Bitsadze-Lavrent'ev problem

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