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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|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 [[#References|[a3]]], [[#References|[a4]]], [[#References|[a5]]], [[#References|[a6]]], [[#References|[a7]]], [[#References|[a8]]]. These problems are important in fluid mechanics (aerodynamics and hydrodynamics, [[#References|[a1]]], [[#References|[a2]]]).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.V. Bitsadze,   "Equations of mixed type" , Macmillan  (1964)  (In Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  C. Ferrari,   F.G. Tricomi,   "Transonic aerodynamics" , Acad. Press  (1968)  (Translated from Italian)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J.M. Rassias,   "Mixed type equations" , '''90''' , Teubner  (1986)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  J.M. Rassias,   "Lecture notes on mixed type partial differential equations" , World Sci.  (1990)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  J.M. Rassias,   "The Bitsadze–Lavrentjev problem"  ''Bull. Soc. Roy. Sci. Liège'' , '''48'''  (1979)  pp. 424–425</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  J.M. Rassias,   "The mixed Bitsadze–Lavrentjev–Tricomi boundary value problem" , ''Texte zur Mathematik'' , '''90''' , Teubner  (1986)  pp. 6–21</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  J.M. Rassias,   "The well posed Tricomi–Bitsadze–Lavrentjev problem in the Euclidean plane"  ''Atti. Accad. Sci. Torino'' , '''124'''  (1990)  pp. 73–83</TD></TR></table>