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The problem related to the existence of solutions for differential equations of mixed elliptic-hyperbolic type with two independent variables in an open domain  $  \Omega $
of special shape. The domain  $  \Omega $
can be decomposed into the union of two subdomains  $  \Omega _ {1} $
and  $  \Omega _ {2} $
by a smooth simple curve  $  AB $
whose end points  $  A $
and  $  B $
are different points of  $  \partial  \Omega $. 
The equation is elliptic in  $  \Omega _ {1} $, 
hyperbolic in  $  \Omega _ {2} $, 
and degenerates to parabolic on the curve  $  AB $. 
The boundary  $  \partial  \Omega _ {1} $
is the union of the curve  $  AB $
and a smooth simple curve  $  \sigma $, 
while  $  \partial  \Omega _ {2} $
is the union of characteristics  $  AC $
and  $  BC $
and the curve  $  AB $. 
The desired solution must take prescribed data on  $  \sigma $
and on only one of the characteristics  $  AC $
and  $  BC $(
see [[Mixed-type differential equation|Mixed-type differential equation]]).

The Tricomi problem for the [[Tricomi equation|Tricomi equation]]

$$ \tag{1 }
Tu  \equiv \ 
yu _ {xx} +
u _ {yy}  =  0
$$

was first posed and studied by F. Tricomi [[#References|[1]]], [[#References|[2]]]. The domain  $  \Omega $
is bounded by a smooth curve  $  \sigma \subset  \{ {( x, y) } : {y = 0 } \} $
with end points  $  A ( 0, 0) $, 
$  B ( 1, 0) $
and characteristics  $  AC $
and  $  BC $:

$$ 
AC : x  =  {
\frac{2}{3}
 } (- y)  ^ {3/2} ,\ \ 
BC : 1 - x  =  {
\frac{2}{3}
 } (- y)  ^ {3/2} .
$$

Under specified restrictions on the smoothness of the given functions and the behaviour of the derivative  $  u _ {y} $
of the solution  $  u $
at the points  $  A $
and  $  B $, 
the Tricomi problem

$$ \tag{2 }
u | _  \sigma   =  \phi ,\  u | _ {AC }   =  \psi
$$

for equation (1) reduces to finding the solution  $  u = u ( x, y) $
of equation (1) that is regular in the domain  $  \Omega  ^ {+} = \Omega \cap \{ {( x, y) } : {y > 0 } \} $
and that satisfies the boundary conditions

$$ \tag{3 }
u \mid  _  \sigma   =  \phi ,
$$

$$ 
u _ {y} ( x, 0)  =  \alpha D _ {0x}  ^ {2/3} u
( x, 0) + \psi _ {1} ( x),\  0 \leq  x \leq  1,
$$

where  $  \alpha = \textrm{ const } > 0 $, 
$  \psi _ {1} ( x) $
is uniquely determined by  $  \psi $, 
$  D _ {0x}  ^ {2/3} $
is the fractional differentiation operator of order  $  2/3 $(
in the sense of Riemann–Liouville):

$$ 
D _ {0x}  ^ {2/3} \tau ( x)  = \ 
{
\frac{1}{\Gamma ( 1/3) }
 }
{
\frac{d}{dx }
 }
\int\limits _ { 0 } ^ { x } 

\frac{\tau ( t)  dt }{( x - t)  ^ {2/3} }
 ,
$$

and  $  \Gamma ( z) $
is the Euler [[Gamma-function|gamma-function]].

The solution of the problem (1), (3) reduces in turn to finding the function  $  \nu ( x) = u _ {y} ( x, 0) $
from some [[Singular integral equation|singular integral equation]]. This equation in the case when  $  \sigma $
is the curve

$$ 
\sigma _ {0}  = \ 
\left \{ {
( x, y) } : {
\left ( x - {
\frac{1}{2}
 } \right )  ^ {2} +
{
\frac{4}{9}
 } y  ^ {3} = {
\frac{1}{4}
 } , y \geq  0
 } \right \}
$$

has the form

$$ 
\nu ( x) +
{
\frac{1}{\pi \sqrt 3 }
 }
\int\limits _ { 0 } ^ { 1 } 
\left ( {
\frac{t}{x}
 } \right )  ^ {2/3}
\left (
{
\frac{1}{t - x }
 } - {
\frac{1}{t + x - 2x }
 }
\right )
\nu ( t)  dt  =  f ( x),
$$

where  $  f ( x) $
is expressed explicitly in terms of  $  \phi $
and  $  \psi $, 
and the integral is understood in the sense of the Cauchy principal value (see [[#References|[1]]]–[[#References|[4]]]).

In the proof of the uniqueness and existence of the solution of the Tricomi problem, in addition to the Bitsadze extremum principle (see [[Mixed-type differential equation|Mixed-type differential equation]]) and the method of integral equations, the so-called  $  a $
$  b $
$  c $
method is used, the essence of which is to construct for a given second-order differential operator  $  L $(
for example,  $  T $) 
with domain of definition  $  D ( L) $, 
a first-order differential operator

$$ 
l  =  a ( x, y)
{
\frac \partial {\partial  x }
 } +
b ( x, y)
{
\frac \partial {\partial  y }
 } + c ( x, y),\ \ 
( x, y) \in \Omega ,
$$

with the property that

$$ 
\int\limits _  \Omega 
lu \cdot Lu  dx  dy  \geq  \ 
C  \| u \|  ^ {2} \ \ 
\textrm{ for }  \textrm{ all } \ 
u \in D ( L),
$$

where  $  C = \textrm{ const } > 0 $
and  $  \| \cdot \| $
is a certain norm (see [[#References|[5]]]).

The Tricomi problem has been generalized both to the case of mixed-type differential equations with curves of parabolic degeneracy (see [[#References|[6]]]) and to the case of equations of mixed hyperbolic-parabolic type (see [[#References|[7]]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  F. Tricomi,   "On second-order linear partial differential equations of mixed type" , Moscow-Leningrad  (1947)  (In Russian; translated from Italian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  F.G. Tricomi,   "Equazioni a derivate parziale" , Cremonese  (1957)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.V. Bitsadze,   "Zum Problem der Gleichungen vom gemischten Typus" , Deutsch. Verlag Wissenschaft.  (1957)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A.V. Bitsadse,   "Equations of mixed type" , Pergamon  (1964)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  L. Bers,   "Mathematical aspects of subsonic and transonic gas dynamics" , Wiley  (1958)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  A.M. Nakhushev,   "A boundary value problem for an equation of mixed type with two lines of degeneracy"  ''Soviet Math. Dokl.'' , '''7''' :  5  (1966)  pp. 1142–1145  ''Dokl. Akad. Nauk SSSR'' , '''170'''  (1966)  pp. 38–40</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  T.D. Dzhuraev,   "Boundary value problems for equations of mixed and mixed-composite type" , Tashkent  (1979)  (In Russian)</TD></TR></table>

====Comments====
Using a functional-analytic method, S. Agmon [[#References|[a5]]] has investigated more general equations. Fourier integral operators were used by R.J.P. Groothuizen [[#References|[a2]]].

For additional references see also [[Mixed-type differential equation|Mixed-type differential equation]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.R. Garabedian,   "Partial differential equations" , Wiley  (1964)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R.J.P. Groothhuizen,   "Mixed elliptic-hyperbolic partial differential operators: a case-study in Fourier integral operators" , ''CWI Tracts'' , '''16''' , CWI , Amsterdam  (1985)  (Thesis Free University Amsterdam)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  M.M. Smirnov,   "Equations of mixed type" , Amer. Math. Soc.  (1978)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  T.V. Gramtcheff,   "An application of Airy functions to the Tricomi problem"  ''Math. Nachr.'' , '''102'''  (1981)  pp. 169–181</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  S. Agmon,   "Boundary value problems for equations of mixed type"  G. Sansone (ed.) , ''Convegno Internaz. Equazioni Lineari alle Derivati Parziali (Trieste, 1954)'' , Cremonese  (1955)  pp. 65–68</TD></TR></table>