The Green function estimates for strongly elliptic systems of second order

Abstract

We establish existence and pointwise estimates of fundamental solutions and Green's matrices for divergence form, second order strongly elliptic systems in a domain Ω ⊆ R n , n ≥ 3, under the assumption that solutions of the system satisfy De Giorgi-Nash type local Hölder continuity estimates. In particular, our results apply to perturbations of diagonal systems, and thus especially to complex perturbations of a single real equation.2000 Mathematics Subject Classification. Primary 35A08, 35B45; Secondary 35J45.

On the Green's matrices of strongly parabolic systems of second order

“…Averaged fundamental matrix. Our approach here is an adaptation of that in Hofmann-Kim [13], which in turn is partly based on the method by Grüter and Widman [12]. Let Y = (s, y) ∈ R n+1 and 1 ≤ k ≤ N be fixed.…”

“…Green's functions of elliptic equations of divergence form in bounded domains have been extensively studied by Littman, Stampacchia, and Weinberger [20] and Grüter and Widman [12], whereas the Green's matrices of the elliptic systems with continuous coefficients in bounded C 1 domains have been discussed by Fuchs [11] and Dolzmann and Müller [5]. Very recently, Hofmann and Kim [13] gave a unified approach in studying Green's functions/matrices in arbitrary domains valid for both scalar equations and systems of elliptic type by considering a class of operators L such that weak solutions of Lu = 0 satisfy an interior Hölder estimate. Some parts of the present article may be considered as a natural follow-up of their work in the parabolic setting.…”

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

“…In fact, Dolzmann and Müller improved the strategy of Fuchs and showed the existence and pointwise estimate for Green's matrix in bounded Lipschitz domains Ω ⊂ R 2 without imposing any regularity on the coefficients (their methods work whenever an L p theory is available for the domain Ω). Recently, Hofmann and Kim [10] gave a unified approach in studying Green's functions/matrices in arbitrary domains Ω ⊂ R n (n ≥ 3) valid for both scalar equations and systems of elliptic type by considering a class of operators L such that weak solutions of Lu = 0 satisfy an interior Hölder estimate. However, like the method used in Grüter and Widman [9], the method of Hofmann and Kim relied heavily on the assumption that n ≥ 3 and could not be applied to the two dimensional case.…”

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

“…We now post-process (103), (104) and (105) to obtain (21), (22) and (23). Reasoning as in Step 2, without loss of generality it suffices to prove (21), (22) and (23) …”

“…bound (21). In order to have also (22)- (23), we consider u(a; y) = ∇G(a; x, y) which, thanks to symmetry (99) and (98), for almost every x ∈ R d and · -almost every a ∈ is a-harmonic in {|y − x| > 2}.…”