General pointwise relations for the relative rearrangement and applications

Abstract

We prove some new Poincaré—Sobolev pointwise relations for the relative rearrangement. The first consequence is the derivation of Polyà—Szëgo pointwise inequalities which imply in particular inequalities in any normed spaces with a rearrangement invariant Fatou norm.

Our method gives an unified approach for many kind of inequalities on bounded or unbounded domains. The second new application for partial differential equation is a simple formalism for having a priori estimates which is based on the pointwise estimate of the relative rearrangement of the gradient with respect to the solution of the P.D.E.

Keywords:

• Relative rearrangement;
• Parabolic elliptic equations
• Monotone rearrangement
• Poincaré—Sobolev pointwise inequalities
• Polyà—Szëgo pointwise inequalities

AMS Classification:

• 26D07
• 35J60
• 35K20
• 35D10
• 46E30
• 46E35