Global estimates for Green's matrix of second order parabolic systems with application to elliptic systems in two dimensional domains

TL;DR

This paper derives global Gaussian estimates for Green's matrices of second order parabolic and elliptic systems in two-dimensional domains, providing a unified approach applicable to scalar and vector cases.

Contribution

It introduces a unified method to obtain global estimates for Green's matrices of parabolic and elliptic systems, extending previous results to more general settings.

Findings

Established global Gaussian estimates for Green's matrices in parabolic systems.

Derived global estimates for elliptic systems in two dimensions.

Applicable to both scalar and vectorial systems with measurable coefficients.

Abstract

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 H\"older 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.