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 $\mathcal{L}_K=-\text{div}(K(x)\nabla )$ on a bounded smooth domain $\Omega\subseteq\mathbb{R}^n (n\geq 2)$ with zero Dirichlet boundary condition, where $K$ is a smooth positively definite matrix-valued function on $\Omega$. 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 $\Omega$, despite the regular part $H_K\notin C^1(\Omega\times\Omega)$ in general.