Hölder Regularity of Solutions to Second-Order Elliptic Equations in Nonsmooth Domains

Abstract

We establish the global Hölder estimates for solutions to second-order elliptic equations, which vanish on the boundary, while the right-hand side is allowed to be unbounded. For nondivergence elliptic equations in domains satisfying an exterior cone condition, similar results were obtained by J. H. Michael, who in turn relied on the barrier techniques due to K. Miller. Our approach is based on special growth lemmas, and it works for both divergence and nondivergence, elliptic and parabolic equations, in domains satisfying a general “exterior measure” condition.