In Chapter 3, we have discussed various versions of the Poincaré-type inequalities in which we have estimated the norm of u – u B in terms of the corresponding norm of du. In this chapter, we develop a series of estimates which provide upper bounds for the norms of ≰ u (if u is a function) or du (if u is a form) in terms of the corresponding norm u – c, where c is any closed form. These kinds of estimates are called the Caccioppoli-type estimates or the Caccioppoli inequalities. In Section 4.2, we study the local A r (Ω)-weighted Caccioppoli inequalities. The local Caccioppoli inequalities with two-weights are discussed in Section 4.3. The global versions of Caccioppoli inequalities on Riemannian manifolds and bounded domains are developed in Sections 4.4 and 4.5, respectively. In Section 4.6, we present Caccioppoli inequalities with Orlicz norms. Finally, in Section 4.7, we address few versions of Caccioppoli inequalities related to the codifferential operator d * .
KeywordsRiemannian Manifold Closed Form Obstacle Problem Weighted Inequality Orlicz Norm
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