Boundary value problem for a generalized Cauchy–Riemann equation with singular coefficients

Hilbert boundary value problem for generalized analytic functions with a singular line

“…Soldatov and A.B. Rasulov [30], who, under certain restrictions on singular coefficients, derived the formula for the general solution of equation (1). This solution was used by the authors in reducing the boundary value problem for equation (1) to a similar problem with a finite index for analytic functions.…”

“…By loosening the restrictions on the coefficients from [30], we reduce the solution of the Hilbert problem for generalized analytic functions with a finite index to the problem for analytic functions, but with an infinite index and two points of vorticity with a new type of singularities. We construct a formula for the general solution of the problem, and conduct a complete study of the solvability.…”

“…Following [30], we will assume that for A(z) there is such an analytic in D function α(z) for that: 2. > ),…”

“…to the Helder class H(D ̅ ), the function ϕ(z) is analytic in the domain G0 ⸦ D. We emphasize that conditions (5) also guarantee a finite index of the boundary value problem for analytic functions, which is obtained when solving the problem considered in [30] with a combined boundary condition.…”

“…We need to make sure that (TεA)(z), z ∈ K, converges uniformly at ε→0 to the limit of Ω(z) on any compact K, K ∈ D + ⋃ D -, where D ± = D ∩ {± Imz > 0}. Following [30], we represent TεA in the form:…”