Singular Integral Equations

Name: Singular Integral Equations | Boundary Problems of Function Theory and Their Application to Mathematical Physics
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Boundary Problems of Function Theory and Their Application to Mathematical Physics

N. I. Muskhelishvili(Autor*in)

Dover Publications (Verlag)

Erschienen am 19. Februar 2013

887 Seiten

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Intro
DOVER BOOKS ON MATHEMATICS
Title Page
Copyright Page
ANNOTATION
EDITOR'S PREFACE
Table of Contents
PREFACE
INTRODUCTION
PART I - FUNDAMENTAL PROPERTIES OF CAUCHY INTEGRALS
CHAPTER 1 - THE HÖLDER CONDITION
§ 1. Smooth and piecewise smooth lines.
§ 2. Some properties of smooth lines.
§ 3. The Hölder Condition (H condition).
§ 4. Generalization to the case of several variables.
§ 5. Two auxiliary inequalities.
§ 6. Sufficient conditions for the H condition to be satisfied.
§ 7. Sufficient conditions for the H condition to be satisfied (continued).
§ 8. Sufficient conditions for the H condition to be satisfied (continued).
CHAPTER 2 - INTEGRALS OF THE CAUCHY TYPE
§ 9. Definitions.
§ 10. The Cauchy integral.
§ 11. Connection with logarithmic potential. Historical remarks.
§ 12. The values of Cauchy integrals on the path of integration.
§ 18. The tangential derivative of the potential of a simple layer.
§ 14. Sectionally continuous functions.
§ 15. Sectionally holomorphic functions.
§ 16. The limiting value of a Cauchy integral.
§ 17. The Plemelj formulae.
§ 18. Generalization of the formulae for the difference in limiting values.
§ 19. The continuity behaviour of the limiting values.
§ 20. The continuity behaviour of the limiting values (continued).
§ 21. On the behaviour of the derivative of a Cauchy integral near the boundary.
§ 22. On the behaviour of a Cauchy integral near the boundary ) .
CHAPTER 3 - SOME COROLLARIES ON CAUCHY INTEGRALS
§ 28. Poincaré-Bertrand transformation formula.
§ 24. On analytic continuation of a function given on the boundary of a region.
§ 25. Generalization of Harnack's theorem.
§ 26. Determination of sectionally holomorphic functions with given discontinuities (case of contours).
§ 27. Inversion of the Cauchy integral (case of contours).
§ 28. The Hilbert inversion formulae.
CHAPTER 4 - CAUCHY INTEGRALS NEAR ENDS OF THE LINE OF INTEGRATION)
§ 29. Statement of the principal results.
§ 80. An auxiliary estimate.
§ 81. Deduction of formula (29.5).
§ 82. Deduction of formula (29.8).
§ 33. On the behaviour of a Cauchy integral near points of discontinuity.
PART II - THE HILBERT AND THE RIEMANN-HILBERT PROBLEMS AND SINGULAR INTEGRAL EQUATIONS (CASE OF CONTOURS)
CHAPTER 5 - THE HILBERT AND RIEMANN-HILBERT BOUNDARY PROBLEMS
§ 34. The homogeneous Hilbert problem.
§ 85. General solution of the homogeneous Hilbert problem. The Index.
§ 36. Associate homogeneous Hilbert problems.
§ 37. The non-homogeneous Hilbert problem.
§ 38. On the extension to the whole plane of analytic functions given on a circle or half-plane.
§ 89. The Riemann-Hilbert problem.
§ 40. Solution of the Rlemann-Hilbert problem for the circle.
§ 41. Example: The Dirichlet problem for a circle.
§ 42. Reduction of the general case to that of a circular region.
§ 48. The Riemann-Hilbert problem for the half-plane.
CHAPTER 6 - SINGULAR INTEGRAL EQUATIONS WITH CAUCHY TYPE KERNELS (CASE OF CONTOURS)
§ 44. Singular equations and singular operators.
§ 45. Fundamental properties of singular operators.
§ 46. Adjoint operators and adjoint equations.
§ 47. Solution of the dominant equation.
§ 48. Solution of the equation adjoint to the dominant equation.
§ 49. Some general remarks).
§ 50. On the reduction of a singular integral equation.
§ 51. On the reduction of a singular integral equation (continued).
§ 52. On the resolvent of the Fredholm equation.
§ 58. Fundamental theorems.
§ 54. Real equations.
§ 55. I. N. Vekua's theorem of equivalence. An alternative proof of the fundamental theorems.
§ 56. Comparison of a singular integral equation with a Fredholm equation. The Quasi-Fredholm singular equation. Reduction to the canonical form.
§ 57. Method of reduction, due to T. Carleman and I.N. Vekua.
§ 58. Introduction of the parameter ?.
§ 59. Brief remarks on some other results.
PART III - APPLICATIONS TO SOME BOUNDARY PROBLEMS
CHAPTER 7 - THE DIRICHLET PROBLEM
§ 60. Statement of the Dirichlet and the modified Dirichlet problems. Uniqueness theorems.
§ 61. Solution of the modified Dirichlet problem by means of the potential of a double layer.
§ 62. Some corollaries.
§ 68. Solution of .the Dirichlet problem.
§ 64. Solution of the modified Dirichlet .problem, using the modified potential of a simple layer.
§ 65. Solution of the Dirichlet problem by the potential of a simple layer. Fundamental problem of electrostatics.
CHAPTER 8 - VARIOUS REPRESENTATIONS OF HOLOMORPHIC FUNCTIONS BY CAUCHY AND ANALOGOUS INTEGRALS
§ 66. General remarks.
§ 67. Representation by a Cauchy integral with real or imaginary density.
§ 68. Representation by a Cauchy integral with density of the form (a + ib)µ.
§ 69. Integral representation by I. N. Vekua.
CHAPTER 9 - SOLUTION OF THE GENERALIZED RIEMANN-HILBERT-POINCARÉ PROBLEM
§ 70. Preliminary remarks.
§ 71. The generalized Riemann-Hilbert-Poincaré problem (Problem V). Reduction to an integral equation.
§ 72. Investigation of the solubility of Problem V.
§ 73. Criteria of solubility of Problem V.
§ 74. The Poincaré Problem (Problem P).
§ 75. Examples.
§ 76. Some generalizations and applications.
PART IV - THE HILBERT PROBLEM IN THE CASE OF ARCS OR DISCONTINUOUS BOUNDARY CONDITIONS AND SOME OF ITS APPLICATIONS
CHAPTER 10 - THE HILBERT PROBLEM IN THE CASE OF ARCS OR DISCONTINUOUS BOUNDARY CONDITIONS)
§ 77. Definitions.
§ 78. Definition of a sectionally holomorphic function for a a given discontinuity.
79. The homogeneous Hilbert problem for open contours.
§ 80. The associate homogeneous Hilbert problem. Associate classes.
§ 81. Solution of the non-homogeneous Hilbert problem for arcs.
§ 82. The concept of the class h of functions given on L.
§ 88. Some generalizations.
§ 84. Examination of the problem F+ + F- = g.
§ 85. The Hilbert problem in the case of discontinuous coefficients.
§ 86. The Hilbert problem in the case of discontinuous coefficients (continued).
§ 87. Connection with the case of arcs.
CHAPTER 11 - INVERSION FORMULAE FOR ARCS )
§ 88. The inversion of a Cauchy integral.
§ 89. Some variations of the inversion problem.
§ 90. Some variations of the inversion problem (continued).
CHAPTER 12 - EFFECTIVE SOLUTION OF SOME BOUNDARY PROBLEMS OF THE THEORY OF HARMONIC FUNCTIONS
§ 91. Dirichlet and analogous problems for the plane with cuts distributed along a straight line.
§ 92. The Dirichlet and analogous problems for the plane with cuts distributed over a circle.
§ 93. The Rlemann-Hilbert problem for discontinuous coefficients.
§ 94. Particular cases: The mixed problem of the theory of holomorphic functions.
§ 95. The mixed problem for the half-plane. Formula of M. V. Keldysh and L. I. Sedov.
CHAPTER 13 - EFFECTIVE SOLUTION OF THE PRINCIPAL PROBLEMS OF THE STATIC THEORY OF ELASTICITY FOR THE HALF-PLANE, CIRCLE AND ANALOGOUS REGIONS
§ 96. General formulae of the plane theory of elasticity.
§ 97. The first, second and mixed boundary problems for an elastic half-plane. )
§ 98. The problem of pressure of rigid stamps on the boundary of an elastic half-plane in the absence of friction.
§ 99. The problem of pressure of rigid stamps on the boundary of an elastic half-plane in the absence of friction (continued).
§ 100. Equilibrium of a rigid stamp on the boundary of an elastic half-plane in the presence of friction.
§ 101. Another method of solution of the boundary problems for the half-plane.
§ 102. The problem of contact of two elastic bodies (the generalized plane problem of Hertz).
§ 108. The fundamental boundary problems for the plane with straight cuts. )
§ 104. The boundary problems for circular regions. )
§ 105. Certain analogous problems. Generalizations.
PART V - SINGULAR INTEGRAL EQUATIONS FOR THE CASE OF ARCS OR DISCONTINUOUS COEFFICIENTS AND SOME OF THEIR APPLICATIONS
CHAPTER 14 - SINGULAR INTEGRAL EQUATIONS FOR THE CASE OF ARCS AND CONTINUOUS COEFFICIENTS
§ 106. Definitions.
§ 107. Solution of the dominant equation.
§ 108. Solution of the equation adjoint to the dominant equation.
§ 109. Reduction of the singular equation K? = f.
§ 110. Reduction of the singular equation K´? = g.
§ 111. Investigation of the equation, resulting from the reduction.
§ 112. Solution of a singular equation. Fundamental theorems.
§ 113. Application to the dominant equation of the first kind.
§ 114. Reduction and solution of an equation of the first kind.
§ 115. An alternative method for the Investigation of singular equations.
CHAPTER 15 - SINGULAR INTEGRAL EQUATIONS IN THE CASE OF DISCONTINUOUS COEFFICIENTS
§ 116. Definitions.
§ 117. Reduction and solution of singular equations in the case of discontinuous coefficients.
CHAPTER 16 - APPLICATION TO THE DIRICHLET AND SIMILAR PROBLEMS )
§ 118. The Dirichlet and similar problems for the plane, cut along arcs of arbitrary shape.
§ 119. Reduction to a Fredholm equation. Examples.
§ 120. The Dirichlet problem for the plane, cut along a finite number of arcs of arbitrary shape.
CHAPTER 17 - SOLUTION OF THE INTEGRO-DIFFERENTIAL EQUATION OF THE THEORY OF AIRCRAFT WINGS OF FINITE SPAN )
§ 121. The integro-differential equation of the theory of aircraft wings of finite span.
§ 122. Reduction to a regular Fredholm equation.
§ 123. Certain generalizations.
PART VI - THE HILBERT PROBLEM FOR SEVERAL UNKNOWN FUNCTIONS AND SYSTEMS OF SINGULAR INTEGRAL EQUATIONS
CHAPTER 18 - THE HILBERT PROBLEM FOR SEVERAL UNKNOWN FUNCTIONS )
§ 124. Definitions.
§ 125. Auxiliary theorems.
§ 126. The homogeneous Hilbert problem.
§ 127. The fundamental system of solutions of the homogeneous Hilbert problem and its general solution.
§ 128. The non-homogeneous Hilbert problem.
§ 129. Supplement to the solution of a dominant system of singular integral equations and of its associate system.
CHAPTER 19 - SYSTEMS OF SINGULAR INTEGRAL EQUATIONS WITH CAUCHY TYPE KERNELS AND SOME SUPPLEMENTS
§ 130. Definitions. Auxiliary theorems.)
§ 131. Reduction of a system of singular equations. Fundamental theorems.
§ 132. Other methods of reduction and the investigation of systems of singular equations.
§ 133. Brief remarks regarding important generalizations and supplements.
APPENDICES
REFERENCES)
REFERENCES AND AUTHORS INDEX
INDEX

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