Green’s Matrices of Second Order Elliptic Systems with Measurable Coefficients in Two Dimensional Domains

Gi j(·, y) = 0 on ∂Ω ∀y ∈ Ω, where δik is the Kronecker delta symbol and δy(·) is the Dirac delta function with a unit mass at y. In the scalar case (i.e., when N = 1), the Green’s matrix becomes a real valued function and is usually called the Green’s function. We prove that if Ω has either finite volume or finite width, then there exists a unique Green’s matrix in Ω; see Theorem 2.12. The same is true when Ω is a domain above a Lipschitz graph (e.g., Ω = R+); see Theorem 2.21. We also establish growth properties of the Green’s matrices including logarithmic pointwise bounds. We emphasize that we do not require Ω to be bounded nor to have a regular boundary in Theorem 2.12. Compared to the result of Dolzmann and Müller [4], where Ω is assumed to be a bounded Lipschitz domain (In fact, their methods work whenever there exists an Lp-theory for the equation under consideration on the domain), our result is quite an improvement in this respect. Although there is no Green’s matrix for Ω = R2, there is a possible definition of a fundamental