Difference between revisions of "Hadamard variational formula"
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The 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|Green function]] | + | 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
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