مسئله معکوس پتانسیل لگاریتمی ۲۰۱۴
Inverse Logarithmic Potential Problem 2014

1. Introduction

2. Chapter 1. Formulation of Inverse Logarithmic Potential Problem. Fundamental Equation

3. §1.1 Formulation of inverse problem

4. §1.2 Nonlinear boundary value problem for mapping function

5. §1.3 The analytic continuation of the potential across a boundary

6. §1.4 The boundary analyticity of domain is a solution to an inverse problem

7. §1.5 The structure of inverse problem solution. final solvability. Examples

8. Chapter 2. Local Solvability of an Inverse Problem

9. §2.1 Univalent function variation

10. §2.2 Local theorem of existence

11. §2.3 Linearization of the boundary value problem

12. §2.4 The auxiliary problem

13. §2.5 The Newton-Kantorovitch method

14. §2.6 The explicit solution of the linear problem

15. §2.7 The local uniqueness theorem

16. §2.8 The density variation. The equivalent solution set of the inverse problems. Remarks

17. §2.9 The complex-valued density case

18. §2.10 Existence theorems for the inverse problem for small constant densities

19. §2.11 Proof of theorems

20. Chapter 3. The Estimate of Bounded Univalent Function Coefficients and Univalent Polynomials

21. §3.1 Classical estimates. Classes of bounded functions and with bounded image area

22. §3.2 The estimate of univalent polynomials coefficients

23. §3.3 The Diedonne-Horowitz inequalities for univalent polynomials

24. §3.4 Numerical estimates of univalent polynomials coefficients

25. Chapter 4. Mass Potential Estimates. Necessary Conditions for Solvability. A Priori Estimates for Inverse Problem Solution

26. §4.1 Exact estimates for a mass potential gradient in the three-dimensional case. Extremal domain

27. §4.2 Exact estimates of logarithmic mass potential

28. §4.3 A priori estimates for inverse potential problem solution

29. §4.4 On zeros of a potential mass gradient

30. §4.5 Estimates of mass potential derivatives in a fixed angle

31. §4.6 The estimates of the mass potential derivatives in the disk

32. §4.7 The estimate of the mass potential based on the Calderon-Zygmund results for the singular integral

33. §4.8 The necessary solvability conditions, a priori estimates — using the univalent function theory

34. Chapter 5. The Continuation by the Parameter of an Inverse Problem Solution

35. §5.1 The dependence of an inverse problem solution on the parameter - a constant density

36. §5.2 The theorem on the continuation of a solution by the parameter

37. §5.3 Inverse potential problems and univalent functions

38. Chapter 6. On the Analyticity and Smoothness of an Inverse Problem Solution

39. §6.1 Theorem on the smoothness of inverse problem solutions

40. §6.2 Applications of the theorem on smoothness, in connection with free boundary smoothness

41. §6.3 Analytical continuation of the potential through the angle points

42. Chapter 7. Inverse Linear Problem. Determination of a Density of the Given Domain by its Exterior Potential

43. §7.1 Existence theorems. The particular solutions of the inhomogeneous problem construction

44. §7.2 Solution of the homogeneous problem. Density of the vanishing external potential

45. §7.3 Special classes of domains

46. §7.4 Determination of a body’s density by the given potential of an elliptic equation

47. §7.5 Linear inverse problem in the classes Lp

48. Chapter 8. Conjugation of Harmonic and Analytic Functions: Direct and Inverse Problems

49. §8.1 Problems of the linear conjugation for harmonic and analytic functions

50. §8.2 The particular cases, modifications, applications and generalization of the base conjugation problem

51. §8.3 Formulation of inverse problems

52. §8.4 Applied inverse problems

53. §8.5 Interior inverse problems

54. Chapter 9. Applications in Gravity Prospecting and in Magnetic Prospecting

55. §9.1 Setting of problems. The algorithm for numerical construction of the equivalent solutions set by the analytically given field

56. §9.2 On one approximation method (analytic continuation) for gravitational fields

57. §9.3 Examples of the numerical constructing of the equivalent solutions family

58. §9.4 Approximation of the anomaly field and determining the object by the random search method

59. Bibliography