MiS Preprint Repository

This paper is divided into two parts: In the main deterministic part, we prove that for an open domain $D\subset {\mathbb{R}}^d$ with $d\ge 2$, 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 $-\mathrm{\nabla }\cdot a\mathrm{\nabla }$ in $D$.

This result implies the existence of the fundamental solution for elliptic systems when $d>2$, i.e. the Green function for $-\mathrm{\nabla }\cdot a\mathrm{\nabla }$ in ${\mathbb{R}}^d$.

In the second part, we introduce a shift-invariant ensemble $⟨\cdot ⟩$ over the set of uniformly elliptic tensor fields, and infer for the fundamental solution $G$ some pointwise bounds for $⟨|G(\cdot ;x,y)|⟩$, $⟨|{\mathrm{\nabla }}_{x}G(\cdot ;x,y)|⟩$ and $⟨|{\mathrm{\nabla }}_x{\mathrm{\nabla }}_{y}G(\cdot ;x,y)|⟩$.

These estimates scale optimally in space and provide a generalisation to systems of the bounds obtained by Delmotte and Deuschel for the scalar case.