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**Analyse mathématique et calcul numérique pour les sciences et les techniques. Tomes 1, 2, 3.**
*(French)*
Zbl 0642.35001

Collection du Commissariat à l’Énergie Atomique. Série Scientifique. Paris etc.: Masson. Tome 1: XXIII, 1410 p.; Tome 2: XV, 1062 p.; Tome 3: XX, 1302 p.; FF 1.840.00/set (1985).

The purpose of the present work is to give a thorough and self-contained treatise on the theory and applications of partial differential equations in mathematics, physics, engineering, and technology.

The contents of the three volumes is the following. Chapter 1 provides the main physical examples which motivated the development of the theory of partial differential equations. First, some typical physical models are presented (classical fluid dynamics, Navier-Stokes equations, linear elasticity, linear visco-elasticity, electromagnetism, Maxwell’s equations, neutron transport equations, diffusion equations, quantum physics, classical mechanics etc.); afterwards, some mathematical notions are discussed (classification of partial differential equations, boundary value problems, initial value problems etc.). Chapter 2 is entirely devoted to the Laplace operator. Harmonic functions, Newton potentials, and classical solutions of the Dirichlet problem are treated in detail, as well as regularity questions and more general second order elliptic equations. One of the basic tools in solving partial differential equations is that of integral transforms which are discussed in detail in Chapter 3. Apart from Mellin and Hankel transforms, particular emphasize is put, of course, on Fourier series and integrals, especially Fast Fourier Transforms. Chapter 4 is concerned with the theory of Sobolev spaces (imbedding theorems, density theorems, trace theorems etc.). Beginning with Chapter 5, the general theory of linear differential operators is treated. First, some basic facts on operators with constant coefficients are discussed, such as the solvability of the Cauchy problem and the regularity of solutions. Afterwards, some aspects of general operator theory in Banach and Hilbert spaces are dealt with in Chapter 6; here the authors succeeded in collecting the basic facts of a first course in functional analysis on less than hundred pages. Chapter 7 is concerned with (linear) variational problems emphasizing, in particular, second order elliptic problems and regularity. The following chapter (on spectral theory) is also purely “functional analytic” and treats the usual functional calculus and spectral decompositions for both bounded and unbounded operators. As an application, Sturm-Liouville problems are discussed. Chapter 9 gives the first “rigorous” applications to two classical fields of mathematical physics. In the first part, some examples in electromagnetism are treated, in the second part some examples in quantum physics, in particular N-particle Hamiltonian systems. The short Chapter 10 is concerned with mixed problems, especially the theory and applications of Tricomi’s problem, while Chapter 11 gives a brief survey on the integral equation method. In view of applications to elliptic problems, the main part is devoted to singular equations.

So far, various important fields of functional analysis, partial differential equations, and mathematical physics are treated. Beginning with Chapter 12, the authors discuss numerical methods as well. Numerical methods for stationary equations are treated, especially finite element methods and approximation methods for eigenvalues and eigenvectors, in Chapter 12, applications of the finite element method to (singular) integral equations in Chapter 13. The next and all following chapters are concerned with evolution problems. In Chapter 14, the Cauchy problem is discussed for various standard equations of mathematical physics, such as diffusion equations, wave equations, and Schrödinger equations, while Chapter 15 deals with the Fourier (“diagonalization”) method for such equations. Apart from application to abstract first and second order (in time) evolution equations, also those to typical physical problems (hydrodynamics, diffusion and transport equations, wave propagation, Schrödinger equations etc.) are treated. The semi-group approach is discussed in Chapter 17, illuminating also important connections with the Cauchy problem. Chapter 18 is concerned with variational methods (vector distributions, Galerkin approximation, first order and second order problems) to evolution equations. Some classes of (linearized) Navier- Stokes equations are discussed in Chapter 19 for both the stationary and evolutionary case. The very detailed Chapter 20 is devoted to numerical method for evolution problems. After providing the necessary preliminaries, the authors discuss first and second order (in time) problems, in general, and the Stokes problem, in particular. Various transport equations (neutron transport etc.) are considered in the last Chapter 21, including “reflection conditions” and approximation by diffusion equations.

Each of these 21 chapters is supplemented with several appendices, a bibliography, and a brief summary which allows the reader to grasp the essentials of the ideas and results by reading first “once slightly over”. The last volume closes with a certain “outlook”, a useful guide to the reader, a list of the main equations, a list of symbols, and a subject index. In spite of the enormous size of the work, the reader will find real pleasure in working with the book by using these facilities. The reviewer should confess that he did not bother to read the three volumes cover-to-cover; certainly, this was not author’s intention. By its encyclopedic character, this work is rather meant as an extremely detailed reference-book on the whole body of the contemporary theory and applications of partial differential equations from both the analytic and numerical viewpoint. Without any doubt, this will become the “Dunford- Schwartz” of differential equations and physics. Except for L. Hörmander’s 4 volumes on the very advanced theory of pseudo- differential operators, perhaps, there is no comparable “chef-d’oeuvre” in the literature on partial differential equations all over the world.

The contents of the three volumes is the following. Chapter 1 provides the main physical examples which motivated the development of the theory of partial differential equations. First, some typical physical models are presented (classical fluid dynamics, Navier-Stokes equations, linear elasticity, linear visco-elasticity, electromagnetism, Maxwell’s equations, neutron transport equations, diffusion equations, quantum physics, classical mechanics etc.); afterwards, some mathematical notions are discussed (classification of partial differential equations, boundary value problems, initial value problems etc.). Chapter 2 is entirely devoted to the Laplace operator. Harmonic functions, Newton potentials, and classical solutions of the Dirichlet problem are treated in detail, as well as regularity questions and more general second order elliptic equations. One of the basic tools in solving partial differential equations is that of integral transforms which are discussed in detail in Chapter 3. Apart from Mellin and Hankel transforms, particular emphasize is put, of course, on Fourier series and integrals, especially Fast Fourier Transforms. Chapter 4 is concerned with the theory of Sobolev spaces (imbedding theorems, density theorems, trace theorems etc.). Beginning with Chapter 5, the general theory of linear differential operators is treated. First, some basic facts on operators with constant coefficients are discussed, such as the solvability of the Cauchy problem and the regularity of solutions. Afterwards, some aspects of general operator theory in Banach and Hilbert spaces are dealt with in Chapter 6; here the authors succeeded in collecting the basic facts of a first course in functional analysis on less than hundred pages. Chapter 7 is concerned with (linear) variational problems emphasizing, in particular, second order elliptic problems and regularity. The following chapter (on spectral theory) is also purely “functional analytic” and treats the usual functional calculus and spectral decompositions for both bounded and unbounded operators. As an application, Sturm-Liouville problems are discussed. Chapter 9 gives the first “rigorous” applications to two classical fields of mathematical physics. In the first part, some examples in electromagnetism are treated, in the second part some examples in quantum physics, in particular N-particle Hamiltonian systems. The short Chapter 10 is concerned with mixed problems, especially the theory and applications of Tricomi’s problem, while Chapter 11 gives a brief survey on the integral equation method. In view of applications to elliptic problems, the main part is devoted to singular equations.

So far, various important fields of functional analysis, partial differential equations, and mathematical physics are treated. Beginning with Chapter 12, the authors discuss numerical methods as well. Numerical methods for stationary equations are treated, especially finite element methods and approximation methods for eigenvalues and eigenvectors, in Chapter 12, applications of the finite element method to (singular) integral equations in Chapter 13. The next and all following chapters are concerned with evolution problems. In Chapter 14, the Cauchy problem is discussed for various standard equations of mathematical physics, such as diffusion equations, wave equations, and Schrödinger equations, while Chapter 15 deals with the Fourier (“diagonalization”) method for such equations. Apart from application to abstract first and second order (in time) evolution equations, also those to typical physical problems (hydrodynamics, diffusion and transport equations, wave propagation, Schrödinger equations etc.) are treated. The semi-group approach is discussed in Chapter 17, illuminating also important connections with the Cauchy problem. Chapter 18 is concerned with variational methods (vector distributions, Galerkin approximation, first order and second order problems) to evolution equations. Some classes of (linearized) Navier- Stokes equations are discussed in Chapter 19 for both the stationary and evolutionary case. The very detailed Chapter 20 is devoted to numerical method for evolution problems. After providing the necessary preliminaries, the authors discuss first and second order (in time) problems, in general, and the Stokes problem, in particular. Various transport equations (neutron transport etc.) are considered in the last Chapter 21, including “reflection conditions” and approximation by diffusion equations.

Each of these 21 chapters is supplemented with several appendices, a bibliography, and a brief summary which allows the reader to grasp the essentials of the ideas and results by reading first “once slightly over”. The last volume closes with a certain “outlook”, a useful guide to the reader, a list of the main equations, a list of symbols, and a subject index. In spite of the enormous size of the work, the reader will find real pleasure in working with the book by using these facilities. The reviewer should confess that he did not bother to read the three volumes cover-to-cover; certainly, this was not author’s intention. By its encyclopedic character, this work is rather meant as an extremely detailed reference-book on the whole body of the contemporary theory and applications of partial differential equations from both the analytic and numerical viewpoint. Without any doubt, this will become the “Dunford- Schwartz” of differential equations and physics. Except for L. Hörmander’s 4 volumes on the very advanced theory of pseudo- differential operators, perhaps, there is no comparable “chef-d’oeuvre” in the literature on partial differential equations all over the world.

Reviewer: J.Appell

### MSC:

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

31-02 | Research exposition (monographs, survey articles) pertaining to potential theory |

65-02 | Research exposition (monographs, survey articles) pertaining to numerical analysis |

34-02 | Research exposition (monographs, survey articles) pertaining to ordinary differential equations |

41-02 | Research exposition (monographs, survey articles) pertaining to approximations and expansions |

42-02 | Research exposition (monographs, survey articles) pertaining to harmonic analysis on Euclidean spaces |

44-02 | Research exposition (monographs, survey articles) pertaining to integral transforms |

70-02 | Research exposition (monographs, survey articles) pertaining to mechanics of particles and systems |

74-02 | Research exposition (monographs, survey articles) pertaining to mechanics of deformable solids |

76-02 | Research exposition (monographs, survey articles) pertaining to fluid mechanics |

78-02 | Research exposition (monographs, survey articles) pertaining to optics and electromagnetic theory |

80-02 | Research exposition (monographs, survey articles) pertaining to classical thermodynamics |