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Difference between revisions of "Bitsadze equation"

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The partial differential equation that can be written in complex form as follows:
 
The partial differential equation that can be written in complex form as follows:
  
<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/b/b016/b016590/b0165901.png" /></td> </tr></table>
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$$
 +
4w _ {\overline{z}\; \overline{z}\; }  \equiv \
 +
w _ {xx} +2 iw _ {xy} -
 +
w _ {yy}  = 0,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016590/b0165902.png" />, and that can be reduced to the elliptic system
+
where $  w(z) = u + iv, z = x + iy $,  
 +
and that can be reduced to the elliptic system
  
<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/b/b016/b016590/b0165903.png" /></td> </tr></table>
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$$
 +
u _ {xx} - u _ {yy} -2v _ {xy}  = 0,
 +
$$
  
<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/b/b016/b016590/b0165904.png" /></td> </tr></table>
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$$
 +
v _ {xx} - v _ {yy} + 2u _ {xy}  = 0,
 +
$$
  
in the real independent variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016590/b0165905.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016590/b0165906.png" />. The homogeneous Dirichlet problem in a disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016590/b0165907.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016590/b0165908.png" />, where the radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016590/b0165909.png" /> is as small as one pleases, for the Bitsadze equation has an infinite number of linearly independent solutions [[#References|[1]]]. The Dirichlet problem for the inhomogeneous equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016590/b01659010.png" /> in the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016590/b01659011.png" /> is normally solvable according to Hausdorff, since it is neither a Fredholm problem nor Noetherian; in a bounded domain containing a segment of the straight line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016590/b01659012.png" />, this problem is not even a Hausdorff problem, even though the homogeneous problem has only one zero solution [[#References|[2]]].
+
in the real independent variables $  x $
 +
and $  y $.  
 +
The homogeneous Dirichlet problem in a disc $  C $:
 +
$  | z - z _ {0} | < \epsilon $,  
 +
where the radius $  \epsilon $
 +
is as small as one pleases, for the Bitsadze equation has an infinite number of linearly independent solutions [[#References|[1]]]. The Dirichlet problem for the inhomogeneous equation $  w _ {\overline{z}\; \overline{z}\; }  = f $
 +
in the disc $  C $
 +
is normally solvable according to Hausdorff, since it is neither a Fredholm problem nor Noetherian; in a bounded domain containing a segment of the straight line $  y = 0 $,  
 +
this problem is not even a Hausdorff problem, even though the homogeneous problem has only one zero solution [[#References|[2]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.V. Bitsadze,  "On the uniqueness of the solution of the Dirichlet problem for elliptic partial differential operators"  ''Uspekhi Mat. Nauk'' , '''3''' :  6  (1948)  pp. 211–212  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.V. Bitsadze,  "Boundary value problems for second-order elliptic equations" , North-Holland  (1968)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  C. Miranda,  "Partial differential equations of elliptic type" , Springer  (1970)  (Translated from Italian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  L. Bers,  F. John,  M. Schechter,  "Partial differential equations" , Interscience  (1964)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.V. Bitsadze,  "On the uniqueness of the solution of the Dirichlet problem for elliptic partial differential operators"  ''Uspekhi Mat. Nauk'' , '''3''' :  6  (1948)  pp. 211–212  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.V. Bitsadze,  "Boundary value problems for second-order elliptic equations" , North-Holland  (1968)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  C. Miranda,  "Partial differential equations of elliptic type" , Springer  (1970)  (Translated from Italian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  L. Bers,  F. John,  M. Schechter,  "Partial differential equations" , Interscience  (1964)</TD></TR></table>

Latest revision as of 10:59, 29 May 2020


The partial differential equation that can be written in complex form as follows:

$$ 4w _ {\overline{z}\; \overline{z}\; } \equiv \ w _ {xx} +2 iw _ {xy} - w _ {yy} = 0, $$

where $ w(z) = u + iv, z = x + iy $, and that can be reduced to the elliptic system

$$ u _ {xx} - u _ {yy} -2v _ {xy} = 0, $$

$$ v _ {xx} - v _ {yy} + 2u _ {xy} = 0, $$

in the real independent variables $ x $ and $ y $. The homogeneous Dirichlet problem in a disc $ C $: $ | z - z _ {0} | < \epsilon $, where the radius $ \epsilon $ is as small as one pleases, for the Bitsadze equation has an infinite number of linearly independent solutions [1]. The Dirichlet problem for the inhomogeneous equation $ w _ {\overline{z}\; \overline{z}\; } = f $ in the disc $ C $ is normally solvable according to Hausdorff, since it is neither a Fredholm problem nor Noetherian; in a bounded domain containing a segment of the straight line $ y = 0 $, this problem is not even a Hausdorff problem, even though the homogeneous problem has only one zero solution [2].

References

[1] A.V. Bitsadze, "On the uniqueness of the solution of the Dirichlet problem for elliptic partial differential operators" Uspekhi Mat. Nauk , 3 : 6 (1948) pp. 211–212 (In Russian)
[2] A.V. Bitsadze, "Boundary value problems for second-order elliptic equations" , North-Holland (1968) (Translated from Russian)
[3] C. Miranda, "Partial differential equations of elliptic type" , Springer (1970) (Translated from Italian)
[4] L. Bers, F. John, M. Schechter, "Partial differential equations" , Interscience (1964)
How to Cite This Entry:
Bitsadze equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bitsadze_equation&oldid=17152
This article was adapted from an original article by A.M. Nakhushev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article