##
**Sobolev spaces of fractional order, Nemytskij operators and nonlinear partial differential equations.**
*(English)*
Zbl 0873.35001

de Gruyter Series in Nonlinear Analysis and Applications. 3. Berlin: de Gruyter. x, 547 p. (1996).

As the title indicates, this book treats three topics, which are not only of interest on their own, but also in view of their interdependencies: Sobolev-type spaces, nonlinear operators, and partial differential equations. It is common sense that, apart from Hölder spaces, Sobolev spaces and their various generalizations provide the most natural and adequate setting for the study of both linear and nonlinear partial differential equations. In practice it is a useful device to reformulate the specific differential equation, possibly subject to initial or boundary conditions, as an abstract operator equation in such spaces. Afterwards, one has to have as much information as possible on the analytical and topological properties of the operators involved in order to apply the large arsenal of methods of linear or nonlinear functional analysis. The purpose of the present book is to provide a systematic treatment of this problem in the setting of Sobolev spaces of fractional order.

To give an idea of the topics covered in the book, let us go through the table of contents. The first chapter is concerned with function spaces of Besov-Triebel-Lizorkin type. Various aspects of these spaces (basic properties, duality, embedding theorems, equivalent characterizations), as well as their role in interpolation theory are discussed. The next chapter treats regular elliptic boundary value problems; in particular, estimates for the Poisson integral in scales of Besov-Triebel-Lizorkin spaces are given. As a m0tter of fact, pointwise multiplication is one of the most important operations on Sobolev-type function spaces; it may also be viewed as a “harmless” nonlinear operator. The authors discuss this problem in detail in the third chapter; more precisely, they are interested in conditions under which the product of several functions from given spaces belongs to another space, and the corresponding embedding is continuous.

Any nonlinearity generates a corresponding Nemytskij operator whose properties depend, interestingly, more on the underlying function space than on the nonlinear function involved. When the first monograph (by P. P. Zabrejko and the reviewer) on Nemytskij operators appeared [Nonlinear superposition operators, Cambridge Univ. Press, Cambridge (1990; Zbl 0701.47041)], almost nothing was known on Nemytskij operators in Sobolev spaces. Now the theory is very advanced and almost complete, mainly due to the authors’ fundamental contributions to the field, and the fifth chapter of this book gives a detailed account of the state-of-the-art.

Finally, the sixth chapter is concerned with applications to semilinear elliptic boundary value problems, including Landesman-Lazer type results, Kazdan-Warner type results, and Ambrosetti-Prodi type results. The book closes with a list of almost 400 references and a subject index.

It is the authors’ merit to have accumulated a vast amount of material which has not yet been presented in book form, but has only been scattered over many research articles, sometimes standard, sometimes not easily accessible. The book is again an impressive proof of the incredible scientific activity of the Jena school, contributing to functional analysis, operator theory, nonlinear analysis, and differential equations. It will certainly become a standard reference for all specialists working in one of these fields, and it should be found at least in every math library.

To give an idea of the topics covered in the book, let us go through the table of contents. The first chapter is concerned with function spaces of Besov-Triebel-Lizorkin type. Various aspects of these spaces (basic properties, duality, embedding theorems, equivalent characterizations), as well as their role in interpolation theory are discussed. The next chapter treats regular elliptic boundary value problems; in particular, estimates for the Poisson integral in scales of Besov-Triebel-Lizorkin spaces are given. As a m0tter of fact, pointwise multiplication is one of the most important operations on Sobolev-type function spaces; it may also be viewed as a “harmless” nonlinear operator. The authors discuss this problem in detail in the third chapter; more precisely, they are interested in conditions under which the product of several functions from given spaces belongs to another space, and the corresponding embedding is continuous.

Any nonlinearity generates a corresponding Nemytskij operator whose properties depend, interestingly, more on the underlying function space than on the nonlinear function involved. When the first monograph (by P. P. Zabrejko and the reviewer) on Nemytskij operators appeared [Nonlinear superposition operators, Cambridge Univ. Press, Cambridge (1990; Zbl 0701.47041)], almost nothing was known on Nemytskij operators in Sobolev spaces. Now the theory is very advanced and almost complete, mainly due to the authors’ fundamental contributions to the field, and the fifth chapter of this book gives a detailed account of the state-of-the-art.

Finally, the sixth chapter is concerned with applications to semilinear elliptic boundary value problems, including Landesman-Lazer type results, Kazdan-Warner type results, and Ambrosetti-Prodi type results. The book closes with a list of almost 400 references and a subject index.

It is the authors’ merit to have accumulated a vast amount of material which has not yet been presented in book form, but has only been scattered over many research articles, sometimes standard, sometimes not easily accessible. The book is again an impressive proof of the incredible scientific activity of the Jena school, contributing to functional analysis, operator theory, nonlinear analysis, and differential equations. It will certainly become a standard reference for all specialists working in one of these fields, and it should be found at least in every math library.

Reviewer: J.Appell (Würzburg)

### MSC:

35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |

46-02 | Research exposition (monographs, survey articles) pertaining to functional analysis |

46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |

47H30 | Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.) |

35Jxx | Elliptic equations and elliptic systems |

35Sxx | Pseudodifferential operators and other generalizations of partial differential operators |