A comparison result related to Gauss measure

M.Francesca Betta; Friedman Brock; Anna Mercaldo; M.Rosaria Posteraro

doi:10.1016/S1631-073X(02)02295-1

A comparison result related to Gauss measure
[Un résultat de comparaison relatif à la mesure de Gauss]

M.Francesca Betta ¹ ; Friedman Brock ² ; Anna Mercaldo ³ ; M.Rosaria Posteraro ³

¹ Dipartimento di Matematica, Seconda Università di Napoli, via Vivaldi 43, 81100 Caserta, Italy
² Department of Mathematics, University of Missouri-Columbia, Columbia, MO 65211, USA
³ Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università di Napoli “Federico II”, Via Cintia, 80126 Napoli, Italy

Comptes Rendus. Mathématique, Volume 334 (2002) no. 6, pp. 451-456.

Abstract (VO)
Résumé

In this paper we prove a comparison result for weak solutions to linear elliptic problems of the type

- {(a_{ij} (x) u_{x_{i}})}_{x_{j}} = f (x) ϕ (x) in Ω, u = 0 on \partial Ω,

where

Ω

is an open set of

ℝ^{n}

(n⩾2), ϕ(x)=(2π)^−n/2exp(−|x|²/2), a_ij(x) are measurable functions such that a_ij(x)ξ_iξ_j⩾ϕ(x)|ξ|² a.e.

x \in Ω

,

ξ \in ℝ^{n}

and f(x) is a measurable function taken in order to guarantee the existence of a solution

u \in H_{0}^{1} (ϕ, Ω)

of (1.1). We use the notion of rearrangement related to Gauss measure to compare u(x) with the solution of a problem of the same type, whose data are defined in a half-space and depend only on one variable.

Dans cette note on démontre un résultat de comparaison pour les solutions faibles de problèmes elliptiques linéaires du type

- {(a_{ij} (x) u_{x_{i}})}_{x_{j}} = f (x) ϕ (x) dans Ω, u = 0 sur \partial Ω,

où

Ω

est un ouvert de

ℝ^{n}

(n⩾2), ϕ(x)=(2π)^−n/2exp(−|x|²/2), a_ij(x) sont des fonctions mesurables telles que a_ij(x)ξ_iξ_j⩾ϕ(x)|ξ|² p.p.

x \in Ω

,

ξ \in ℝ^{n}

et f(x) est une fonction mesurable telle qu'il existe une solution u de (0.1), dans

H_{0}^{1} (ϕ, Ω)

. On utilise la notion de rearrangement relatif à la mesure de Gauss pour comparer u(x) avec la solution d'un problème du même type, dont les données sont définies dans un demi plan et dépendent d'une variable seulement.

Reçu le : 2001-07-10
Accepté le : 2002-02-04
Publié le : 2002-01-01

DOI : 10.1016/S1631-073X(02)02295-1

Affiliations des auteurs :

M.Francesca Betta ¹ ; Friedman Brock ² ; Anna Mercaldo ³ ; M.Rosaria Posteraro ³

¹ Dipartimento di Matematica, Seconda Università di Napoli, via Vivaldi 43, 81100 Caserta, Italy
² Department of Mathematics, University of Missouri-Columbia, Columbia, MO 65211, USA
³ Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università di Napoli “Federico II”, Via Cintia, 80126 Napoli, Italy

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     title = {A comparison result related to {Gauss} measure},
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M.Francesca Betta; Friedman Brock; Anna Mercaldo; M.Rosaria Posteraro. A comparison result related to Gauss measure. Comptes Rendus. Mathématique, Volume 334 (2002) no. 6, pp. 451-456. doi : 10.1016/S1631-073X(02)02295-1. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)02295-1/

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