Bitsadze Equation with Supersingularities in the Lowest Coefficients

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

“…In order for the solution ϕ(z) of problem (17) to be bounded in the domain D, it is necessary and sufficient that the function F(z), which is included in the formula of the general solution (20), satisfies in D the growth constraints (21) and on the boundary conditions (19) and (22).…”

“…It is clear that the existence and set of bounded solutions to problem (17) depends on the existence and set of analytic functions in D, and they satisfy conditions (19), (21), and (22). In [35], it is proved that a homogeneous Hilbert problem with n points of vorticity is solvable if and only if all n homogeneous problems with a single point of vorticity are solvable, that is, the solvability of the problem is affected only by the parameters of each point of vorticity.…”

“…To prove the theorem, we will make sure that formula ( 20) is meaningful. To do this, we will make sure that there is a function F(z) analytic in the disk D that satisfies conditions ( 19), (21), and (22). For this we need a whole function:…”

“…The construction of the solution of the generalized Cauchy-Riemann system with strong singularities of the coefficients at an isolated point is also devoted to the work [20]. The study of solutions of equation (1) in the case when the coefficients of this equation have singular lines is given in the monograph [21], also [22,23]. We emphasize that the presence of strong singularities in the coefficients of equation ( 1) is not only a natural development of the classical theory of generalized analytic functions, but is also in demand as models for problems of thin momentless shells, axisymmetric field theory, and deformation problems [9,3].…”