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We establish global pointwise bounds for the Green's matrix for divergenceform, second order elliptic systems in a domain under the assumption that weaksolutions of the system vanishing on a portion of the boundary satisfy acertain local boundedness estimate. Moreover, we prove that such a localboundedness estimate for weak solutions of the system is equivalent to theusual global pointwise bound for the Green's matrix. In the scalar case, suchan estimate is a consequence of De Giorgi-Moser-Nash theory and holds forequations with bounded measurable coefficients in arbitrary domains. In thevectorial case, one need to impose certain assumptions on the coefficients ofthe system as well as on domains to obtain such an estimate. We present aunified approach valid for both the scalar and vectorial cases and discussseveral applications of our result.

Global pointwise estimates for Green's matrix of second order elliptic systems