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 -