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.
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Acknowledgements Sungwon Cho was supported by the National Research Foundation of Korea Grant funded by the Korean Government (Ministry of Education, Science and Technology)[NRF-2010-359-C00003]. Hongjie Dong was partially supported by the National Science Foundation under agreement No. DMS-0800129. Seick Kim was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology (2010-0008224) and also WCU(World Class University) program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology (R31-2008-000-10049-0).
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