Abstract
We study the existence of the Green function for an elliptic system in divergence form -nabla cdot anabla in {mathbb {R}}^d, with d>2. The tensor field a=a(x) is only assumed to be bounded and lambda -coercive. For almost every point y in {mathbb {R}}^d, the existence of a Green’s function G(cdot , y) centered in y has been proven in Conlon et al. (Calc Var PDEs 56(6), (2017))[2]. In this paper we show that the set of points y in {mathbb {R}}^d for which G( cdot , y) does not exist has zero p-capacity, for an exponent p >2 depending only on the dimension d and the ellipticity ratio of a.
Highlights
This paper is an extension of [2] and further investigates the existence of a Green’s function for the second-order elliptic operator −∇ · a∇ in Rd, with d > 2
We focus on the case of systems of m equations, namely when a is a measurable tensor field a : Rd → L(Rm×d ; Rm×d ), with m being any positive integer
In [2], Conlon and the authors show that a Green’s function G(·, y) centered in y exists for every coefficient field a satisfying (0.1) and for (Lebesgue-)almost every point y ∈ Rd
Summary
This paper is an extension of [2] and further investigates the existence of a Green’s function for the second-order elliptic operator −∇ · a∇ in Rd , with d > 2. We improve this result by showing that the exceptional set of points y ∈ Rd for which G(a; ·, y) does not exist has p-capacity zero [4, Definition 4.10], for an exponent p > 2 depending only on the dimension d and the ellipticity ratio λ.
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