Global Hölder continuity of weak solutions to quasilinear elliptic equations with Morrey data

The note deals with solutions to the Dirichlet problem for general quasilinear divergence-form elliptic operators whose prototype is the p-Laplacean operator. The nonlinear terms are given by Carathéodory functions and satisfy the natural structure conditions of Ladyzhenskaya and Ural’tseva with data belonging to suitable Morrey spaces. The fairly non-regular boundary of the underlying domain is supposed to satisfy a capacity density condition which allows domains with exterior cone (or even corkscrew) property. We prove Hölder continuity up to the boundary for the boundedweak solutions of such equations, generalizing thisway the classical L^p -result of Ladyzhenskaya and Ural’tseva to the settings of the Morrey spaces.