Maz’ya V.G., Poborchi S.V. Differentiable Functions on Bad Domains

World Scientific Publ., 1997. — 502 p. — ISBN: 9810227671.The spaces of functions with derivatives in Lp, 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 primarily fills this gap, certain aspects of the theory of Sobolev spaces for domains with singularities are studied. The text focuses on the so-called imbedding theorems, extension theorems and trace theorems that have numerous applications to partial differential equations. Some 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 should be interesting not only to specialists in analysis and applied mathematics but also to postgraduate students.Introduction to sobolev spaces for domains
Basic Properties of Sobolev Spaces
Examples of "Bad" Domains in the Theory of Sobolev Spaces
Sobolev spaces for domains depending on parameters
Extension of Functions Defined on Parameter Dependent Domains
Boundary Values of Functions with First Derivatives in LP 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 Dom