# Hadamard variational formula

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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=47159
This article was adapted from an original article by I.A. Aleksandrov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article