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 $ \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.