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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 } $.

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/g043640a.gif" />

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 [[#References|[1]]] by methods developed by F. Tricomi [[#References|[2]]] for the [[Tricomi problem|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 [[#References|[3]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Gellerstedt,   "Quelques problèmes mixtes pour l'équation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043640/g04364032.png" />"  ''Ark. Mat. Astr. Fysik'' , '''26A''' :  3  (1937)  pp. 1–32</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  F.G. Tricomi,   "Integral equations" , Interscience  (1957)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M.M. Smirnov,   "Equations of mixed type" , Amer. Math. Soc.  (1978)  (Translated from Russian)</TD></TR></table>

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L. Bers,   "Mathematical aspects of subsonic and transonic gas dynamics" , Wiley  (1958)</TD></TR></table>