Gradient estimates for a class of higher-order elliptic equations of p-growth over a nonsmooth domain

Abstract

This article is devoted to a global Calderón-Zygmund estimate in the framework of Lorentz spaces for the m m -order gradients of weak solution to a higher-order elliptic equation with p p -growth. We prove the main result based on a proper power decay estimation of the upper-level set by the principle of layer cake representation for the L γ , q {L}^{\gamma ,q} -estimate of D m u {D}^{m}u , while the coefficient satisfies a small BMO semi-norm and the boundary of underlying domain is flat in the sense of Reifenberg. In particular, a tricky ingredient is to establish the normal component of higher derivatives controlled by the horizontal component of higher derivatives of solutions in the neighborhood at any boundary point, which is achieved by comparing the solution under consideration with that for some reference problems.