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The formula
 
The formula
  
<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/h/h046/h046100/h0461001.png" /></td> </tr></table>
+
$$
 +
g  ^ {*} ( z , \zeta )  = g ( z , \zeta ) +
 +
$$
  
<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/h/h046/h046100/h0461002.png" /></td> </tr></table>
+
$$
 +
- \sum _ { k = 1 } ^ { n }  \epsilon _ {k} \int\limits _ { 0 } ^ { {l _ k } }
 +
\frac{
 +
\partial  g ( \phi _ {k} ( s ) , z ) }{\partial  n  ^ {(k)}}
 +
 +
\frac{
 +
\partial  g ( \phi _ {k} ( s ) , \zeta ) }{\partial  n
 +
^ {(k)} }
 +
\phi _ {k} ( s )  ds + O ( \epsilon  ^ {2} )
 +
$$
  
for the [[Green function|Green function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046100/h0461003.png" /> of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046100/h0461004.png" />-connected domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046100/h0461005.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046100/h0461006.png" />) in the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046100/h0461007.png" />-plane. Hadamard's variational formula is applicable if: 1) the boundary components <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046100/h0461008.png" /> of the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046100/h0461009.png" /> are twice-differentiable closed Jordan curves, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046100/h04610010.png" /> is the arc length on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046100/h04610011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046100/h04610012.png" />; 2) the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046100/h04610013.png" /> are so small that the ends of the segments of the interior normals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046100/h04610014.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046100/h04610015.png" /> of length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046100/h04610016.png" /> lying in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046100/h04610017.png" /> form continuously-differentiable curves, bounding an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046100/h04610018.png" />-connected domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046100/h04610019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046100/h04610020.png" />; and 3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046100/h04610021.png" /> is a fixed point in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046100/h04610022.png" />. Hadamard's variational formula represents the Green function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046100/h04610023.png" /> of the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046100/h04610024.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046100/h04610025.png" /> with a uniform estimate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046100/h04610026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046100/h04610027.png" />, of the remainder term in the direct product of the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046100/h04610028.png" /> and an arbitrary compact set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046100/h04610029.png" />. Hadamard's variational formula can also be used for the Green function of a finite Riemann surface with boundary.
+
for the [[Green function|Green function]] $  g( z, \zeta ) $
 +
of an $  n $-
 +
connected domain $  G $(
 +
$  n = 1, 2 , . . . $)  
 +
in the complex $  z $-
 +
plane. Hadamard's variational formula is applicable if: 1) the boundary components $  \Gamma _ {k} = \{ {z } : {z = \phi _ {k} ( s) } \} $
 +
of the domain $  G $
 +
are twice-differentiable closed Jordan curves, where $  s $
 +
is the arc length on $  \Gamma _ {k} $,  
 +
0 \leq  s \leq  l _ {k} $;  
 +
2) the numbers $  \epsilon _ {k} > 0 $
 +
are so small that the ends of the segments of the interior normals $n^{(k)} $
 +
to $  \Gamma _ {k} $
 +
of length $  \epsilon _ {k} \phi _ {k} ( s ) $
 +
lying in $  G $
 +
form continuously-differentiable curves, bounding an $  n $-
 +
connected domain $  G  ^ {*} $,  
 +
$  \overline{ {G  ^ {*} }}\; \subset  G $;  
 +
and 3) $  \zeta $
 +
is a fixed point in $  G  ^ {*} $.  
 +
Hadamard's variational formula represents the Green function $  g  ^ {*} ( z, \zeta ) $
 +
of the domain $  G  ^ {*} $
 +
by $  g( z, \zeta ) $
 +
with a uniform estimate $  O ( \epsilon  ^ {2} ) $,
 +
$  \epsilon = \max \{ \epsilon _ {k} , 0\leq  k \leq  n \} $,  
 +
of the remainder term in the direct product of the domain $  G  ^ {*} $
 +
and an arbitrary compact set in $  G $.  
 +
Hadamard's variational formula can also be used for the Green function of a finite Riemann surface with boundary.
  
 
The formula was proposed by J. Hadamard [[#References|[1]]].
 
The formula was proposed by J. Hadamard [[#References|[1]]].
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====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J. Hadamard,  "Memoire sur le problème d'analyse relatif a l'équilibre des plagues élastiques eucastrées"  ''Mém. prés. par divers savants à l'Acad. Sci.'' , '''33'''  (1907)  (Also: Oeuvres, Vol. II, C.N.R.S. (1968), pp. 515–631)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M. Schiffer,  D.C. Spencer,  "Functionals of finite Riemann surfaces" , Princeton Univ. Press  (1954)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J. Hadamard,  "Memoire sur le problème d'analyse relatif a l'équilibre des plagues élastiques eucastrées"  ''Mém. prés. par divers savants à l'Acad. Sci.'' , '''33'''  (1907)  (Also: Oeuvres, Vol. II, C.N.R.S. (1968), pp. 515–631)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M. Schiffer,  D.C. Spencer,  "Functionals of finite Riemann surfaces" , Princeton Univ. Press  (1954)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Latest revision as of 20:37, 16 January 2024


The formula

$$ g ^ {*} ( z , \zeta ) = g ( z , \zeta ) + $$

$$ - \sum _ { k = 1 } ^ { n } \epsilon _ {k} \int\limits _ { 0 } ^ { {l _ k } } \frac{ \partial g ( \phi _ {k} ( s ) , z ) }{\partial n ^ {(k)}} \frac{ \partial g ( \phi _ {k} ( s ) , \zeta ) }{\partial n ^ {(k)} } \phi _ {k} ( s ) ds + O ( \epsilon ^ {2} ) $$

for the Green function $ g( z, \zeta ) $ of an $ n $- connected domain $ G $( $ n = 1, 2 , . . . $) in the complex $ z $- plane. Hadamard's variational formula is applicable if: 1) the boundary components $ \Gamma _ {k} = \{ {z } : {z = \phi _ {k} ( s) } \} $ of the domain $ G $ are twice-differentiable closed Jordan curves, where $ s $ is the arc length on $ \Gamma _ {k} $, $ 0 \leq s \leq l _ {k} $; 2) the numbers $ \epsilon _ {k} > 0 $ are so small that the ends of the segments of the interior normals $n^{(k)} $ to $ \Gamma _ {k} $ of length $ \epsilon _ {k} \phi _ {k} ( s ) $ lying in $ G $ form continuously-differentiable curves, bounding an $ n $- connected domain $ G ^ {*} $, $ \overline{ {G ^ {*} }}\; \subset G $; and 3) $ \zeta $ is a fixed point in $ G ^ {*} $. Hadamard's variational formula represents the Green function $ g ^ {*} ( z, \zeta ) $ of the domain $ G ^ {*} $ by $ g( z, \zeta ) $ with a uniform estimate $ O ( \epsilon ^ {2} ) $, $ \epsilon = \max \{ \epsilon _ {k} , 0\leq k \leq n \} $, of the remainder term in the direct product of the domain $ G ^ {*} $ and an arbitrary compact set in $ G $. Hadamard's variational formula can also be used for the Green function of a finite Riemann surface with boundary.

The formula was proposed by J. Hadamard [1].

References

[1] J. Hadamard, "Memoire sur le problème d'analyse relatif a l'équilibre des plagues élastiques eucastrées" Mém. prés. par divers savants à l'Acad. Sci. , 33 (1907) (Also: Oeuvres, Vol. II, C.N.R.S. (1968), pp. 515–631)
[2] M. Schiffer, D.C. Spencer, "Functionals of finite Riemann surfaces" , Princeton Univ. Press (1954)

Comments

For a proof of Hadamard's variational formula under minimal hypotheses, plus further references, see [a1].

References

[a1] S.E. Warschawski, "On Hadamard's variation formula for Green's function" J. Math. Mech. , 9 (1960) pp. 497–511
How to Cite This Entry:
Hadamard variational formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hadamard_variational_formula&oldid=12275
This article was adapted from an original article by I.A. Aleksandrov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article