Global Caccioppoli-Type and Poincaré Inequalities with Orlicz Norms

The L-theory of solutions of the homogeneous A-harmonic equation d A x, dω 0 for differential forms has been very well developed in recent years. Many L-norm estimates and inequalities, including the Hardy-Littlewood inequalities, Poincaré inequalities, Caccioppoli-type estimates, and Sobolev imbedding inequalities, for solutions of the homogeneous A-harmonic equation have been established; see 1–11 . Among these results, the Caccioppoli-type inequalities and the Poincaré inequalities for differential forms have become more and more important tools in analysis and related fields, including partial differential equations and potential theory. However, the study of the nonhomogeneous A-harmonic equation d A x, dω B x, dω just began 4, 6 . Roughly, the Caccioppoli-type inequalities or estimates provide upper bounds for the norms of ∇u or du in terms of the corresponding norm of u or u−c, where u is a differential form or a function satisfying certain conditions. For example, u may be a solution of an A-harmonic equation or a minimizer of a functional, and c is some constant if u is a function or a closed form if u is a differential form. Different versions of the Caccioppoli-type inequalities and the Poincaré inequalities have been established during the past several decades. For instance, Sbordone proved in 12 the following version of the Caccioppoli-type inequality: