Green’s function for elliptic systems: existence and Delmotte–Deuschel bounds

This paper is divided into two parts: In the main deterministic part, we prove that for an open domain D ⊂ R d with d ≥ 2 , for every (measurable) uniformly elliptic tensor field a and for almost every point y ∈ D , there exists a unique Green’s function centred in y associated to the vectorial operator - ∇ · a ∇ in D . This result implies the existence of the fundamental solution for elliptic systems when d > 2 , i.e. the Green function for - ∇ · a ∇ in R d . In the second part, we introduce a shift-invariant ensemble ⟨ · ⟩ over the set of uniformly elliptic tensor fields, and infer for the fundamental solution G some pointwise bounds for ⟨ | G ( · ; x , y ) | ⟩ , ⟨ | ∇ x G ( · ; x , y ) | ⟩ and ⟨ | ∇ x ∇ y G ( · ; 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.