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

Joseph G. Conlon, Arianna Giunti, and Felix Otto

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Submission date: 14. Jun. 2016
Pages: 50
published in: Calculus of variations and partial differential equations, 56 (2017) 6, art-no. 163 
DOI number (of the published article): 10.1007/s00526-017-1255-0
Bibtex
MSC-Numbers: 35J08, 35B27, 35J47, 60H25
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Abstract:
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)|⟩, ⟨|∇_xG(⋅;x,y)|⟩ and ⟨|∇_x∇_yG(⋅;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.