Hadamard variational formula

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