View source for Hadamard variational formula

<!--
h0461001.png
$#A+1 = 29 n = 0
$#C+1 = 29 : ~/encyclopedia/old_files/data/H046/H.0406100 Hadamard variational formula
Automatically converted into TeX, above some diagnostics.
Please remove this comment and the {{TEX|auto}} line below,
if TeX found to be correct.
-->

{{TEX|auto}}
{{TEX|done}}

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|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]]].

====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>

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

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S.E. Warschawski,   "On Hadamard's variation formula for Green's function"  ''J. Math. Mech.'' , '''9'''  (1960)  pp. 497–511</TD></TR></table>