Green’s function for elliptic systems: existence and Delmotte–Deuschel bounds
Tóm tắt
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
$$-\nabla \cdot a\nabla $$
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.
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