Let the domain

be bounded by the line

with the ends at the points

and

of the real axis and by the characteristics

and

of Lavrent'ev-Bitsadze equation

Let
and
be the elliptic and the hyperbolic parts of the mixed domain
.

The unknown solution of (2.1) in the jump problem as in the Tricomi problem should satisfy the following boundary conditions:

(here
is arc abscissa of arc
being measured from the point
to the point
).

As it is commonly accepted in the theory of the boundary value problems for the mixed-type equations we denote

Assume that on the segment
there is the finite number of points in which functions
can have discontinuities of the first kind and functions
can have singularities of integrable order. We say that such points are the exclusive points.

Denote by
is the set of points of the segment
which are not the exclusive ones.

We can seek a solution of the problem

in the different classes of solutions [

2, Section

]. The regular solution

and satisfies (2.1) in

. The generalized solution of the class

belongs to

, satisfies (2.1) in

and is the generalized solution of (2.1) in

in the sense that

(the D'Alembert formula) where
. Here
is a class of functions which can have singularities of integrable order at the points
and
, but satisfy Hoelder's condition with some index at any part of the interval
. As it is known the generalized solution of the class
will be regular if we assume in addition that
.

In the jump problem on the type change line for Lavrent'ev-Bitsadze equation, we need to seek a function
which

(1)is regular or generalized solution of (2.1) in
;

(2)satisfies the boundary conditions (2.2);

(3)has the limiting values
on
and

are fulfilled.

Let us obtain functional correlations at the segment
which connect functions
, and
. The general scheme of reasoning is just the same as under solving the Tricomi problem.

Consider auxilliary boundary value problem

in the domain

. Let

be the Green function of the problem

for (2.1) in the domain

. Then in

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

. Subtracting (2.9) from (2.7) we get

It follows from here that function

should be a solution of integral equation

Similar equation can be obtained for the function
.

If the domain

is a semidisc

, then

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
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
. As it is shown in [13, page 111], there is a unique opportunity to choose numbers
(
being stationary points of group of the homographic transformations).

So index of problem (2.20)

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

By this if the elliptic part

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

The function to be found
can be obtained by differentiation, but as it will be shown later it is not obligatory.

By the main correlation (2.7)

here it is taken into account that function
satisfies (2.17). Expressions of functions
can be obtained from conditions (2.5).

The solution of the jump problem in the domain
can be easily derived by the D'Alembert formula (2.4), and it is not necessary to seek the expression of the function
for this, it is sufficient to have formula to calculate its primitive. The solution of the jump problem in the domain
can be obtained by two methods: either as a solution of problem
or as a solution of the Dirichlet problem.

Let

be a simple connected domain bounded by piecewise-smooth curve and let function

conformally map by variable

in

onto unit disc in such way that

. Then (see, [

14, page 464]) function

is the Green function of the Dirichlet problem for the domain

. If

is conformal mapping of the domain

onto unit disc then

More general statement is formulated in [2, page 30]. If the function
maps the domain
of the plane
onto the domain
of the plane
and
is the Green function of the Dirichlet problem for the domain
, then
is the Green function of the Dirichlet problem for the domain
.

In the case of problem
it is also possible to use the method of conformal mappings [2, Section
]. Let domains
and
be bounded by segment
of real axis and by curves
and
placed in the upper half-plane. Let function
map the domain
onto the domain
in such way that
goes over into
and ends of this segment remain stationary. If
is the Green function of problem
for the domain
, then
is the Green function of problem
for the domain
.

By this way, the Green function of problem

for the domain

in the jump problem can be derived from the Green function of problem

for some simple canonical domain

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

is the Green function (2.13) of the problem

for semidisc and

is a mapping of any other domain

onto this semidisc satisfying the above mentioned conditions. Then for the Green function of the problem

for the domain

, we have

So the integral equation (2.11) can be transformed into equation of the form (2.17) by substitution of variables.