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

In this paper, we study an inhomogeneous Hilbert boundary value problem with a finite index and a boundary condition on a circle for a generalized Cauchy-Riemann equation with a singular coefficient. To solve this problem, we conducted a complete study of the solvability of the Hilbert boundary value problem of the theory of analytic functions with an infinite index due to a finite number of points of a special type of vorticity. Based on these results, we have derived a formula for the general solution and studied the existence and number of solutions to the boundary value problem of the theory of generalized analytic functions.


Introduction
Analytical and generalized analytical functions are widely used in modeling various physical and mechanical processes (for example [1,2]). In particular, the theory of generalized analytic functions has deep applications to mechanical problems of infinitesimal surface bends [3] and problems of stress state of membrane shell theory [3,4]. The Riemann boundary value problems (in particular, the jump problem) [5,6] and Hilbert [7,8] play a significant role in this case. We can interpret any infinitesimal bending of the surface as a certain state of stress equilibrium of the shell. This state of the shell can be described by solving a homogeneous equation. We reduce this equation to the Hilbert boundary value problem for analytic functions. The determination of the hydrodynamic pressure on the side surfaces of the shell in the presence of a surface load is also reduced to the Riemann-Hilbert problem of the theory of generalized analytical functions [3], p. 491. So, the boundary value problems for generalized analytical functions are the apparatus for solving mechanical problems, and the methods developed for the boundary value problems of the theory of generalized analytical functions can also be used to solve many nonlinear problems of the general bending problem.
In classical theory [3] generalized analytic functions U(z), z = x + iy, satisfy in some domain a linear elliptic system of differential equations, which is usually given in the following complex notation: where the coefficients and the right-hand side of the equation are summable functions with some degree p > 2. Fundamental results in the theory of boundary value problems for solutions of the system of the type (1) are obtained by I.N. Vekua [3,4]. Notable contributions to the development of this theory were made by J. Nitsche [14], B.V. Boyarsky [15], L.G. MihaĭLov [16], A.P. Soldatov [17,18].
The relaxation of restrictions on the coefficients of equation (1) is an important direction in the development of the theory of generalized analytic functions. Z.D. Usmanov [19] constructed a complete theory of generalized Cauchy-Riemann systems, whose coefficients have a polar singularity of the 1st order at an isolated inner point of the domain. On the basis of the developed analytical apparatus, he investigated local and global problems of the theory of infinitesimal bends of surfaces with an isolated flattening point. 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].
Boundary value problems for generalized Cauchy-Riemann equations with singular coefficients were solved by N.R. Radjabov, A.B. Rasulov, A.P. Soldatov, U.S. Fedorov, Bobodzhanova M.A., and others (for example, [24][25][26][27]). In these papers, the method for solving the boundary value problem for equation (1) is based on a reduction to a similar problem for analytic functions.In this case (for example, [28,29]), it is possible to formulate boundary value problems for generalized analytic functions with a singular line, when the coefficients of equation (1) delegate their features to the boundary condition of the problem for analytic functions and turn the latter into a problem with an infinite index. A.B. Rasulov investigated some situations related to this effect, when the boundary value problem of the theory of generalized analytic functions with a finite index has an infinite set of solutions [28,29].
This article is written in line with these works of A.B. Rasulov. The paper contains a detailed study of the solvability of the Hilbert boundary value problem with finite index for a class of generalized analytic functions with a singular line. The results of the article supplement the research of A.P. 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.
In the unit circle D = {z = re iθ :0 < r < 1, 0 ≤ θ < 2π}, L = ∂D, the plane of a complex variable z = x + iy = re iθ we consider the Hilbert boundary value problem about finding by the boundary condition: solution U(z) of the generalized Cauchy-Riemann system with a singular line: We highlight that equation (3) is a special case of the studied by A.P. Soldatov and A.B. Rasulov in [14] of the generalized Cauchy-Riemann equation: in a singly connected domain G with a smooth border, with singular (n=1) or super-singular (n>1) line. For this equation, a formula for the general solution is derived, and a boundary value problem with a combined boundary condition is set and solved. These conditions combine the features of the linear conjugation problem and the Hilbert problem.
Following [30], we will assume that for A(z) there is such an analytic in D function α(z) for that: In [30] under additional conditions: where τ0, τ1 -points of intersection of the contour L with a singular line (in this article τ0 = 1, τ1 = -1), a formula for the general solution of equation (3) is derived, which for the domain D takes the form: function: Vekua integral operator: 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.
In this paper, instead of condition (5), we assume that the following asymptotic formulas are satisfied: Accordingly, we will consider the boundary values of the function a0(z) to be Heldercontinuous on the upper L + and lower Larcs of the unit circle, including the ends. This leads to the following changes in the proof of formula (6).
As in [14], we introduce Dε ± = D ∩ {± Imz > ε}, ε -as a small positive number and denote Dε = Dε + ⋃ Dε -, the boundary of the region D/Dε is made up of the union lε of two segments lε ± and the union γε of two arcs of the unit circle γε,0, γε,1. These circles contain the points τ0 = 1, τ1 = -1, respectively. We denote by the symbol TεA the integral Vekua operator on the union of domains Dε. 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: the segment lεis oriented negatively. After moving to the limit in δ → 0 we get: We will take into account that: now, for z ∈ K and sufficiently small values of ε by Cauchy's theorem, we have: For z ∈ K and for ε → 0, using the Helder continuity on the arcs (including ends) L + and Lof the function a0(t), we derive: in this case, the Cauchy-type integral with two points of discontinuity of the density of the logarithmic type has [31] near the point τj of the form: where Ψ(z) -is an analytic in D function tending to a certain limit at z → τj, j =0, j = 1. In accordance with equality (7), (9) the function Ω(z) has the form of a point τj the following asymptotic representation: Here, by arg(z -τj) we mean a continuous branch in the domain D, whose boundary values satisfy the equalities: Now we repeat the calculations from [14] and derive the formula for the general solution of equation (3) in the form (6).
Using formula (6), we derive a formula for the general solution of the Hilbert boundary value problem (2) for solutions U(z) of differential equation (3) in the class A of functions U(z) with a product Ue -CΩ , bounded in D ̅ with some C ≥ 1.

Results
After we substitute function (6) into condition (2), we obtain the boundary condition for the Hilbert problem for the analytic function ϕ(z) in the disk: After we pass in formula (9) to the limit in z → t, t ∈ L, near singular points and introduce the notation , 4 Thus, we reduced the Hilbert problem with a finite index for solving equation (1) to the Hilbert problem (11) with an infinite index and two logarithmic vorticity points for the analytic function. Since the Hilbert boundary value problem with such a vorticity character has not yet been studied, we will carry out the solution of the problem in detail. The boundary condition of the problem, taking into account the formulas (9), (12)- (15) and the equality ln|t -t̅ | = 2ln|t -t0| + 2ln|t -t1| -2ln2, we will rewrite in the form: where Γ + (t) -is the limit value on the contour L of the integral: We will first consider a homogeneous problem: We will introduce an analytic function in D: This function on L by virtue of (17) satisfies the condition: ImF + (t) = 0. (19) We will express from equality (18) the desired function: (20) is a bounded solution to problem (17), then there must be a function F(z), analytic in D. This function satisfies the inequality: That is, this function is an exact growth of T in a semi-neighborhood of the point T, and its boundary values must satisfy condition (19) and the inequality: The validity of the inverse statement follows from the generalized maximum principle for analytic functions ( [32] p. 456, 457, see also [33]). Thus, it is proved.

Theorem 1
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. In this case, the solution of the problem with n points of vorticity can be represented as the products of the solutions of these problems with a single point of vorticity. Thus, the solution of problem (17) ϕ(z) = ϕ0(z)ϕ1(z) where ϕj(z) is the general solution of a homogeneous problem with a single point of vorticity τj, j = 0, j = 1, is determined by the formula: where Fj(z) is an analytic function in the domain D that satisfies the conditions: Besides F(z) = F0(z)F1(z). If we use these remarks, the following theorem will be proved.
We will prove point a) of the theorem. We will assume that βj --βj + < 0 and problem (17) are solvable. The solvability of the homogeneous problem (17) is equivalent to [35] the solvability of the homogeneous Hilbert problem with a point of vorticity τ1 and the homogeneous Hilbert problem with a point of vorticity τ0. The general solution of a homogeneous problem with one singular point τj, j = 0.1 is represented by the formula (23), in which the function Fj(z) is subject to the conditions (24). Now we will consider the question of the existence of the function Fj(z), for which we will transfer it to the upper halfplane H + . We will get the function: It follows from these conditions that the function f1(ζ) is a narrowing to the upper halfplane of the whole function: a refined zero-order ρ(|ζ|) ≤ ln ln 2 |ζ |/ln|ζ | with a restriction on the growth of its boundary values in the form of inequalities (25). For a function ) ( and a plane with a cut along the real semiaxis, we apply the Phragmén -Lindelöf theorem and deduce that under condition βj --βj + < 0 follows f1(ζ) ≡ 0, F1(z) ≡ 0, that is, problem (17) has only a trivial solution. Similarly, we consider the case of the function и F0(z). Now let βj --βj + = 0. We will consider in detail the case j = 1. Conditions (21), (22) after passing to the half-plane using the mapping z1(ζ) = τ1(iζ + 1)/(iζ -1) now take the form of the inequalities: , From this, as above, we deduce that if Δ1 + > 0 or Δ1 -> 0, then the problem has only a trivial solution. , 4 ln (30) in formula (27), then the inequality (22) holds for the function . It is obvious that the function (27) takes real values on the real axis and, therefore, the functions ) ( ), ( satisfy the condition (19). Then, using the formula (23) , we find ϕ0(z), ϕ1(z) and the desired solution of the homogeneous problem ϕ(z) = ϕ0(z)ϕ1(z).
Finally we consider the case β1 --β1 + = 0, Δ1 -< 0, Δ1 + < 0. To prove the theorem in this subsection, it suffices to verify the existence of a solution ϕ1(z) to the homogeneous problem with vorticity at the point τ1. As above, this is equivalent to the existence in H + of an analytic function f1(ζ). The refined zero order of this function satisfies the inequality | | ln , This function takes real values on the boundary and satisfying condition (26). As such a function, we can take Similarly, but using the conditions β0 --β0 + = 0, Δ0 + < 0, Δ0 -< 0, the existence of the solution ϕ0(z) is justified. We now turn to the solution of the inhomogeneous Hilbert problem (11). We will assume that the conditions © of Theorem 2 are satisfied, under which the homogeneous boundary value problem (17) is solvable (the situation with condition (b) is elementary).
We will look for the general solution of the inhomogeneous problem in the form of the sum of the general solution of the corresponding homogeneous problem and the particular solution of the inhomogeneous problem. To find the latter, we need some solution of the homogeneous problem (17). For this solution, the function F, which is included in the formula (20) and is defined by the conditions (19), (21) and (22), has the following additional properties: a) everywhere (except, perhaps, for the points τ1, τ2) on L, the condition F is satisfied We will search for the function satisfy the conditions (24). We will construct these functions in the following two cases. We will first consider the case of βj --βj + > 0, j = 0.1. We introduce a function ) (  j f , which we define in the complex plane ζ by formula (27). We'll take λj = (βj --βj + )/4π.
Next, using the mapping (28) , we construct a function: for the module of which the asymptotic representation (29) is valid. Now we will denote the integer part of the number x with the symbol  x . We will consider the following restrictions fulfilled: We define the function: It is easy to check that this function takes real values at the points of the unit circle, and if we compare formulas (29) and (12), (14), it is not difficult to verify the validity of the other two conditions (24). Obviously, also for the constructed function, the fulfillment of additional constraints. Moreover, the function is an integer. Now we will consider the case when βj --βj + = 0, and the numbers Δj + ≤ 0, Δj -≤ 0, and at least one of the inequalities is strict. Here we put 2, = 1, where rj is defined as above.
We will look for a particular solution to the inhomogeneous problem. This problem has the same sequences of zeros as the function ) ( z F and hence ). ( z  Therefore, the relation ) ( )/ ( z z   for the desired function will be an analytic and bounded function in D, with the exception, perhaps, of the points τ0, τ1, in which a power singularity of order less than one is allowed. Since , and at the points of τ1, τ2 can have power-law singularities of order less than one, the function can be represented by the Schwarz formula, therefore: The last formula gives a particular solution to the inhomogeneous boundary value problem (11), the general solution of which is represented as the sum of the general solution of the corresponding homogeneous problem and the given particular solution. The following is true.

Theorem 3
An inhomogeneous problem (11) is solvable in the class A of analytic functions in D if the corresponding homogeneous problem (17) is solvable. The general solution of the inhomogeneous problem is represented as the sum of the general solution (20) of the homogeneous problem and the particular solution (33) of the inhomogeneous problem.
After we substitute the found solution of the inhomogeneous problem (11) in the formula (6), we get the general solution of the boundary value problem (2).

Discussion
The article contains a solution and a study of the solvability of the Hilbert boundary value problem for the generalized Cauchy-Riemann equation with a singular coefficient of the form (3). The solution of the boundary value problem is based on the construction of the formula (6) of the general solution of equation (3). This formula allows us to reduce the solution of the boundary value problem (2) with a finite index for a generalized analytic function to the boundary value problem (11) with an infinite index for an analytic function. The main content of our paper is the solution and investigation of the solvability of the boundary value problem (11) with an infinite index and two points of a new type of vorticity. The picture of the solvability of problem (11) is contained in Theorems 2, 3. It is described in terms of the characteristics of the singularities of the coefficient and is a picture of the solvability of the boundary value problem (2). If we use the works [20] and [37][38][39], then we can get a similar result for the Riemann problem.

Conclusions
We have formulated and solved the inhomogeneous Hilbert boundary value problem for an important special case (3) of the generalized Cauchy-Riemann equation with a singular coefficient. We have obtained a formula for the general solution of this problem. We have found the conditions for the existence and uniqueness of the solution of the boundary value problem. In the case of non-uniqueness of the solution, a complete description of the set of solutions is given. The solution of the Hilbert boundary value problem with an infinite index of the theory of analytic functions is also of independent importance.