On Regularity and Extension of Green's Operator on Bounded Smooth Domains

摘要

We prove regularity and extension results for Green's operators that are associated to strictly elliptic second order divergence-type linear PDO's with coefficients in C ~(1,α) (Ω?). Here α ∈ (0, 1) and Ω ? R ~n, n ≥ 3, is a bounded C ~(2,α) domain. The regularity result gives boundary estimates for the derivatives up to order (2 + α) of the associated Green's function. With the aid of this regularity result, we then extend the Green's operator to a globally defined integral operator whose second order partial derivatives are Calderón-Zygmund singular integrals. We also show that, under reasonable a priori assumptions, the C ~(2,α) regularity of the domain is necessary for the aforementioned extension of the Green's operator to a weakly singular integral operator, belonging to the class SK ~(-2) _(Rn)(α).