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

Hideo Kozono, Erika Ushikoshi

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    6 Citations (Scopus)

    Abstract

    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.

    Original languageEnglish
    Pages (from-to)1005-1055
    Number of pages51
    JournalArchive for Rational Mechanics and Analysis
    Volume208
    Issue number3
    DOIs
    Publication statusPublished - 2013

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    ASJC Scopus subject areas

    • Analysis
    • Mechanical Engineering
    • Mathematics (miscellaneous)

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