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

Abstract

We establish global Gaussian estimates for the Green's matrix of divergence form, second order parabolic systems in a cylindrical domain under the assumption that weak solutions of the system vanishing on a portion of the boundary satisfy a certain local boundedness estimate and a local Hölder estimate. From these estimates, we also derive global estimates for the Green's matrix for elliptic systems with bounded measurable coefficients in two dimensional domains. We present a unified approach valid for both the scalar and vectorial cases and discuss several applications of our result.

Gradient estimates and the fundamental solution for higher-order elliptic systems with rough coefficients

“…In this case Meyers's estimate (1.2) implies that solutions u locally satisfy ∇ u ∈ L p for some p > d; Morrey's inequality then implies that solutions are necessarily locally Hölder continuous. The papers [DK09, KK10,CDK12] investigate the related topic of Green's functions in domains; they too require local boundedness of solutions (either as an explicit assumption or by virtue of working in dimension d = 2).…”

Green's functions for elliptic and parabolic systems with Robin-type boundary conditions

“…The following assumption (H3) is used to obtain global Gaussian estimates for the Robin Green's function. We point out that the integral appearing in (H3) is different from those in the condition (A3) of [11] and the condition (LB) of [10].…”

“…Their argument is based on a careful analysis on the resolvent and semigroup of a self-adjoint realization of the corresponding elliptic operator in L 2 (Ω). In this article, we follow an approach that is different from theirs and based on techniques developed in recent papers [9,10,11]. We construct Green's function for L in Ω × (−∞, ∞) satisfying the Robin boundary condition (1.2); i.e.…”

Green’s function for elliptic systems: existence and Delmotte–Deuschel bounds

“…if a is VMO). In [22], Kang and Kim (see also Cho et al [7] for the case d = 2) further develop the previous theory and in addition provide a necessary and sufficient condition on a in order to have for the Green function an optimal pointwise bound. We also mention that a result similar to [22] has been proved by Auscher and Tchamitchian [2] in the parabolic case via the introduction of a criterion [the Dirichlet Property (D)] for the parabolic Green function to have Gaussian bounds.…”

“…In a series of works, Hoffman and Kim [21] and Kim and collaborators (see e.g. [22] and [7]) considerably weaken the assumptions on the domain D and on the regularity of a ( both in the elliptic and in the corresponding parabolic setting): In [21], they establish the existence of the Green function for an arbitrary open domain D ⊂ R d with d > 2, provided that the coefficient field is such that a-harmonic functions satisfy an interior Hölder continuity estimate (e.g. if a is VMO).…”