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.

Résumé
Abstract

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.

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.

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

@article{CRMATH_2002__334_6_451_0,
     author = {M.Francesca Betta and Friedman Brock and Anna Mercaldo and M.Rosaria Posteraro},
     title = {A comparison result related to {Gauss} measure},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {451--456},
     publisher = {Elsevier},
     volume = {334},
     number = {6},
     year = {2002},
     doi = {10.1016/S1631-073X(02)02295-1},
     language = {en},
}

TY  - JOUR
AU  - M.Francesca Betta
AU  - Friedman Brock
AU  - Anna Mercaldo
AU  - M.Rosaria Posteraro
TI  - A comparison result related to Gauss measure
JO  - Comptes Rendus. Mathématique
PY  - 2002
SP  - 451
EP  - 456
VL  - 334
IS  - 6
PB  - Elsevier
DO  - 10.1016/S1631-073X(02)02295-1
LA  - en
ID  - CRMATH_2002__334_6_451_0
ER  -

%0 Journal Article
%A M.Francesca Betta
%A Friedman Brock
%A Anna Mercaldo
%A M.Rosaria Posteraro
%T A comparison result related to Gauss measure
%J Comptes Rendus. Mathématique
%D 2002
%P 451-456
%V 334
%N 6
%I Elsevier
%R 10.1016/S1631-073X(02)02295-1
%G en
%F CRMATH_2002__334_6_451_0

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/

Bibliographie
Cité par

[1] A. Alvino; P.-L. Lions; G. Trombetti Comparison results for elliptic and parabolic equations via Schwarz symmetrization, Ann. Inst. H. Poincaré Anal. Non Linéaire, Volume 7 (1990), pp. 37-65

[2] A. Alvino; G. Trombetti Sulle migliori costanti di maggiorazione per una classe di equazioni ellittiche degeneri, Ricerche Mat., Volume 27 (1978), pp. 413-428

[3] M.F. Betta; F. Brock; A. Mercaldo; M.R. Posteraro A weighted isoperimetric inequality and application to symmetrization, J. Inequal. Appl., Volume 4 (1999), pp. 215-240

[4] M.F. Betta, F. Brock, A. Mercaldo, M.R. Posteraro, Weighted isoperimetric inequalities and comparison results for elliptic equations, in preparation

[5] A. Ehrhard Éléments extremaux pour les inégalités de Brunn–Minkowski gaussennes, Ann. Inst. H. Poincaré Anal. Non Linéaire, Volume 22 (1986), pp. 149-168

[6] H. Federer Geometric Measure Theory, Grundlehren Math. Wiss., 153, Springer-Verlag, 1969

[7] J.M. Rakotoson; B. Simon Relative rearrangement on a measure space. Application to the regularity of weighted monotone rearrangement. Part 1–Part 2, Rev. Real Acad. Cienc. Exact. Fis. Natur Madrid, Volume 91 (1997), pp. 17-31 (33–45)

[8] G. Talenti Linear elliptic P.D.E.'s: Level sets, rearrangements and a priori estimates of solutions, Boll. Un. Mat. Ital. B, Volume 4 (1985), pp. 917-949

[9] G. Talenti A weighted version of a rearrangement inequality, Ann. Univ. Ferrara Sez. VII (N.S.), Volume 43 (1997), pp. 121-133

[10] G. Trombetti Metodi di simmetrizzazione nelle equazioni a derivate parziali, Boll. Un. Mat. Ital. B, Volume 3 (2000), pp. 601-634

[11] N.S. Trudinger Linear elliptic operators with measurable coefficients, Ann. Scuola Norm. Sup. Pisa, Volume 27 (1973), pp. 265-308

Cité par Sources :

Commentaires - Politique