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

TL;DR

This paper proves that for elliptic systems in divergence form with bounded, coercive coefficients, the set of points lacking a Green's function has zero p-capacity, extending previous results to almost all points.

Contribution

It demonstrates that the set of points without a Green's function has zero p-capacity, generalizing earlier existence results to a full measure set.

Findings

Green's function exists for almost every point in 

The set of points without Green's function has zero p-capacity

Results depend only on dimension and ellipticity ratio

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 $a$.