## 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:

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