DIFFERENTIABLE FUNCTIONS ON BAD DOMAINS
Vladimir G Maz'ya (Linköping Univ.),
Sergei V Poborchi (St Petersburg State Univ.),
World
Scientific, 1997.
The spaces of functions with derivatives in L_{p}, 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 socalled 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 nonLipschitz 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 NonLipschitz 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
Readership: Mathematicians.
504pp 
Pub. date: Jan 1998 
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