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 and applied mathematics but also to postgraduate students.

Contents:

• Introduction to Sobolev Spaces for Domains: Basic Properties of Sobolev Spaces
• Examples of "Bad" Domains in the Theory of Sobolev Space
• Sobolev Spaces for Domains Depending on Parameters: Extension of Functions Defined on Parameter Dependent Domains
• Boundary Values of Functions with First Derivatives L_p on Parameter Dependent Domains
• Sobolev Spaces for Domains with Cusps: Extension of Functions to the Exterior of a Domain with the Vertex of a Peak on the Boundary
• Boundary Values of Sobolev Functions on Non-Lipschitz Domains Bounded by Lipschitz Surfaces
• Boundary Values of Functions in Sobolev Spaces for Domains with Peaks
• Imbedding and Trace Theorems for Domains with Outer Peaks and for General Domains

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