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 $C^{1,alpha}(overline{Omega})$ . 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 ${rm{SK}}^{-2}_{{bf{R}}^n}(alpha)$ .