MiS Preprint Repository

Delve into the future of research at MiS with our preprint repository. Our scientists are making groundbreaking discoveries and sharing their latest findings before they are published. Explore repository to stay up-to-date on the newest developments and breakthroughs.

MiS Preprint

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

Joseph G. Conlon, Arianna Giunti and Felix Otto


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.

Jun 14, 2016
Jun 17, 2016
MSC Codes:
35J08, 35B27, 35J47, 60H25

Related publications

2017 Journal Open Access
Joseph G. Conlon, Arianna Giunti and Felix Otto

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

In: Calculus of variations and partial differential equations, 56 (2017) 6, p. 163