Green's function for elliptic systems: existence and Delmotte-Deuschel bounds

TL;DR

This paper establishes the existence of Green's functions for elliptic systems with measurable coefficients in open domains and derives optimal pointwise bounds for their ensemble averages, generalizing scalar case results.

Contribution

It proves the existence and uniqueness of Green's functions for elliptic systems in arbitrary domains and extends Delmotte-Deuschel bounds to vectorial systems under stationarity.

Findings

Existence of Green's functions for elliptic systems in open domains.

Pointwise bounds for ensemble averages of Green's functions and their derivatives.

Generalization of scalar bounds to vectorial elliptic systems.

Abstract

We prove that for an open domain $D \subset \mathbb{R}^d $ with $d \geq 2 $ , for every (measurable) uniformly elliptic tensor field $a$ and for almost every point $y \in D$ , there exists a unique Green's function centred in $ y $ associated to the vectorial operator $ -\nabla \cdot a\nabla $ in D. In particular, when $d > 2$ this result also implies the existence of the fundamental solution for elliptic systems, i.e. the Green function for $ -\nabla \cdot a\nabla $ in $ \mathbb{R}^d $. Moreover, introducing an ensemble $\langle\cdot \rangle$ over the set of uniformly elliptic tensor fields, under the assumption of stationarity we infer for the fundamental solution $G$ some pointwise bounds for $\langle |G(\cdot; x,y)|\rangle$, $\langle|\nabla_x G(\cdot; x,y)|\rangle$ and $\langle |\nabla_x\nabla_y G(\cdot; x,y)|\rangle$. These estimates scale optimally in space and provide a generalization to systems of the bounds obtained by Delmotte and Deuschel for the scalar case.