A generalization of the Pólya-Szegő inequality for one-dimensional functionals
type: Item
instance of: scholarly article
title: A generalization of the Pólya-Szegő inequality for one-dimensional functionals
zbMATH Open document ID: 1275.26033
DOI: 10.1134/S1064562411030021
published in: Doklady Mathematics
publication date: 2012-01-17T00:00:00Z
review text: Let \(F:\mathbb{R}\times \mathbb{R}\to \overline{\mathbb{R}}_+\) be a continuous function which is even and convex with respect to the second argument. Also let \(a: A\times\mathbb{R}\to\overline{\mathbb{R}}_+\) be a continuous function, where \(A= [-1,1]\). Let the functional \(I\) be given by \[ I(u)= \int^1_{-1} F(u(x),\;|a(x,u(x))\,u'(x)|)\,dx\tag{1} \] and defined on the set \(M\) of functions \(u\in W^1_1(A)\), for which the integral converges. Monotonicity properties of functionals under rearrangements were studied by a number of authors. Motivated by a paper by \textit{F. Brock} [Calc. Var. Partial Differ. Equ. 8, No. 1, 15--25 (1999; Zbl 0947.35056)], where the behaviour of a functional \(I\) and its multidimensional analogue under the Steiner symmetrization was studied, the authors of the present paper establish the absence of the Lavrentiev phenomenon for this functional (1). The multidimensional case however remains open. For Lipschitz functions \(u\) a stronger inequality is proved under only necessary conditions on the weight function. For an arbitrary function \(u\in M\) they succeed in doing it under a certain additional restriction.
reviewed by: József Sándor
Mathematics Subject Classification ID: 26D15
Mathematics Subject Classification ID: 49K05
zbMATH DE Number: 5998391
zbMATH Keywords: rearrangements
zbMATH Keywords: Lavrentiev's phenomenon
author: Sergeĭ V. Bankevich
author: A. I. Nazarov
MaRDI profile type: Publication
cites work: Analysis.
cites work: Rearrangements and convexity of level sets in PDE
cites work: Weighted Dirichlet-type inequalities for Steiner symmetrization
cites work: Q4210481
cites work: Some remarks on rearrangements and functionals with non‐constant density
Recommended article: The Steiner rearrangement in any codimension
Recommended article: Second-order derivatives and rearrangements
Recommended article: Weighted Dirichlet-type inequalities for Steiner symmetrization
Recommended article: Q4667210
Recommended article: Rearrangement and polarization
Recommended article: A new rearrangement inequality and its application for \(L^2\)-constraint minimizing problems
Recommended article: STEFFENSEN TYPE INEQUALITIES OVER GENERAL MEASURE SPACES
Recommended article: Q4312128
Recommended article: Q5437214
Recommended article: Cases of equality and strict inequality in the extended Hardy–Littlewood inequalities
description: scientific article; zbMATH DE number 5998391