Mikhlin / Sneddon / Stark Multidimensional Singular Integrals and Integral Equations

1. Auflage 2014
ISBN: 978-1-4831-6449-6
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 172 Seiten, Web PDF

Reihe: International Series in Pure and Applied Mathematics

Multidimensional Singular Integrals and Integral Equations presents the results of the theory of multidimensional singular integrals and of equations containing such integrals. Emphasis is on singular integrals taken over Euclidean space or in the closed manifold of Liapounov and equations containing such integrals. This volume is comprised of eight chapters and begins with an overview of some theorems on linear equations in Banach spaces, followed by a discussion on the simplest properties of multidimensional singular integrals. Subsequent chapters deal with compounding of singular integrals; properties of the symbol, with particular reference to Fourier transform of a kernel and the symbol of a singular operator; singular integrals in Lp spaces; and singular integral equations. The differentiation of integrals with a weak singularity is also considered, along with the rule for the multiplication of the symbols in the general case. The final chapter describes several applications of multidimensional singular integral equations to boundary problems in mathematical physics. This book will be of interest to mathematicians and students of mathematics.

Autoren/Hrsg.

Mikhlin, S. G.

Sneddon, I. N.

Stark, M.

Ulam, S.

Inhaltsverzeichnis

1;Front Cover;1
2;Multidimensional Singular Integrals and Integral Equations;4
3;Copyright Page;5
4;Table of Contents;8
5;Dedication;6
6;Preface;12
7;CHAPTER I. INTRODUCTION;14
7.1;1. Review 0f previous work;14
7.2;2. Some theorems on linear equations in Banach spaces;34
7.3;3. Stereographic projection;43
7.4;4. Completely continuous operators;45
8;CHAPTER II. SIMPLEST PROPERTIES OF MULTIDIMENSIONAL SINGULAR INTEGRALS;50
8.1;5. Basic concepts;50
8.2;6. Lipschitz Conditions;59
8.3;7. Order of singular integrals at infinity;63
8.4;8. Differentiation of integrals with a weak singularity;72
9;CHAPTER III. COMPOUNDING OF SINGULAR INTEGRALS;76
9.1;9. Compounding 0f singular and ordinary integrals;76
9.2;10. Compounding of double singular integrals;80
9.3;11. The concept of a singular operator;83
9.4;12. Compounding of double singular integrals. The symbol;84
9.5;13. Compounding of multidimensional singular integrals;85
9.6;14. Formulae for reference;87
9.7;15. Product of the Operators A1 and An;90
9.8;16. Product it the operators A2 and An;94
9.9;17. Calculation of x1, m;96
9.10;18. Symbol of a multidimensional singular integral;99
10;CHAPTER IV. PROPERTIES OF THE SYMBOL;106
10.1;19. Fourier transform of a singular kernel;106
10.2;20. Fourier transform of a kernel and the symbol of a singular operator;110
10.3;21. Transformation of the symbol under change of variables;117
10.4;22. Differentiability of the symbol;122
10.5;23. The conditions for the continuity of the symbol;125
11;CHAPTER V. SINGULAR INTEGRALS IN Lp SPACES;129
11.1;24. The simplest corollaries from the Fourier transform. First theorem on boundedness in L2;129
11.2;25. Symbol dependent on the pole. Second theorem on bound edness in L2;132
11.3;26. On the boundedness of a singular integral operator in Lp,space;136
11.4;27. Integrals taken over any manifold;143
11.5;28. Differential properties of singular integrals;144
12;CHAPTER VI. FURTHER INVESTIGATION OF THE SYMBOL;147
12.1;29. More about the differentiation of integrals with a weak singularity;147
12.2;30. Polyharmonic potentials;148
12.3;31. Series of spherical functions;149
12.4;32. Differential properties of the symbol and the characteristic;161
12.5;33. Rule for multiplication of the symbols in the general case;163
12.6;34. Conjugate singular operator;167
13;CHAPTER VII. SINGULAR INTEGRAL EQUATIONS;170
13.1;35. The case where the symbol is independent of the pole;170
13.2;36. The case where the symbol is dependent on the pole. Regularization and domains of constancy of the index;171
13.3;37. Equivalent regularization. Index theorem;173
13.4;38. Equations with an integral taken over a closed manifold;185
13.5;39. Extension by means of the parameter;193
13.6;40. Systems of singular integral equations;197
13.7;41. Singular integral equations in classes of Lipschitz functions;203
14;CHAPTER VIII. MISCELLANEOUS APPLICATIONS;212
14.1;42. Leading derivatives of volume potential;212
14.2;43. Problem it the oblique derivative;216
14.3;44. Inequality involving the tangential and normal components of the gradient of a harmonic function;221
14.4;45. Equilibrium of an isotropic elastic body;223
14.5;46. Diffraction of stationary elastic waves;233
15;APPENDIX: MULTIPLIERS OF FOURIER INTEGRALS;238
16;BIBLIOGRAPHY;254
17;INDEX;264
18;OTHER TITLES IN THE SERIES IN PURE AND APPLIED MATHEMATICS;270