Differentiable functions on bad domains
The spaces of functions with derivatives in L<sub>p</sub>, called the Sobolev spaces, play an important role in modern analysis. During the last decades, these spaces have been intensively studied and by now many problems associated with them have been solved. However, the theory of these function classes for domains with nonsmooth boundaries is still in an unsatisfactory state. In this book, which partially fills this gap, certain aspects of the theory of Sobolev spaces for domains with singul…
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- Full title :Differentiable functions on bad domains / Vladimir G. Mazʹya, Sergei V. Poborchi.
Summary
The spaces of functions with derivatives in L<sub>p</sub>, called the Sobolev spaces, play an important role in modern analysis. During the last decades, these spaces have been intensively studied and by now many problems associated with them have been solved. However, the theory of these function classes for domains with nonsmooth boundaries is still in an unsatisfactory state. In this book, which partially fills this gap, certain aspects of the theory of Sobolev spaces for domains with singularities are studied. We mainly focus on the so-called imbedding theorems, extension theorems and trace theorems that have numerous applications to partial differential equations. Some of such applications are given. Much attention is also paid to counter examples showing, in particular, the difference between Sobolev spaces of the first and higher orders. A considerable part of the monograph is devoted to Sobolev classes for parameter dependent domains and domains with cusps, which are the simplest non-Lipschitz domains frequently used in applications. This book will be interesting not only to specialists in analysis but also to postgraduate students.
Read More ↓Edition Statements
- Publication Date:©1997.
Publication Statements
- Publisher(s):World Scientific
- Place of Publication:Singapore
- Language:eng
- Publication Date:©1997.
The Physical Object
- Description:1 online resource (xviii, 481 pages) : : illustrations
Edition Identifiers
- ISBN:9789812817280, 981281728X
- oclc:ocn841809909
- LC Classification:QA323
- Dewey Decimal:515/.353
Table of Contents
I. Introduction to Sobolev spaces for domains. 1. Basic properties of Sobolev spaces. 1.1. Preliminaries. 1.2. Functions with generalized derivatives. 1.3. Classes of domains. 1.4. Density of smooth functions in Sobolev spaces. 1.5. Poincare's inequality and equivalent norms in Sobolev spaces. 1.6. Extendability of functions in Sobolev spaces. 1.7. Change of coordinates for Sobolev functions. 1.8. Summability and continuity of functions in Sobolev spaces. 1.9. Equivalence of integral and isoperimetric inequalities. 1.10. Compactness theorems. 1.11. The maximal algebra in [symbol]. 1.12. Application to the Neumann problem for elliptic operators of arbitrary order -- 2. Examples of "bad" domains in the theory of Sobolev spaces. 2.1. The property [symbol] does not ensure the density of [symbol] in Sobolev spaces. 2.2. Functions with bounded gradients are not always dense in [symbol]. 2.3. A planar bounded domain for which [symbol] is not dense in [symbol]. 2.4. On density of bounded functions in [symbol] for paraboloids in [symbol]. 2.5. Imbedding and compactness properties may fail for the intersection of "good" domains. 2.6. A domain for which the imbedding [symbol] is continuous b…
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