Global Hoelder continuity of solutions to quasilinear equations with Morrey data

Abstract

We deal with general quasilinear divergence-form coercive operators whose prototype is the $m$-Laplacean operator. The nonlinear terms are given by Carath\'eodory functions and satisfy controlled growth structure conditions 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 corkscrew property. We prove global boundedness and Hoelder continuity up to the boundary for the weak solutions of such equations, generalizing this way the classical $L^p$-result of Ladyzhenskaya and Ural'tseva to the settings of the Morrey spaces.