On extension of Green's operator on bounded smooth domains

TL;DR

This paper establishes boundary regularity estimates for Green's functions of elliptic PDEs with smooth coefficients on bounded domains, extending the Green's operator to a Calderón-Zygmund type singular integral.

Contribution

It provides new boundary regularity results for Green's functions and extends the Green's operator to a global singular integral of Calderón-Zygmund type.

Findings

Boundary derivatives up to order (2+α) are estimated.

Green's operator is extended to a Calderón-Zygmund type singular integral.

Results apply to elliptic PDEs with C^{1,α} coefficients on C^{2,α} domains.

Abstract

We prove a regularity result for Green's functions that are associated to elliptic second order divergence-type linear PDO's with coefficients in C^{1,\alpha}(\bar{\Omega}). Here \alpha\in (0,1) and \Omega\subset \R^n is a bounded C^{2,\alpha} domain in dimension n\ge 3. The regularity result gives boundary estimates for derivatives up to order (2+\alpha) and, by using these estimates, we extend the associated Green's operator to a globally defined singular integral which of Calder\'on--Zygmund type.