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A boundary value problem for a Chaplygin-type equation
 
A boundary value problem for a Chaplygin-type equation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043640/g0436401.png" /></td> </tr></table>
+
$$
 +
K ( y) z _ {xx} + z _ {yy}  = 0,
 +
$$
  
in which the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043640/g0436402.png" /> increases, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043640/g0436403.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043640/g0436404.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043640/g0436405.png" />. The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043640/g0436406.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043640/g0436407.png" />, parabolic on the line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043640/g0436408.png" />, and hyperbolic for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043640/g0436409.png" />. The half-plane of hyperbolicity is covered by two families of characteristics, which satisfy the equations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043640/g04364010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043640/g04364011.png" />.
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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" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/g043640a.gif" />
Line 9: Line 32:
 
Figure: g043640a
 
Figure: g043640a
  
The characteristics of one of these families merge with the characteristics of the other on the line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043640/g04364012.png" />.
+
The characteristics of one of these families merge with the characteristics of the other on the line $  y = 0 $.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043640/g04364013.png" /> be a simply-connected domain with as boundary a sufficiently-smooth contour <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043640/g04364014.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043640/g04364015.png" /> or pieces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043640/g04364016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043640/g04364017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043640/g04364018.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043640/g04364019.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043640/g04364020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043640/g04364021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043640/g04364022.png" /> being the characteristics of one family, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043640/g04364023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043640/g04364024.png" /> of the other (see Fig.). The theorem on the existence and the uniqueness of solutions of the following boundary value problems is valid in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043640/g04364025.png" />: the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043640/g04364026.png" /> is given on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043640/g04364027.png" />; the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043640/g04364028.png" /> is given on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043640/g04364029.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043640/g04364030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043640/g04364031.png" />) 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]]].
+
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====
 
====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>
 
<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====
 
====Comments====
 
  
 
====References====
 
====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>
 
<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>

Latest revision as of 19:41, 5 June 2020


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)

Comments

References

[a1] L. Bers, "Mathematical aspects of subsonic and transonic gas dynamics" , Wiley (1958)
How to Cite This Entry:
Gellerstedt problem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Gellerstedt_problem&oldid=47064
This article was adapted from an original article by L.P. Kuptsov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article