Sobolev Spaces: with Applications to Elliptic Partial Differential Equations (Grundlehren der mathematischen Wissenschaften Book 342)

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. 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 Annotation 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 rst appeared in German as three booklets of Teubner-Texte zur Mathematik (1979, 1980). In the Springer volume €œSobolev 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 signicantly augmented list of references aim to create a broader and modern view of the area 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.