A Method for Studying Singular Integral Equations

Abstract

We examine a singular integral equation of the first kind on a bounded open set of an n-dimensional space. Open subsets with a common (contact) (n − 1)-dimensional piecewise smooth part of boundaries are selected. The underdetermined case is treated in which the unknown part of the integrand depends on 2n independent variables whereas a given integral depends only on n variables. In this situation we pose the problem of finding the contact part of the boundaries and prove unique solvability of the problem. We use the following notation: ρ(A, B) is the distance between sets A and B in R n ; ∂A is the boundary ofConsider a function g(x, y, ω) measurable on T ×T ×Ω and bounded almost everywhere in the following sense: There exist T and Ω ( T ∈ T and Ω ∈ Ω) coinciding in measure with T and Ω and such that |g(x, y, ω)| ≤ const for (x, y, ω) ∈ T × T × Ω. Here and in what follows, the notation const is used to denote a positive constant. In order to reduce the number of denotations, the bounded functions are often written below as O(1). We put T i = T i ∩ T and T 0 = T 1 ∪ T 2 , i = 1, 2. A point of the unit sphere Ω is denoted by ω if it designates an independent variable. If this point is expressed through other variables then we employ the letter s; for example, s = (y − x)/|y − x|. Consider the integral(1.1)The functions V i (x) for x ∈ T i , i = 1, 2, are understood to be singular integrals. We assume that T i , i = 1, 2, and g(x, y, ω) satisfy the following condition: for every u ∈ T i ,Of course, this restriction means that the integrand is a summable function. For convenience, g(x, y, ω) is assumed to be extended by zero in y beyond T . Note that if we use the spherical coordinates y − u = tν, ν ∈ Ω, dy = t n−1 dtdν then (1.2) is a consequence of the condition Ω dν ∞ 0 |g(x, u + tν, ω) − g(x, u, ω)| t dt ≤ const, ω ∈ Ω, x ∈ T .