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A perturbed integral geometry problem in a three-dimensional space. (English. Russian original) Zbl 0952.45001

Sib. Math. J. 41, No. 1, 1-12 (2000); translation from Sib. Mat. Zh. 41, No. 1, 3-14 (2000).
The article contains uniqueness theorems for the following integral equation in a function \(u(\xi)\): \[ \int_{p(x)}u(\xi) d {\overline {\xi}}+\int_{P(x)}W(x,\xi)u(\xi) d \xi = f(x), \tag{1} \] where \(W(x,\xi)\) is a given weight function, \( x\in \mathbb R^3 \), \( \xi \in \mathbb R^3 \), \({\overline {\xi}}=(\xi_1,\xi_2) \), \( p(x) \) is the projection of the paraboloid \({\mathcal {P}}(x)=\{\xi : x_3-\xi_3=|{\overline x}-{\overline \xi}|^2, \;0 \leq \xi_3 \leq h \}\) onto the plane \( x_3= 0 \), \( {\overline {x}}=(x_1,x_2) \), \( 0 < h < \infty \). It is supposed that the vertices \(x\) of the paraboloids \({\mathcal {P}}(x)\) lie in the layer \( S=\{x \in \mathbb R^3_+: 0 < x_3 < h \}\), \( \mathbb R^3_+=\{x=(x_1,x_2,x_3): x_3 \geq 0\} \). In (1), \( P(x) \) is the part of the layer \( \overline{S} \) which is bounded by the surface of \({\mathcal {P}}(x)\) and the plane \(x_3=0\), where \({\overline{S}}=\{x \in \mathbb R^3_+: 0 \leq x_3 \leq h \}\). Let \({\mathcal {A}}=\{x \in \mathbb R^3_+: |x_k|\leq a_k\) \((k=1,2)\), \(a_k < \infty\), \(0 < x_3 < h \}\).
The following theorem is the main result: A solution to (1) in the class \(C_0^3({\mathcal {A}})\) is unique if (a) the function \(f(x)\) is known for all \(x\in {\overline{S}}\) and (b) the weight function \(W\) has all continuous derivatives up to the second order and vanishes together with its derivatives on the surface of the paraboloid \({\mathcal {P}}(x)\).

MSC:

45A05 Linear integral equations
53C65 Integral geometry
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References:

[1] Lavrent’ev M. M., ”Integral geometry problems with perturbation on the plane,” Sibirsk. Mat. Zh.,37, No. 4, 851–857 (1996).
[2] Bukhgeîm A. L., ”On some integral geometry problems,” Sibirsk. Mat. Zh.,13, No. 1, 34–42 (1972).
[3] Lavrent’ev M. M. andSavel’ev L. Ya., Linear Operators and Ill-Posed Problems [in Russian], Nauka, Moscow (1991).
[4] Begmatov Akbar K., ”On one class of integral geometry problems on the plane,” Dokl. Akad. Nauk,331, No. 3, 261–262 (1993). · Zbl 0822.53040
[5] Begmatov Akbar K., ”Reduction of an integral geometry problem in three-dimensional space to a perturbed polysingular integral equation,” Dokl. Akad. Nauk,360, No. 5, 583–585 (1998). · Zbl 0976.53086
[6] Gradshteîn I. S. andRyzhik I. M., Tables of Integrals, Sums, Series, and Products [in Russian], Fizmatgiz, Moscow (1962).
[7] Handbook of Mathematical Functions with Formulas, M. Abramowitz and I. A. Stegun, eds [Russian translation], Nauka, Moscow (1979).
[8] Prudnikov A. P., Brychkov Yu. A., andMarichev O. A., Integrals and Series. Elementary Functions [in Russian], Nauka, Moscow (1983). · Zbl 0626.00033
[9] Lavrent’ev M. M., Romanov V. G., andShishatskiî S. P., Ill-Posed Problems of Mathematical Physics and Analysis [in Russian], Nauka, Moscow (1980).
[10] Kreîn S. G., Linear Differential Equations in Banach Space [in Russian], Nauka, Moscow (1967).
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