Abstract
We present a survey paper on boundary value problems for analytic and harmonic functions in weighted classes of Cauchy type integrals in a simply connected domain not containing z = 8 and having a density from variable exponent Lebesgue spaces. It is assumed that the domain boundary is a piecewise smooth curve. The solvability conditions are established and solutions are constructed. The solution is found to essentially depend on the coefficients from the boundary condition, the weight, space exponent values at the angular points of the boundary curve and also on the angle values. The non-Fredholmian case is investigated. An application of the obtained results to the Neumann problem is given. This survey is based on the joint research with V. Paatashvili.
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