LAUSR

Let Γ be a simple closed curve that bounds the finite domain D, z=z(ζ)=z(reiϑ) be the conformal mapping of the circle {ζ:|ζ|<1} onto the domain D. Furthermore, let the functions A(z), B(z) be given on D and Us,2(A;B;D) be the set of regular solutions of the equation LW=∂zW+A(z)W+B(z)W‾=0. We call the Smirnov class Ep(t)(A;B;D) the set of those generalized functions Win D for which W∈Us,2(A;B;D),s>2andsup0<ρ<1∫02π|W(z(ρeiϑ)|p(z(eiϑ))|z′(ρeiϑ)|dϑ<∞, where p(t) is a positive measurable function on Γ. We consider the Riemann-Hilbert problem: Define a function W(z) from the class Ep(t)(A;B;D) for which the equality, Re[λ(t)W+(t)]=b(t),b(t)∈Lp(t)(Γ), is fulfilled almost everywhere on Γ. It is assumed that Γ is a piecewise-smooth curve without external peaks; A,B∈L∞(D), p is Log Holder continuous and p_=minp(t)>1,b(t)∈Lp(t)(Γ), the function a(t)=λ(t)‾λ(t) belongs to the class A(p(t);Γ), which is the generalization of the well-known Simonenko class A(p;Γ), where p is constant. The solvability conditions are established, and solutions are constructed.