Global regularity for quasilinear elliptic equations with Morrey data

Abstract

In this thesis, we deal with general quasilinear $m$-Laplacian type operators in divergence form. The nonlinear terms are given by Carath\'eodory functions which are controlled within data belonging to suitable Morrey spaces. We first prove global boundedness and H\"older 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. The boundary of the underlying domain is supposed to satisfy a capacity density condition. We also derive global gradient estimates in Morrey spaces for the weak solutions to quasilinear equations having $(\delta,R)$-vanishing nonlinearity. In this case, we assume the boundary of the domain considered is Reifenberg flat which includes boundaries with rough fractal structure.