##
**The analysis of solutions of elliptic equations. Transl. from the Russian by P. M. Gauthier. Rev. and upd. transl.
Rev. and upd. transl.**
*(English)*
Zbl 0877.35002

Mathematics and its Applications (Dordrecht). 406. Dordrecht: Kluwer Academic Publishers. xx, 479 p. (1997).

This monograph is a translation by P. M. Gauthier from the Russian original, published in Nauka, Novosibirsk in 1991 with the title Laurent Series for Solutions of Elliptic Equations (Zbl 0743.35021).

The central concern of the book is with general elliptic equations, but also leads into some modern aspects of PDE and functional analysis, such as removable singularities, Laurent expansions, approximation by solutions, Carleman formulas, quasiconformality, etc. The author points out the distinction between results which hold in a very general setting (arbitrary elliptic equations with the unique continuation property) and those which hold under more restrictive assumptions on the differential operators (homogeneous, of first order).

The book starts with an interesting discussion of the subject of removable singularities and its relation to the theory of Hausdorff measures, capacity or metric properties of the capacity associated to Hölder spaces.

In the second chapter the author treats the theory of Laurent series and gives several interesting examples. In fact, the greater part of the material presented in this book is related to applications of Laurent series for a solution of a system of differential equations, which is a convenient way of writing the Green formula. The author goes into careful detail in discussing whether the concept of Laurent series was, in fact, beginning to take the shape of what has come to be known as analyticity of solutions of homogeneous systems (Petrovskii’s theorem). The concept of Laurent series for solutions of homogeneous elliptic differential equations with constant coefficients was introduced and developed by R. Harvey and J. C. Polking [Trans. Am. Math. Soc. 180, 407-413 (1973; Zbl 0285.35024)]. Moreover, they suggested how to choose the coefficients in order to guarantee their uniqueness. Also in the second chapter, Tarkhanov shows that very conclusive results in this direction hold in the context of elliptic complexes. Following an idea of Khavin, he constructs Laurent series for solutions of a homogeneous elliptic system with a compact set of singularities.

Chapter 3 is about the representation of solutions with non-discrete singularities. After introducing the topology in spaces of solutions of elliptic systems the author discusses the structure of solutions with compact singularities. We remark here Theorem 3.2.1 which is a wonderful representation result, initially due to Khavin, then generalized by Baernstein, Trutnev, and Elin in the framework of holomorphic functions of several variables and stated by Tarkhanov in the general case. We also mention in this chapter the treatment of the Green-type integral in a ball with applications to the solvability of the system \(Pu=f\) in such a case.

Chapter 4, entitled “Uniform Approximation”, starts with the Runge theorem and continues with Walsh type theorems. The culminating application is an analogue of a theorem of Vitushkin for uniform and mean approximation by solutions of an elliptic system. The constructive technique of A. G. Vitushkin was developed in [Usp. Mat. Nauk 22, No. 6(138), 141-199 (1967); English translation in Russ. Math. Surv. 22, 139-200 (1967; Zbl 0164.37701)]. The last part of Chapter 4 deals with a fundamental result (Theorem 4.5.3) which is initially due to Vitushkin for compact sets and Cauchy-Riemann operators in the plane, and which was completed by Sinanyan for nowhere dense compact sets. This theorem gives a complete description in terms of the capacity with respect to \(C^s(X)\) of those compacta \(K\subset\mathbb{R}^n\) on which each vector-valued function \(u\in \text{Sol} (\text{Int} K)\cap C^s(K)^k\) can be approximated with respect to the norm of \(C^s(K)^k\) by “rational” solutions of an appropriate system, with poles outside \(K\). An analogue of this result is given in the next chapter (Theorem 5.5.2) which is formulated in Sobolev spaces, instead of spaces of continuous functions.

The space of functions of bounded mean oscillation (BMO) was introduced by F. John and L. Nirenberg in their paper [Commun. Pure Appl. Math. 14, 415-426 (1961; Zbl 0102.04302)]. In Chapter 6 the author examines in what sense approximation within BMO may be seen as intermediate between the approximation theories in uniform and Sobolev spaces. The analysis which is carried out shows that the localized form of BMO is closely related to the local Hardy spaces introduced by D. Goldberg [Duke Math. J. 46, No. 1, 27-42 (1979; Zbl 0409.46060)].

Chapter 7, which is entitled “Conditional Stability”, starts with considerations on linear problems. The standard example of a conditionally stable, but not necessarily stable problem is due to Hadamard and concerns the Cauchy problem for the Laplace equation, in the case where the Cauchy data is given on a piece of the boundary. The author proves two Carleman type formulas and an inversion formula for Toeplitz operators.

In Chapter 8 the solvability of the Cauchy problem in Hardy spaces is studied. There are considered both cases of problems with data given on the whole boundary and on a part of the boundary.

The last chapter of this monograph deals with the notion of quasiconformality, which is closely linked to the theory of stability of conformal mappings. The author summarizes the essence of these studies as follows: “let \(f\) be a quasiconformal mapping. When can we state that \(f\) is globally close to a conformal map, in the same sense or in a different sense?” We point out here the study of the Beltrami equation and the variants of the classical theorems of Cauchy, Morera, and Liouville with respect to this framework.

The material included in this monograph is nicely organized and the book is pleasant to read. Many illustrative examples are presented. The text is largely self-contained but requires basic knowledge on distributions and pseudodifferential operators. It is a welcome addition to the literature and can be recommended to those who wish to learn more about the modern theory of elliptic equations.

The central concern of the book is with general elliptic equations, but also leads into some modern aspects of PDE and functional analysis, such as removable singularities, Laurent expansions, approximation by solutions, Carleman formulas, quasiconformality, etc. The author points out the distinction between results which hold in a very general setting (arbitrary elliptic equations with the unique continuation property) and those which hold under more restrictive assumptions on the differential operators (homogeneous, of first order).

The book starts with an interesting discussion of the subject of removable singularities and its relation to the theory of Hausdorff measures, capacity or metric properties of the capacity associated to Hölder spaces.

In the second chapter the author treats the theory of Laurent series and gives several interesting examples. In fact, the greater part of the material presented in this book is related to applications of Laurent series for a solution of a system of differential equations, which is a convenient way of writing the Green formula. The author goes into careful detail in discussing whether the concept of Laurent series was, in fact, beginning to take the shape of what has come to be known as analyticity of solutions of homogeneous systems (Petrovskii’s theorem). The concept of Laurent series for solutions of homogeneous elliptic differential equations with constant coefficients was introduced and developed by R. Harvey and J. C. Polking [Trans. Am. Math. Soc. 180, 407-413 (1973; Zbl 0285.35024)]. Moreover, they suggested how to choose the coefficients in order to guarantee their uniqueness. Also in the second chapter, Tarkhanov shows that very conclusive results in this direction hold in the context of elliptic complexes. Following an idea of Khavin, he constructs Laurent series for solutions of a homogeneous elliptic system with a compact set of singularities.

Chapter 3 is about the representation of solutions with non-discrete singularities. After introducing the topology in spaces of solutions of elliptic systems the author discusses the structure of solutions with compact singularities. We remark here Theorem 3.2.1 which is a wonderful representation result, initially due to Khavin, then generalized by Baernstein, Trutnev, and Elin in the framework of holomorphic functions of several variables and stated by Tarkhanov in the general case. We also mention in this chapter the treatment of the Green-type integral in a ball with applications to the solvability of the system \(Pu=f\) in such a case.

Chapter 4, entitled “Uniform Approximation”, starts with the Runge theorem and continues with Walsh type theorems. The culminating application is an analogue of a theorem of Vitushkin for uniform and mean approximation by solutions of an elliptic system. The constructive technique of A. G. Vitushkin was developed in [Usp. Mat. Nauk 22, No. 6(138), 141-199 (1967); English translation in Russ. Math. Surv. 22, 139-200 (1967; Zbl 0164.37701)]. The last part of Chapter 4 deals with a fundamental result (Theorem 4.5.3) which is initially due to Vitushkin for compact sets and Cauchy-Riemann operators in the plane, and which was completed by Sinanyan for nowhere dense compact sets. This theorem gives a complete description in terms of the capacity with respect to \(C^s(X)\) of those compacta \(K\subset\mathbb{R}^n\) on which each vector-valued function \(u\in \text{Sol} (\text{Int} K)\cap C^s(K)^k\) can be approximated with respect to the norm of \(C^s(K)^k\) by “rational” solutions of an appropriate system, with poles outside \(K\). An analogue of this result is given in the next chapter (Theorem 5.5.2) which is formulated in Sobolev spaces, instead of spaces of continuous functions.

The space of functions of bounded mean oscillation (BMO) was introduced by F. John and L. Nirenberg in their paper [Commun. Pure Appl. Math. 14, 415-426 (1961; Zbl 0102.04302)]. In Chapter 6 the author examines in what sense approximation within BMO may be seen as intermediate between the approximation theories in uniform and Sobolev spaces. The analysis which is carried out shows that the localized form of BMO is closely related to the local Hardy spaces introduced by D. Goldberg [Duke Math. J. 46, No. 1, 27-42 (1979; Zbl 0409.46060)].

Chapter 7, which is entitled “Conditional Stability”, starts with considerations on linear problems. The standard example of a conditionally stable, but not necessarily stable problem is due to Hadamard and concerns the Cauchy problem for the Laplace equation, in the case where the Cauchy data is given on a piece of the boundary. The author proves two Carleman type formulas and an inversion formula for Toeplitz operators.

In Chapter 8 the solvability of the Cauchy problem in Hardy spaces is studied. There are considered both cases of problems with data given on the whole boundary and on a part of the boundary.

The last chapter of this monograph deals with the notion of quasiconformality, which is closely linked to the theory of stability of conformal mappings. The author summarizes the essence of these studies as follows: “let \(f\) be a quasiconformal mapping. When can we state that \(f\) is globally close to a conformal map, in the same sense or in a different sense?” We point out here the study of the Beltrami equation and the variants of the classical theorems of Cauchy, Morera, and Liouville with respect to this framework.

The material included in this monograph is nicely organized and the book is pleasant to read. Many illustrative examples are presented. The text is largely self-contained but requires basic knowledge on distributions and pseudodifferential operators. It is a welcome addition to the literature and can be recommended to those who wish to learn more about the modern theory of elliptic equations.

Reviewer: V.Rădulescu (Craiova)

### MSC:

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

35Jxx | Elliptic equations and elliptic systems |

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

47G30 | Pseudodifferential operators |

58J40 | Pseudodifferential and Fourier integral operators on manifolds |