Comptes Rendus
no. 6
A comparison result related to Gauss measure
[Un résultat de comparaison relatif à la mesure de Gauss]
M.Francesca Betta1 ; Friedman Brock2 ; Anna Mercaldo3 ; M.Rosaria Posteraro3
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.

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 ∂Ω,
où Ω est un ouvert de ℝ n (n⩾2), ϕ(x)=(2π)−n/2exp(−|x|2/2), aij(x) sont des fonctions mesurables telles que aij(x)ξiξj⩾ϕ(x)|ξ|2 p.p. x∈Ω, ξ∈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.

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 ∂Ω,
where Ω is an open set of ℝ n (n⩾2), ϕ(x)=(2π)−n/2exp(−|x|2/2), aij(x) are measurable functions such that aij(x)ξiξj⩾ϕ(x)|ξ|2 a.e. x∈Ω, ξ∈ℝ n and f(x) is a measurable function taken in order to guarantee the existence of a solution u∈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.

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