Advances in Differential Equations

Some new applications of the pointwise relations for the relative rearrangement

J. M. Rakotoson

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Let $V=V(\Omega)$ be a set satisfying the Poincaré-Sobolev pointwise inequalities for the relative rearrangement and $\rho$ an arbitrary norm on the set of measurable functions on the interval $\Omega_*=(0,measure(\Omega))$. Then using the associate norm of $\rho$, we give some sufficient conditions to ensure that a function $u$ of $V$ has to be bounded or integrable in an Orlicz space. We show similar results for a solution of quasilinear equation under some conditions between the data and $\rho$. We then give an unified approach for many kinds of estimates. Using a main relation involving some pointwise relations for the relative rearrangement, we prove a regularity result for the time derivative of a parabolic system well posed in a Grand Sobolev space.

Article information

Adv. Differential Equations, Volume 7, Number 5 (2002), 617-640.

First available in Project Euclid: 27 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46E30: Spaces of measurable functions (Lp-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
Secondary: 26D07: Inequalities involving other types of functions 35D10 35J60: Nonlinear elliptic equations 35K20: Initial-boundary value problems for second-order parabolic equations


Rakotoson, J. M. Some new applications of the pointwise relations for the relative rearrangement. Adv. Differential Equations 7 (2002), no. 5, 617--640.

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