# Tricomi problem

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).

The Tricomi problem for the Tricomi equation

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

was first posed and studied by F. Tricomi , . 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.

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. 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 ).

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) 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 ).

The Tricomi problem has been generalized both to the case of mixed-type differential equations with curves of parabolic degeneracy (see ) and to the case of equations of mixed hyperbolic-parabolic type (see ).

How to Cite This Entry:
Tricomi problem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tricomi_problem&oldid=49034
This article was adapted from an original article by A.M. Nakhushev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article