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