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.

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Abstract. We establish existence and various estimates of fundamental matrices and Green's matrices for divergence form, second order strongly parabolic systems in arbitrary cylindrical domains under the assumption that solutions of the systems satisfy an interior Hölder continuity estimate. We present a unified approach valid for both the scalar and the vectorial cases.

We construct Green's function for second order elliptic operators of the form Lu = −∇ · (A∇u + bu) + c · ∇u + du in a domain and obtain pointwise bounds, as well as Lorentz space bounds. We assume that the matrix of principal coefficients A is uniformly elliptic and bounded and the lower order coefficients b, c, and d belong to certain Lebesgue classes and satisfy the condition d − ∇ · b ≥ 0. In particular, we allow the lower order coefficients to be singular. We also obtain the global pointwise bounds for the gradient of Green's function in the case when the mean oscillations of the coefficients A and b satisfy the Dini conditions and the domain is C 1,Dini .2010 Mathematics Subject Classification. 35A08, 35J08.

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.