Expansion of Green's function and regularity of Robin's function for elliptic operators in divergence form

We consider Green’s function $G_K$ of the elliptic operator in divergence form ${\mathcal{L}}_K=-\text{div}(K(x)\mathrm{\nabla })$ on a bounded smooth domain $\mathrm{\Omega }\subseteq {\mathbb{R}}^n(n\ge 2)$ with zero Dirichlet boundary condition, where $K$ is a smooth positively definite matrix-valued function on $\mathrm{\Omega }$. We obtain a high-order asymptotic expansion of $G_K(x,y)$, which defines uniquely a regular part $H_K(x,y)$. Moreover, we prove that the associated Robin’s function $R_K(x)=H_K(x,x)$ is smooth in $\mathrm{\Omega }$, despite the regular part $H_K\mathrm{\notin }{C}^1(\mathrm{\Omega }\times \mathrm{\Omega })$ in general.