### 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)

### Cite this

**Hadamard Variational Formula for the Green's Function of the Boundary Value Problem on the Stokes Equations.** / Kozono, Hideo; Ushikoshi, Erika.

Research output: Contribution to journal › Article

*Archive for Rational Mechanics and Analysis*, vol. 208, no. 3, pp. 1005-1055. https://doi.org/10.1007/s00205-013-0611-2

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TY - JOUR

T1 - Hadamard Variational Formula for the Green's Function of the Boundary Value Problem on the Stokes Equations

AU - Kozono, Hideo

AU - Ushikoshi, Erika

PY - 2013

Y1 - 2013

N2 - For every ε > 0,we consider the Green's matrix Gε(x,y) of the Stokes equations describing the motion of incompressible fluids in a bounded domain Ωε ⊂ ℝd, which is a family of perturbation of domains from Ω ≡ Ω0 with the smooth boundary ∂Ω. Assuming the volume preserving property, that is, vol.Ωε = vol.Ω for all ε > 0, we give an explicit representation formula for δG(x,y) ≡ limε→+0 ε-1(Gε(x,y) - G0)) in terms of the boundary integral on ∂Ω of G0(x,y). Our result may be regarded as a classical Hadamard variational formula for the Green's functions of the elliptic boundary value problems.

AB - For every ε > 0,we consider the Green's matrix Gε(x,y) of the Stokes equations describing the motion of incompressible fluids in a bounded domain Ωε ⊂ ℝd, which is a family of perturbation of domains from Ω ≡ Ω0 with the smooth boundary ∂Ω. Assuming the volume preserving property, that is, vol.Ωε = vol.Ω for all ε > 0, we give an explicit representation formula for δG(x,y) ≡ limε→+0 ε-1(Gε(x,y) - G0)) in terms of the boundary integral on ∂Ω of G0(x,y). Our result may be regarded as a classical Hadamard variational formula for the Green's functions of the elliptic boundary value problems.

UR - http://www.scopus.com/inward/record.url?scp=84877052465&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84877052465&partnerID=8YFLogxK

U2 - 10.1007/s00205-013-0611-2

DO - 10.1007/s00205-013-0611-2

M3 - Article

VL - 208

SP - 1005

EP - 1055

JO - Archive for Rational Mechanics and Analysis

JF - Archive for Rational Mechanics and Analysis

SN - 0003-9527

IS - 3

ER -