Regularity estimates for higher order elliptic systems on Reifenberg flat domains

Abstract

Consider a higher order elliptic system{Dα(aijαβ(x)Dβuj)=DαfiαinΩ,|ui|+|Dui|+…+|Dmi−1ui|=0on∂Ω, for all i=1,…,N with N∈N+, and all multi-indices |α|=mi, |β|=mj with mi∈N+ for all i=1,…,N, and the standard summation notation is understood. We assume that the leading coefficients aijαβ(x) have small BMO norms and the domain Ω⊂Rn is open, bounded and flat in the Reifenberg's sense. This article is to prove the regularity estimates of this system in weighted Lorentz spaces and in Lorentz–Morrey spaces. Our results require weak assumptions on the regularity of the coefficients aijαβ(x) and the boundary ∂Ω, and they are new even for scalar higher order elliptic equations.