##### Actions

The problem of finding a function which satisfies (a1)

in a mixed domain that is simply connected and bounded by a Jordan (non-self-intersecting) "elliptic" arc (for ) with end-points and and by the "real" characteristics (for ) of the Bitsadze–Lavrent'ev equation (a1), which satisfy the characteristic equation and meet at the point , and which assumes prescribed continuous boundary values (a2)

where is the arc length reckoned from the point and Consider the aforementioned domain (denoted by ). Then a function is a regular solution of the Bitsadze–Lavrent'ev problem if it satisfies the following conditions:

1) is continuous in  , ;

2) are continuous in (except, possibly, at the points and , where they may have poles of order less than , i.e., they may tend to infinity with order less than as and );

3) , are continuous in (except possibly on , where they need not exist);

4) satisfies (a1) at all points (i.e., without );

5) satisfies the boundary conditions (a2).

Consider the normal curve (of Bitsadze–Lavrent'ev) Note that it is the upper semi-circle and can also be given by (the upper part of) where . The curve contains in its interior.

The idea of A.V. Bitsadze and M.A. Lavrent'ev for finding regular solutions of the above problem is as follows. First, solve the problem N (in , ). That is, find a regular solution of equation (a1) satisfying the boundary conditions: on ; on , where is continuous for , , and may tend to infinity of order less than as and .

Secondly, solve the Cauchy–Goursat problem (in , ). That is, find a regular solution of (a1) satisfying the boundary conditions: on ; on , where is continuous for , , and may tend to infinity of order less that as and .

Finally, take into account the boundary condition Therefore, one has a Goursat problem (in , ) for (a1) with boundary conditions: on ; on .

Several extensions and generalizations of the above boundary value problem of mixed type have been established [a3], [a4], [a5], [a6], [a7], [a8]. These problems are important in fluid mechanics (aerodynamics and hydrodynamics, [a1], [a2]).