For every \documentclass[12pt]{minimal}
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\begin{document}$${\varepsilon > 0}$$\end{document}, we consider the Green’s matrix \documentclass[12pt]{minimal}
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\begin{document}$${G_{\varepsilon}(x, y)}$$\end{document} of the Stokes equations describing the motion of incompressible fluids in a bounded domain \documentclass[12pt]{minimal}
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\begin{document}$${\Omega_{\varepsilon} \subset \mathbb{R}^d}$$\end{document}, which is a family of perturbation of domains from \documentclass[12pt]{minimal}
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\begin{document}$${\Omega\equiv \Omega_0}$$\end{document} with the smooth boundary \documentclass[12pt]{minimal}
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\begin{document}$${\partial\Omega}$$\end{document}. Assuming the volume preserving property, that is, \documentclass[12pt]{minimal}
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\begin{document}$${\mbox{vol.}\Omega_{\varepsilon} = \mbox{vol.}\Omega}$$\end{document} for all \documentclass[12pt]{minimal}
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\begin{document}$${\varepsilon > 0}$$\end{document}, we give an explicit representation formula for \documentclass[12pt]{minimal}
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\begin{document}$${\delta G(x, y) \equiv \lim_{\varepsilon\to +0}\varepsilon^{-1}(G_{\varepsilon}(x, y) - G_0(x, y))}$$\end{document} in terms of the boundary integral on \documentclass[12pt]{minimal}
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\begin{document}$${\partial \Omega}$$\end{document} of \documentclass[12pt]{minimal}
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\begin{document}$${G_0(x, y)}$$\end{document}. Our result may be regarded as a classical Hadamard variational formula for the Green’s functions of the elliptic boundary value problems.
