On the existence of the Green function for elliptic systems in divergence form

2019/11/05 by Giunti, Arianna, Otto, Felix
#28A12 #35J08 #35J47 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1911.02110

Abstract

We study the existence of the Green function for an elliptic system in divergence form -∇⋅ a∇ in ℝ^d, with d>2. The tensor field a=a(x) is only assumed to be bounded and λ-coercive. For almost every point y ∈ ℝ^d, the existence of a Green's function G(a; ⋅, y) centered in y has been proven in [J. Conlon, A. Giunti and F.Otto, "Green's function for elliptic systems: Delmotte-Deuschel bounds", 2017]. In this paper, we show that the set of points y ∈ ℝ^d for which G(a; ⋅, 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.

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