Green\u2019s function for elliptic systems: existence and Delmotte\u2013Deuschel bounds

Arianna Giunti,Joseph G Conlon,Felix Otto

doi:10.1007/s00526-017-1255-0

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 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 anabla 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 anabla 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 _xnabla _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.

Highlights

In this work we shall be concerned with the study of the Green function for the second order vectorial operator in divergence form −∇ · a∇, on a general open domain D ⊂ Rd with d ≥ 2
That is we prove for every a and almost every y ∈ D the function G D(a; ·, y) exists
In this paper we mainly provide two existence results for the Green function in a domain D ⊂ Rd with d ≥ 2

Summary

Introduction

In this work we shall be concerned with the study of the Green function for the second order vectorial operator in divergence form −∇ · a∇, on a general open domain D ⊂ Rd with d ≥ 2. In the scalar case it is a well-known result (see e.g. Grüter and Widman [20], Littman et al [23]) that for any measurable and strongly uniformly elliptic a, the Green function exists and has optimal pointwise decay, e.g. as the Green function associated to the Laplacian (c.f. the r.h.s. in (1)) This bound on the decay is a consequence of the De Giorgi-Nash-Moser theory, which does not hold in the case of systems. One is motivated to expect pointwise bounds on averages of second derivatives of Green functions for certain parabolic PDE in divergence form with random coefficients, a conjecture formulated by Spencer [29] and proven in [11]

Notation and setting

Main result and remarks

Proof of Theorem 1

Proof of Corollary 1 and 2

Fourier approach