# Bitsadze-Lavrent'ev problem

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Tricomi–Bitsadze–Lavrent'ev problem

The problem of finding a function $u = u ( x,y )$ which satisfies

$$\tag{a1 } { \mathop{\rm sgn} } ( y ) u _ {xx } + u _ {yy } = 0$$

in a mixed domain that is simply connected and bounded by a Jordan (non-self-intersecting) "elliptic" arc $g _ {1}$( for $y > 0$) with end-points $O = ( 0,0 )$ and $A = ( 1,0 )$ and by the "real" characteristics (for $y < 0$)

$$g _ {2} : x - y = 1, \quad g _ {3} : x + y = 0$$

of the Bitsadze–Lavrent'ev equation (a1), which satisfy the characteristic equation

$$- ( d y ) ^ {2} + ( d x ) ^ {2} = 0$$

and meet at the point $P = ( {1 / 2 } , - {1 / 2 } )$, and which assumes prescribed continuous boundary values

$$\tag{a2 } u = p ( s ) \textrm{ on } g _ {1} , \quad u = q ( x ) \textrm{ on } g _ {3} ,$$

where $s$ is the arc length reckoned from the point $A$ and

$${ \mathop{\rm sgn} } ( y ) = \left \{ \begin{array}{l} {1 \ \textrm{ for } y > 0, } \\ {0 \ \textrm{ for } y = 0, } \\ {-1 \ \textrm{ for } y < 0. } \end{array} \right .$$

Consider the aforementioned domain (denoted by $D$). Then a function $u = u ( x,y )$ is a regular solution of the Bitsadze–Lavrent'ev problem if it satisfies the following conditions:

1) $u$ is continuous in ${\overline{D}\; }$ $= D \cup \partial D$, $\partial D = g _ {1} \cup g _ {2} \cup g _ {3}$;

2) $u _ {x} ,u _ {y}$ are continuous in ${\overline{D}\; }$( except, possibly, at the points $O$ and $A$, where they may have poles of order less than $1$, i.e., they may tend to infinity with order less than $1$ as $x \rightarrow 0$ and $x \rightarrow 1$);

3) $u _ {xx }$, $u _ {yy }$ are continuous in $D$( except possibly on $OA$, where they need not exist);

4) $u$ satisfies (a1) at all points $D \setminus OA$( i.e., $D$ without $OA$);

5) $u$ satisfies the boundary conditions (a2).

Consider the normal curve (of Bitsadze–Lavrent'ev)

$$g _ {1} ^ {0} : \left ( x - { \frac{1}{2} } \right ) ^ {2} + y ^ {2} = { \frac{1}{4} } , y > 0.$$

Note that it is the upper semi-circle and can also be given by (the upper part of)

$$g _ {1} ^ {0} : \left | {z - { \frac{1}{2} } } \right | = { \frac{1}{2} } ,$$

where $z = x + iy$. The curve $g _ {1}$ contains $g _ {1} ^ {0}$ 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 $D$, $y > 0$). That is, find a regular solution of equation (a1) satisfying the boundary conditions:

$u = p ( s )$ on $g _ {1}$;

$u _ {y} = r ( x )$ on $OA$, where $r = r ( x )$ is continuous for $x$, $0 < x < 1$, and may tend to infinity of order less than $1$ as $x \rightarrow 0$ and $x \rightarrow 1$.

Secondly, solve the Cauchy–Goursat problem (in $D$, $y < 0$). That is, find a regular solution of (a1) satisfying the boundary conditions:

$u = t ( x )$ on $OA$;

$u _ {y} = r ( x )$ on $OA$, where $t = t ( x )$ is continuous for $x$, $0 < x < 1$, and may tend to infinity of order less that $1$ as $x \rightarrow 0$ and $x \rightarrow 1$.

Finally, take into account the boundary condition

$$u = q ( x ) \textrm{ on } g _ {3} .$$

Therefore, one has a Goursat problem (in $D$, $y < 0$) for (a1) with boundary conditions:

$u = t ( x )$ on $OA$;

$u = q ( x )$ on $g _ {3}$.

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
How to Cite This Entry:
Bitsadze-Lavrent'ev problem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bitsadze-Lavrent%27ev_problem&oldid=46077
This article was adapted from an original article by J.M. Rassias (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article