Green’s function for elliptic systems: existence and Delmotte–Deuschel bounds
Joseph G. Conlon, Arianna Giunti, Félix Otto · Calculus of Variations and Partial Differential Equations · 2017
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 $$- abla \cdot a abla $$ in D. This result implies the existence of the fundamental solution for elliptic systems when $$d>2$$ , i.e. the Green function for $$- abla \cdot a abla $$ 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 $$\langle |G(\cdot ; x,y)|\rangle $$ , $$\langle | abla _x G(\cdot ; x,y)|\rangle $$ and $$\langle | abla _x abla _y G(\cdot ; x,y)|\rangle $$ . These estimates scale optimally in space and provide a generalisation to systems of the bounds obtained by Delmotte and Deuschel for the scalar case.