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Selected Works of S. L. Sobolev: Volume I: Equations of Mathematical Physics, Computational Mathematics, and Cubature Formulas

Gennadii V. Demidenko and Vladimir L. Vaskevich, editors
Springer Verlag
Publication Date: 
Number of Pages: 
[Reviewed by
Fernando Q. Gouvêa
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Even I, who know very little about partial differential equations and related mathematics, recognize the name of S. L. Sobolev (1908–1989) as one of the most influential contributors to the field in the 20th century. This book is the first volume of his "selected works," first published in Russian on the 95th anniversary of Sobolev's birth and presented here in English translation.

The preface of this volume gives a survey of the papers here included, and the first article, "Academician S. L. Sobolev is a Founder of New Directions in Functional Analysis" by Yu. G. Reshetnyak, gives a short sketch of Sobolev's life and work. The papers are grouped into two sections, "Equations of Mathematical Physics" and "Computational Mathematics and Cubature Formulas"; within each section, the articles are printed in chronological order. Original publication information is given only on the first page of each article, which may make life harder for some people. On the other hand, they are all in English, which probably more than makes up for that. An index is provided, and will be welcome.

There is no question that all good research libraries will want to have a copy of this one.

Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College in Waterville, ME.



Academician S. L. Sobolev is a Founder of New Directions of Functional Analysis (by Yu. G. Reshetnyak)


Part I. Equations of Mathematical Physics

1. Application of the Theory of Plane Waves to the Lamb Problem

2. On a New Method in the Plane Problem on Elastic Vibrations

3. On Application of a New Method to Study Elastic Vibrations in a Space with Axial Symmetry

4. On Vibrations of a Half-Plane and a Layer with Arbitrary Initial Conditions

5. On a New Method of Solving Problems about Propagation of Vibrations

6. Functionally Invariant Solutions of the Wave Equation

7. General Theory of Diffraction of Waves on Riemann Surfaces

8. The Problem of Propagation of a Plastic State

9. On a New Problem of Mathematical Physics

10. On Motion of a Symmetric Top with a Cavity Filled with Fluid

11. On a Class of Problems of Mathematical Physics


Part II. Computational Mathematics and Cubature Formulas

1. Schwarz’s Algorithm in Elasticity Theory

2. On Solution Uniqueness of Difference Equations of Elliptic Type

3. On One Difference Equation

4. Certain Comments on the Numeric Solutions of Integral Equations

5. Certain Modern Questions of Computational Mathematics

6. Functional Analysis and Computational Mathematics

7. Formulas of Mechanical Cubatures in n-Dimensional Space

8. On Interpolation of Functions of n Variables

9. Various Types of Convergence of Cubature and Quadrature Formulas

10. Cubature Formulas on the Sphere Invariant under Finite Groups of Rotations

11. The Number of Nodes in Cubature Formulas on the Sphere

12. Certain Questions of the Theory of Cubature Formulas

13. A Method for Calculating the Coefficients in Mechanical Cubature Formulas

14. On the Rate of Convergence of Cubature Formulas

15. Theory of Cubature Formulas

16. Convergence of Approximate Integration Formulas for Functions from L2^(m)

17. Evaluation of Integrals of Infinitely Differentiable Functions

18. Cubature Formulas with Regular Boundary Layer

19. A Difference Analogue of the Polyharmonic Equation

20. Optimal Mechanical Cubature Formulas with Nodes on a Regular Lattice

21. Constructing Cubature Formulas with Regular Boundary Layer

22. Convergence of Cubature Formulas on Infinitely Differentiable Functions

23. Convergence of Cubature Formulas on the Elements of L2^(m)

24. The Coefficients of Optimal Quadrature Formulas

25. On the Roots of Euler Polynomials

26. On the End Roots of Euler Polynomials

27. On the Asymptotics of the Roots of the Euler Polynomials

28. More on the Zeros of Euler Polynomials

29. On the Algebraic Order of Exactness of Formulas of Approximate Integration