### Abstract

For every ε > 0,we consider the Green's matrix G_{ε}(x,y) of the Stokes equations describing the motion of incompressible fluids in a bounded domain Ω_{ε} ⊂ ℝ^{d}, which is a family of perturbation of domains from Ω ≡ Ω_{0} with the smooth boundary ∂Ω. Assuming the volume preserving property, that is, vol.Ω_{ε} = vol.Ω for all ε > 0, we give an explicit representation formula for δG(x,y) ≡ lim_{ε}→+0 ε^{-1}(G_{ε}(x,y) - G_{0})) in terms of the boundary integral on ∂Ω of G_{0}(x,y). Our result may be regarded as a classical Hadamard variational formula for the Green's functions of the elliptic boundary value problems.

Original language | English |
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Pages (from-to) | 1005-1055 |

Number of pages | 51 |

Journal | Archive for Rational Mechanics and Analysis |

Volume | 208 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2013 |

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### ASJC Scopus subject areas

- Analysis
- Mechanical Engineering
- Mathematics (miscellaneous)