The Jump Problem for Mixed-Type Equations with Defects on the Type Change Line

The jump problem and problems with defects on the type change line for model mixed-type equations in the mixed domains are investigated. The explicit solutions of the jump problem are obtained by the method of integral equations and by the Fourier transformation method. The problems with defects are reduced to singular integral equations. Some results for the solution of the equation under consideration are discussed concerning the existence and uniqueness for the solution of the suggested problem.


Introduction
Consider the jump problem and problems with defects on the type change line for the mixed- are the jump problems on the type change line for 1.1 . Obviously, the classical Tricomi problem is the jump problem with zero jump. Two methods are used in this papre to solve the jump problem: the method of integral equations and the method of integral Fourier transformation. It is shown that explicit solutions of the jump problem can be used as potentials under researching boundary value problems with defects.

The Jump Problem for Lavrent'ev-Bitsadze Equation: The Method of Integral Equations
Let the domain D be bounded by the line Γ with the ends at the points A 0, 0 and B 1, 0 of the real axis and by the characteristics AC : x y 0 and BC : x − y 1 of Lavrent'ev-Bitsadze equation Let D 1 and D 2 be the elliptic and the hyperbolic parts of the mixed domain D.

Boundary Value Problems 3
The unknown solution of 2.1 in the jump problem as in the Tricomi problem should satisfy the following boundary conditions: here s is arc abscissa of arc Γ being measured from the point B to the point A .
As it is commonly accepted in the theory of the boundary value problems for the mixed-type equations we denote

2.3
Assume that on the segment AB 0, 1 there is the finite number of points in which functions τ j x can have discontinuities of the first kind and functions ν j x can have singularities of integrable order. We say that such points are the exclusive points.
Denote by AB * is the set of points of the segment AB which are not the exclusive ones. We can seek a solution of the problem T in the different classes of solutions 2, Section 15 . The regular solution u x, y ∈ C 2 D 1 ∪ D 2 and satisfies 2.1 in D 1 ∪ D 2 . The generalized solution of the class R * belongs to C 2 D 1 , satisfies 2.1 in D 1 and is the generalized solution of 2.1 in D 2 in the sense that the D'Alembert formula where τ 2 x , ν 2 x ∈ H * . Here H * H * 0, 1 is a class of functions which can have singularities of integrable order at the points 0 and 1, but satisfy Hoelder's condition with some index at any part of the interval 0, 1 . As it is known the generalized solution of the class R * will be regular if we assume in addition that τ 2 x , ν 2 x ∈ C 1 0, 1 .
In the jump problem on the type change line for Lavrent'ev-Bitsadze equation, we need to seek a function u x, y which are fulfilled.

Boundary Value Problems
Let us obtain functional correlations at the segment AB which connect functions τ 1 x , ν 1 x , τ 2 x , and ν 2 x . The general scheme of reasoning is just the same as under solving the Tricomi problem.
Consider auxilliary boundary value problem N in the domain D 1 . Let G 2 ζ, z G 2 ξ, η, x, y be the Green function of the problem N for 2.1 in the domain D 1 . Then in By this From D'Alembert's formula 2.4 and condition on the characteristic it follows immediately that Consider system of 2.7 , 2.9 , and 2.5 for functions τ j x , ν j x . Subtracting 2.9 from 2.7 we get It follows from here that function ν 1 x should be a solution of integral equation Similar equation can be obtained for the function ν 2 x . If the domain D 1 is a semidisc |2z − 1| < 1, Im z > 0, then 2.14 Boundary Value Problems 5 Let us transform the integral equation 2.11 with logarithmic singularity in the kernel into the integral equation with the analogue of the Cauchy kernel. As so the new function to be found, should satisfy the equation Generally speaking 2.17 is the complete singular integral equation with the Cauchy kernel, but special form of the kernel enables us to construct its solution in the explicit form. By this it is advisible to use the method of reduction to the Riemann boundary value problem for automorphic functions 12, Chapter III . Consider an auxiliary piecewise-holomorphic function satisfying the automorphic type condition It follows from the analogues of the Sohotski formulas that integral equation 2.17 is equivalent to the Riemann boundary value problem for analytic functions satisfying the condition 2.19 . The solutions of problem 2.20 should be limited at the points z 0, z 1 and at infinity. The canonical function of the Riemann problem in class of the automorphic functions has the form is simple automorphic function of group z, z/ 2z − 1 . As it is shown in 13, page 111 , there is a unique opportunity to choose numbers κ 0 0 κ 1 0 z 0 z 1 being stationary points of group of the homographic transformations .
So index of problem 2.20 κ κ 0 κ 1 0 and its unique solution limited at infinity By condition 2.19 an arbitrary constant in the right-hand side is equal to zero. Since the boundary value of the canonical function from the upper half-plane on 0, 1 dt.

2.25
By this if the elliptic part D 1 of the mixed domain is a semidisc then by formula 2.25 , we can write down the solution of the integral equation 2.11 in the form dt.

2.26
The function to be found ν 1 x can be obtained by differentiation, but as it will be shown later it is not obligatory.
Boundary Value Problems 7 By the main correlation 2.7

2.29
More general statement is formulated in 2, page 30 . If the function w w z maps the domain D of the plane z onto the domain D 0 of the plane w and G 0 ζ, z is the Green function of the Dirichlet problem for the domain D 0 , then G ζ, z G 0 w ζ , w z is the Green function of the Dirichlet problem for the domain D.
In the case of problem N it is also possible to use the method of conformal mappings 2, Section 7 . Let domains D 0 and D 1 be bounded by segment AB of real axis and by curves Γ 0 and Γ 1 placed in the upper half-plane. Let function w w z map the domain D 1 onto the domain D 0 in such way that AB goes over into AB and ends of this segment remain stationary. If G 0 ζ, z is the Green function of problem N for the domain D 0 , then G ζ, z G 0 w ζ , w z is the Green function of problem N for the domain D 1 .
By this way, the Green function of problem N for the domain D 1 in the jump problem can be derived from the Green function of problem N for some simple canonical domain D 0 by conformal mapping. In 2 the upper half-plane is chosen as a canonical domain but in our case a semidisc is more convenient to be considered as such domain. Hence if G 2 ζ, z is the Green function 2.13 of the problem N for semidisc and w w z is a mapping of any other domain D 1 onto this semidisc satisfying the above mentioned conditions. Then for the Green function of the problem N for the domain D 1 , we have So the integral equation 2.11 can be transformed into equation of the form 2.17 by substitution of variables.

The Jump Problem for Lavrent'ev-Bitsadze Equation: The Method of Fourier Transformation
Let us construct the solution of the jump problem for Lavrent'ev-Bitsadze equation in the unbounded mixed domain by the method of the integral Fourier transformation. Preliminary, we consider two auxiliary Cauchy problems in the upper and lower halfplanes using some results of the works 15, 16 . We will use the following denotions: under Fourier transformation function a x goes over into function distribution A ξ .
Note that the boundary value problems in the half-space for partial differential equations have been investigated quite adequately see, 17 . If the Cauchy problem in the half-space is overdetermined, then analysis of the algebraic equation for the Fourier transform of the unknown solution gives necessary and sufficient conditions for the boundary functions.
We seek a solution of 2.1 in the upper half-plane y > 0 satisfying the boundary conditions The Fourier transform of the unknown solution will be a solution of the equation This solution exists if and only if when the right-hand side of 3.2 vanishes under η i|ξ|, that is, the condition  1 x and ν 1 x . We seek a solution of 2.1 in the lower half-plane y < 0 satisfying the boundary conditions The Fourier transform of the unknown solution satisfies the equation and boundary functions can be given arbitrary. It follows from 3.7 , that In the particular case under a x ≡ 0, b x ≡ 0, the jump problem coincides with the Tricomi problem in the unbounded mixed domain. Without loss of generality we can assume that ϕ 0 ψ 0 0. We will seek a solution of the jump problem in the upper and in the lower half-planes as solutions of the Cauchy problems. Let us continue the unknown solution in D 2 onto the whole lower half-plane so that The Fourier transforms of values of the unknown solution on the axis y 0 should satisfy the transformed conditions 3.12 3.14 and the condition 3.3 . Here A ξ A ξ , B ξ B ξ are the Fourier transforms of functions a x , b x being completed by zero up to the whole axis.
Denote by T ξ T 1 ξ , N ξ N 1 ξ and represent each of these functions as a sum of the Fourier transforms of one-side-functions. Hence

3.15
By condition 3.10 , where Φ ξ Φ − ξ is the Fourier transform of the boundary function ϕ x being completed by zero up to the whole axis.
Condition 3.11 on the characteristic can be written down in the form where Ψ ξ Ψ ξ is the Fourier transform of the boundary function ψ x being completed by zero up to the whole axis. Actually, by the D'Alembert formula After Fourier transformation subject to the evident identity we obtain 3.17 . Condition 3.3 in the new denotions has the form Excluding function T ξ from 3.17 and 3.20 we get in view of 3.16 Equality 3.21 is the condition of the Riemann boundary value problem with discontinuous coefficient given on the real axis this equality is being understood as the equality of distributions .
Note that solution of the jump problem in the whole plane without condition on the characteristic 3.11 is not unique but is determined within the arbitrary function.
The canonical function has the form Denote by Then By the Sohotski formulas

3.25
Now we can easily obtain the expressions of the other auxiliary functions N 2 ξ N 1 ξ − B ξ , T 1 ξ −N 1 ξ /|ξ|, T 2 ξ T 1 ξ − A ξ and consequently, the solution of the jump problem in the domains D 1 and D 2 .
The technique of the integral Fourier transformation can be used also in the cases when the mixed domain in the jump problem has another form.
If the elliptic part of the mixed domain is, for example, a semidisc then the Fourier transformation method can be modificated in the following way. Assume that the unknown solution of the jump problem on the semidisc is equal to zero. Continue the function u x, y to the whole upper half-plane symmetrically about Γ, that is, in such way that values of function are equal at the points symmetrical about semidisc. Besides the solution and its normal derivative should be continuous on the semidisc. Then all formulas obtained at the beginning of the section remain valid but after substitution of variable integrals on infinite intervals can be transformed into integrals on segment. This method can be used in more general case when the elliptic part of the mixed domain is a half of the symmetrical about real axis fundamental domain of group of homographic transformations 12, Chapter III .

The Boundary Value Problems with Defect on the Line of Type Change for Lavrent'ev-Bitsadze Equation
Let the mixed domain D be bounded by the line Γ with the ends at the points A 0, 0 and Later on for simplicity we will assume that in the set M there is only one segment α, β and 0 < α < β < 1.
If in the problem with defect the values of the unknown solution are given on the type change line, then we say that such defect is the defect of the first kind. If on M the values of the derivative ∂u/∂y of the unknown solution are given the defect of the 2nd kind , then by the main correlation 2.9 nothing changes in fact. By the same reason the problem with defect of the 3d kind when on M the linear combination of the solution and its derivative are given can be reduced to the problem with defect of the 1st kind. Note that defect can be considered as a cut and independent boundary conditions can be given on every side of the cut.