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MiS Preprint
34/2016
Green's function for elliptic systems: Existence and Delmotte-Deuschel bounds
Joseph G. Conlon, Arianna Giunti and Felix Otto
Abstract
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 $d \geq 2$, 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 $D$.
This result implies the existence of the fundamental solution for elliptic systems when $d>2$, i.e. the Green function for $-\nabla \cdot a\nabla$ in $\mathbb{R}^d$.
In the second part, we introduce a shift-invariant ensemble $\langle\cdot \rangle$ over the set of uniformly elliptic tensor fields, and 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 generalisation to systems of the bounds obtained by Delmotte and Deuschel for the scalar case.
