Green’s matrices of second order elliptic systems with measurable coefficients in two dimensional domains

Abstract

Abstract. We study Green's matrices for divergence form, second order strongly elliptic systems with bounded measurable coefficients in two dimensional domains. We establish existence, uniqueness, and pointwise estimates of Green's matrices.

Diffusion Coefficients Estimation for Elliptic Partial Differential Equations

“…A proof of this fact in the case d ≥ 3 can be found in [13, Theorem 1.1]. The same proof holds also in the case d = 2, utilizing the regularity properties of the two dimensional Green's function discussed in [7]. Now, given any x ∈ D, let B(x, ρ(x)/2) ⊂ D be the ball centered at x with radius ρ(x)/2.…”

Global Estimates for Green’s Matrix of Second Order Parabolic Systems with Application to Elliptic Systems in Two Dimensional Domains

“…Proof It is known that the condition (LH) is satisfied by [12,Lemma 4.4]. Therefore, the corollary follows from Theorem 6.9.…”

“…8 We point out that the assumption = ( ) < ∞ in Theorem 6.5 can be somehow relaxed. In fact, the quantity is related to the constant K = K( ) in the following Poincaré's inequality (see [12,Lemma 3.1]):…”

“…However, again, the global estimate similar to Eq. 1.6 was established only when is the open set above a Lipschitz graph ϕ : R → R. In this sense, the present article may be considered as a sequel of both articles [8] and [12] with a considerable improvement.…”

“…1.4 was proved only in the case when = R n . In another recent article [12], the second and third named authors proved the existence and local estimates of Green's matrix for elliptic systems with bounded measurable coefficients in a two dimensional domain with finite area or width, by utilizing the estimates established in [8]. However, again, the global estimate similar to Eq.…”