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