On the approximation of singular integral equations by equations with smooth kernels (Q1891856)

Singular integral equations with Cauchy kernel and piecewise-continuous matrix coefficients on open and closed smooth curves \(\Gamma\) are replaced by integral equations with smooth kernels of the form \((t- \tau)[(t- \tau)^2- n^2(t) \varepsilon^2]^{- 1}\), \(\varepsilon\to 0\), where \(n(t)\), \(t\in \Gamma\), is a continuous field of unit vectors non-tangential to \(\Gamma\). Necessary and sufficient conditions are given, under which the approximating equations have unique solutions and these solutions converge to the solution of the original equation. For the scalar case and the space \(L_2(\Gamma)\) these conditions coincide with the strong ellipticity of the given equation.