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.
Bibliographical noteFunding Information:
We thank the referee for useful comments. Kyungkeun Kang was supported by the Korean Research Foundation Grant (MOEHRD, Basic Research Promotion Fund, KRF-2008-331-C00024) and the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2009-0088692). Kyungkeun Kang appreciates the hospitality of the Department of Computational Science and Engineering, Yonsei University. Seick Kim was supported by the Korea Science and Engineering Foundation grant (MEST, No. R01-2008-000-20010-0) and also by WCU (World Class University) program through the Korea Science and Engineering Foundation (MEST, No. R31-2008-000-10049-0).
All Science Journal Classification (ASJC) codes
- Applied Mathematics