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 [1], [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.

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 [1]–[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) 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 [5]).

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

References

[1]	F. Tricomi, "On second-order linear partial differential equations of mixed type" , Moscow-Leningrad (1947) (In Russian; translated from Italian)
[2]	F.G. Tricomi, "Equazioni a derivate parziale" , Cremonese (1957)
[3]	A.V. Bitsadze, "Zum Problem der Gleichungen vom gemischten Typus" , Deutsch. Verlag Wissenschaft. (1957) (Translated from Russian)
[4]	A.V. Bitsadse, "Equations of mixed type" , Pergamon (1964) (Translated from Russian)
[5]	L. Bers, "Mathematical aspects of subsonic and transonic gas dynamics" , Wiley (1958)
[6]	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
[7]	T.D. Dzhuraev, "Boundary value problems for equations of mixed and mixed-composite type" , Tashkent (1979) (In Russian)

Comments

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

For additional references see also Mixed-type differential equation.

References

[a1]	P.R. Garabedian, "Partial differential equations" , Wiley (1964)
[a2]	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)
[a3]	M.M. Smirnov, "Equations of mixed type" , Amer. Math. Soc. (1978) (Translated from Russian)
[a4]	T.V. Gramtcheff, "An application of Airy functions to the Tricomi problem" Math. Nachr. , 102 (1981) pp. 169–181
[a5]	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