Application of Cauchy integrals and singular integral equations in scattered data problems

A new approach is suggested for constructing “smooth” surfaces which contain discontinuities in functional values or derivatives at prescribed locations. The approach is based on solving singular integral equations with Cauchy-type kernels. It is applicable in the interpolation or approximation of scattered data. We investigate, in particular, several two-dimensional biharmonic and triharmonic problems which have smooth solutions except on given line segments, across which different types of discontinuities occur. We also discuss some issues concerning the application of the approach in practical problems.