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 \(G(\cdot , y)\) centered in y has been proven in Conlon et al. (Calc Var PDEs 56(6), (2017))[2]. In this paper we show that the set of points \(y \in {\mathbb {R}}^d\) for which \(G( \cdot , y)\) does not exist has zero p-capacity, for an exponent \(p >2\) depending only on the dimension d and the ellipticity ratio of a.

我们研究了格林函数的存在性散度型椭圆系统\（ - \ nabla \ CDOT一个\ nabla \）在\（{\ mathbb {R}} ^ d \） ，用\（d> 2 \）。张量字段\（a = a（x）\）仅假定为有界且\（\ lambda \）为强制。对于几乎每个点\（y \ in {\ mathbb {R}} ^ d \），以y为中心的格林函数\（G（\ cdot，y）\）的存在已在Conlon等人中得到了证明。（Calc Var PDEs 56（6），（2017））[2]。在本文中，我们表明，该组点\（在Y \ {\ mathbb {R}} ^ d \）为其\（G（\ CDOT，Y）\）不存在具有零p-容量，对于指数\（P> 2 \）仅在尺寸取决于d和的椭圆率比一个。