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

Abstract

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 $\hat{I}\text{c◯}\subseteq {\mathbb{R}}^n(n\ge 2)$ with zero Dirichlet boundary condition, where $K$ is a smooth positively definite matrix-valued function on $\hat{I}\text{c◯}$. 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 $\hat{I}\text{c◯}$, despite the regular part $H_K\notin C^1(\hat{I}\text{c◯}\times \hat{I}\text{c◯})$ in general.

16 pages