On the existence of the Green function for elliptic systems in divergence form

Abstract

We study the existence of the Green function for an elliptic system in divergence form $-\nabla\cdot a\nabla$ in $\mathbb{R}^d$, with $d>2$. The tensor field $a=a(x)$ is only assumed to be bounded and $\lambda$-coercive. For almost every point $y \in \mathbb{R}^d$, the existence of a Green's function $G(a; \cdot, y)$ centered in $y$ has been proven in [J. Conlon, A. Giunti and F.Otto, "Green's function for elliptic systems: Delmotte-Deuschel bounds", 2017]. In this paper, we show that the set of points $y \in \mathbb{R}^d$ for which $G(a; \cdot, y)$ does not exist has zero $p$-capacity, for an exponent $p >2$ depending only on the dimension $d$ and the ellipticity ratio of $a$.