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

TL;DR

This paper derives global pointwise bounds for Green's matrices of second order elliptic systems, linking boundary behavior of solutions to the Green's function estimates, with applications in scalar and vector cases.

Contribution

It establishes a unified approach to obtain Green's matrix bounds for elliptic systems, connecting local boundedness of solutions to global estimates, applicable to scalar and vector cases.

Findings

Global pointwise bounds for Green's matrix are established.

Local boundedness of solutions is equivalent to Green's matrix bounds.

The approach applies to both scalar and vectorial elliptic systems.

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.