Singapore: World Scientific. xviii, 481 p. £54.00 (1997).
The theory, methods, and applications of Sobolev spaces over “nicely behaved” domains in Euclidean space is very advanced. Less is known in the case of “badly behaved” domains, e.g. those with nonsmooth boundary, although such domains arise quite naturally in some applications. The aim of this monograph is to give a self-contained presentation of Sobolev and related spaces over such domains.
The contents is described by the authors in the preface as follows. The book consists of three parts. The first part contains two chapters and deals with Sobolev functions on either “nice” or general domains. Chapter 1 is an introduction to the theory of Sobolev spaces. The basic properties given here (including Sobolev’s imbedding theorem, density and extension theorems) are sufficient for many applications. In the second chapter, the authors give counterexamples showing that these properties may fail for unrestricted domains. Some of the counterexamples demonstrate the difference between Sobolev spaces of the first- and higher-orders, the latter being more sensitive to singularities of the boundary. The first part of the book is supplied with exercises that complement the text.
Part II is devoted to Sobolev spaces for domains depending on small or large parameters in such a way that the boundary of the domain loses smoothness as the parameters approach their limit values. Here, the authors are concerned with the speed at which extension and boundary trace operators degenerate. This requires that one determines explicitly the dependence of appropriate characteristics of these operators on the parameters. This analysis proves to be useful in attempts to justify the formal asymptotics of solutions to boundary value problems in domains with singularly perturbed boundaries as described, for example, the book by the first author, S. A. Nazarov and B. A. Plamenevskij [“Asymptotic of solutions to elliptic boundary value problems under singular perturbations of domains” (Russian) (1981; Zbl 0462.35001)]. In particular, in Chapter 3 sharp two-sided -dependent estimates for the norms of extension operators acting in Sobolev spaces are given for a small domain (of diameter ) and for a thin cylindrical layer (of width ). Chapter 4 contains (among other results) two-sided parameter dependent estimates for the norms of boundary trace operators for the exterior and the interior of a small domain and a thin cylinder. These estimates are used in subsequent chapters.
In Part III (Chapter 5-8) the authors study properties of Sobolev spaces for domains with cusps – the simplest non-Lipschitz domains frequently used in applications. The analysis is focused on extension, imbedding and compactness theorems, and on the description of boundary values of Sobolev functions. These theorems directly lead to conditions for the solvability of boundary value problems for partial differential equations and to theorems on the structure of the spectrum of corresponding differential operators as those discussed in the books by D. Gilbarg and N. S. Trudinger [“Elliptic partial differential equations of second-order” (1983; Zbl 0562.35001)] or the first author [“Sobolev spaces” (1985; Zbl 0692.46023)]. Some applications of this type are considered in Sections 7.5 and 8.3.2.
In the last chapter, the authors present imbedding and trace theorems for Sobolev spaces of the first-order for general domains. Necessary and sufficient conditions for the continuity and compactness of trace operators are given in terms of isoperimetric inequalities, an approach which was initiated by the first author in 1960.
This monograph contains a wealth of theorems, examples, and counterexamples on Sobolev spaces over non-standard domains. It should be useful to specialists in the theory of function spaces, in particular to those interested in more exotic aspects of partial differential equations.