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

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Abstract

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.

Original languageEnglish
Pages (from-to)2643-2662
Number of pages20
JournalJournal of Differential Equations
Volume249
Issue number11
DOIs
Publication statusPublished - 2010 Dec

All Science Journal Classification (ASJC) codes

  • Analysis
  • Applied Mathematics

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