\varOmega, \nu) \subset W^{m}_{r}(\varOmega)$ ....Pages 737-753
Approximation in Weighted Sobolev Spaces....Pages 755-768
Spectrum of the Schrödinger Operator and the Dirichlet Laplacian....Pages 769-801
Back Matter....Pages 803-866
