The indicator of contact boundaries for an integral geometry problem

Abstract

We pose and study a rather particular integral geometry problem. In the two-dimensional space we consider all possible straight lines that cross some domain. The known data consist of the integrals over every line of this kind of an unknown piecewise smooth function that depends on both points of the domain and the variables characterizing the lines. The object we seek is the discontinuity curve of the integrand. This problem arose in the author's previous research in X-ray tomography. In essence, it is a generalization of one mathematical aspect of flaw detection theory, but seems of interest in its own right. The main result of this article is the construction of a special function that can be unbounded only near the required curve. Precisely for this reason we call the function the indicator of contact boundaries. A uniqueness theorem for the solution follows rather easily from the property of indicators.Consider some bounded domain D in R 2 and denote the diameter of D by d. Consider a system of domains D i , for i = 1, . . . , p, such thatIt is not difficult to see that the union of the boundaries ∂D i of D i , for i = 1, . . . , p, coincides with the boundary ∂D 0 of their union. We assume that the closed curves ∂D i are piecewise smooth of class C 2 . More exactly, we make the following assumptions: Denote the coordinates of points x, y, z, ω in R 2 in the main coordinate system by (x 1 , x 2 ), (y 1 , y 2 ), (z 1 , z 2 ), (ω 1 , ω 2 ) respectively. Define z ∈ ∂D 0 as a contact point for domains D j and D l , where 1 ≤ j and l ≤ p, if in some neighborhood V (z) of z there are no points of D i except for D j and D l , and the common part of the two boundaries V (z) ∩ ∂D j = V (z) ∩ ∂D l represents the graph of a smooth function in local coordinates. We assume that in this local coordinate system centered at z the first axis is directed along the tangent to ∂D j or ∂D l at z. The second axis is directed along the unit vector of the interior normal n j (z) to ∂D j at z so that the smallest angle of rotation from the first axis to the second goes counterclockwise, as in the main coordinate system. Using some local coordinates (ξ, η) for every point y ∈ R 2 , we assume thatwhere a positive number δ and a function ψ(ξ) may depend on z in general. Assume that all but possibly finitely many points of ∂D 0 \ ∂D are contact points.It is easy to see that |ψ(ξ)| ≤ const |ξ| 2 . Here and below the symbol const stands for some positive number.

An integral geometry underdetermined problem for a family of curves

“…Under these conditions, we had to confine exposition to searching only the discontinuity surfaces of the integrand which constitute the important part of information about the unknown function. The present article is the result of the authors' study into similar and related problems in [9][10][11][12]. Here we substantially use our previous results and especially [12], and for the clarity of references, we try to maximally keep the previous notation, in particular: As before, R m stands for the m-dimensional arithmetic space; ∂T is the boundary of an arbitrary set T in R m ; ρ(x, T ) is the distance from x ∈ R m to T ; T x = T \ {x} is the set of all points in T except for x; αT is the set of the vectors of T multiplied by a real α; det(A) is the determinant of a matrix A; Ω = {ω : ω ∈ R n , |ω| = 1} is the unit sphere in R n ; the symbol const is a positive real; O(1) is a bounded function; C k (T ) is the space of bounded and continuous functions on T with all their partial derivatives up to order k. For reducing the notation, agree to denote the partial derivatives of a function φ(ν 1 , .…”

A Method for Studying Singular Integral Equations

“…For the first time, similar ideas were used by the author in [3,4]. Then this method was developed in the study of problems of X-ray tomography and integral geometry [5][6][7][8][9][10]. The statement of the problems with unknown boundaries is not rare these days.…”