Gellerstedt problem

A boundary value problem for a Chaplygin-type equation

$$ K ( y) z _ {xx} + z _ {yy} = 0, $$

in which the function $ K( y) $ increases, $ K( 0) = 0 $ and $ yK ( y) > 0 $ for $ y \neq 0 $. The function $ z( x, y) $ which is to be found is specified on the boundary. This boundary consists of a sufficiently-smooth contour and pieces of characteristics. This equation is elliptic in the half-plane $ y > 0 $, parabolic on the line $ y = 0 $, and hyperbolic for $ y < 0 $. The half-plane of hyperbolicity is covered by two families of characteristics, which satisfy the equations $ y ^ \prime = {[ - K( y) ] } ^ {- 1/2 } $ and $ y ^ \prime = {-[- K( y)] } ^ {- 1/2 } $.

Figure: g043640a

The characteristics of one of these families merge with the characteristics of the other on the line $ y = 0 $.

Let $ E $ be a simply-connected domain with as boundary a sufficiently-smooth contour $ \Gamma $ if $ y \geq 0 $ or pieces $ \Gamma _ {1} $, $ \Gamma _ {2} $, $ \Gamma _ {3} $, and $ \Gamma _ {4} $ if $ y \leq 0 $, $ \Gamma _ {1} $ and $ \Gamma _ {3} $ being the characteristics of one family, and $ \Gamma _ {2} $ and $ \Gamma _ {4} $ of the other (see Fig.). The theorem on the existence and the uniqueness of solutions of the following boundary value problems is valid in $ E $: the function $ z( x, y) $ is given on $ \Gamma + \Gamma _ {1} + \Gamma _ {4} $; the function $ z( x, y) $ is given on $ \Gamma + \Gamma _ {2} + \Gamma _ {3} $.

These problems were first studied (for $ K( y) = \mathop{\rm sgn} y \cdot | y | ^ \alpha $, $ \alpha > 0 $) by S. Gellerstedt [1] by methods developed by F. Tricomi [2] for the Tricomi problem, and represent a generalization of that problem. Gellerstedt's problem has important applications in gas dynamics with velocities around the velocity of sound. These and related problems were studied for certain multiply-connected domains and for linear equations containing lower-order terms [3].

References

[1]	S. Gellerstedt, "Quelques problèmes mixtes pour l'équation " Ark. Mat. Astr. Fysik , 26A : 3 (1937) pp. 1–32
[2]	F.G. Tricomi, "Integral equations" , Interscience (1957)
[3]	M.M. Smirnov, "Equations of mixed type" , Amer. Math. Soc. (1978) (Translated from Russian)

References

[a1]	L. Bers, "Mathematical aspects of subsonic and transonic gas dynamics" , Wiley (1958)