Estimates of the Green’s Functions for the Elasto-static Equations and Stokes Equations in a Three Dimensional Lipschitz Domain

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In this paper, we study a Green’s functions GE, GS for an elasto-static equations and Stokes equations in a three-dimensional bounded Lipschitz domain Ω. We prove that there is a positive constant c > 0 depending on the Lipschitz constant such that \(|G^E(X,Y)|, |G^S(X,Y)| \leq \frac{c}{|X-Y|}\) for all \(X,Y \in \overline {\Omega}\). Furthermore, we show that there is a positive constant η ∈ (0,1) depending on the Lipschitz constant such that \(|G^E(X,Y)|, |G^S(X,Y)| \leq c\frac{min(dist(X,{\partial} {\Omega}), dist(Y, {\partial} {\Omega}))^\eta}{|X-Y|^{1 + \eta}}\) for all \(X,Y \in \overline {\Omega}\).