Bitsadze-Lavrent'ev problem

Tricomi–Bitsadze–Lavrent'ev problem

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]).

References

[a1]	A.V. Bitsadze, "Equations of mixed type" , Macmillan (1964) (In Russian)
[a2]	C. Ferrari, F.G. Tricomi, "Transonic aerodynamics" , Acad. Press (1968) (Translated from Italian)
[a3]	J.M. Rassias, "Mixed type equations" , 90 , Teubner (1986)
[a4]	J.M. Rassias, "Lecture notes on mixed type partial differential equations" , World Sci. (1990)
[a5]	J.M. Rassias, "The Bitsadze–Lavrentjev problem" Bull. Soc. Roy. Sci. Liège , 48 (1979) pp. 424–425
[a6]	J.M. Rassias, "The bi-hyperbolic Bitsadze–Lavrentjev–Rassias problem in three-dimensional Euclidean space" C.R. Acad. Sci. Bulg. Sci. , 39 (1986) pp. 29–32
[a7]	J.M. Rassias, "The mixed Bitsadze–Lavrentjev–Tricomi boundary value problem" , Texte zur Mathematik , 90 , Teubner (1986) pp. 6–21
[a8]	J.M. Rassias, "The well posed Tricomi–Bitsadze–Lavrentjev problem in the Euclidean plane" Atti. Accad. Sci. Torino , 124 (1990) pp. 73–83