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

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.

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