New Hardy inequalities and behaviour of linear elliptic equations

In this study, we want to emphasize the role of some Hardy inequalities in the blow-up phenomena of the very weak solution of a linear equation in the sense of Brezis. Thus we present here some new Hardy inequalities related to some extended Sobolev spaces such that Sobolev–Hardy spaces, Sobolev– Zygmund spaces, or other non-standard weighted spaces. Firstly we apply those results then provide two applications of these inequalities. Secondly we improve recent results by showing that the blow-up phenomena of the gradient can also occur in Hardy spaces. The Hardy inequalities for Sobolev–Zygmund spaces are obtained via an integral formula estimating the oscillation in a ball of radius r of a general function u in the usual Sobolev space. This formula involves the notion of relative rearrangement. We shall give a pointwise estimate for the solution u of linear equation − u=−div(F ) for a bounded function F, using the distance function δ. © 2012 Elsevier Inc. All rights reserved