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 - 2013/6

Y1 - 2013/6

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.

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U2 - 10.1007/s00205-013-0611-2

DO - 10.1007/s00205-013-0611-2

M3 - Article

AN - SCOPUS:84877052465

SN - 0003-9527

VL - 208

SP - 1005

EP - 1055

JO - Archive for Rational Mechanics and Analysis

JF - Archive for Rational Mechanics and Analysis

IS - 3

ER -