A comparison result related to Gauss measure
Comptes Rendus. Mathématique, Volume 334 (2002) no. 6, pp. 451-456.

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

 $-{\left({\mathrm{a}}_{\mathrm{ij}}\left(\mathrm{x}\right){\mathrm{u}}_{{x}_{i}}\right)}_{{x}_{j}}=\mathrm{f}\left(\mathrm{x}\right)\varphi \left(\mathrm{x}\right)\phantom{\rule{10.0pt}{0ex}}\mathrm{in}\phantom{\rule{3.30002pt}{0ex}}\Omega ,\phantom{\rule{10.0pt}{0ex}}\phantom{\rule{10.0pt}{0ex}}\mathrm{u}=0\phantom{\rule{10.0pt}{0ex}}\mathrm{on}\phantom{\rule{3.30002pt}{0ex}}\partial \Omega ,$
where $\Omega$ 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)ξiξjϕ(x)|ξ|2 a.e. $\mathrm{x}\in \Omega$, $\xi \in {ℝ}^{n}$ and f(x) is a measurable function taken in order to guarantee the existence of a solution $\mathrm{u}\in {\mathrm{H}}_{0}^{1}\left(\varphi ,\Omega \right)$ 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

 $-{\left({\mathrm{a}}_{\mathrm{ij}}\left(\mathrm{x}\right){\mathrm{u}}_{{x}_{i}}\right)}_{{x}_{j}}=\mathrm{f}\left(\mathrm{x}\right)\varphi \left(\mathrm{x}\right)\phantom{\rule{10.0pt}{0ex}}\mathrm{dans}\phantom{\rule{3.30002pt}{0ex}}\Omega ,\phantom{\rule{10.0pt}{0ex}}\phantom{\rule{10.0pt}{0ex}}\mathrm{u}=0\phantom{\rule{10.0pt}{0ex}}\mathrm{sur}\phantom{\rule{3.30002pt}{0ex}}\partial \Omega ,$
$\Omega$ 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)ξiξjϕ(x)|ξ|2 p.p. $\mathrm{x}\in \Omega$, $\xi \in {ℝ}^{n}$ et f(x) est une fonction mesurable telle qu'il existe une solution u de (0.1), dans ${\mathrm{H}}_{0}^{1}\left(\varphi ,\Omega \right)$. 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.

Accepted:
Published online:
DOI: 10.1016/S1631-073X(02)02295-1
Betta, M.Francesca 1; Brock, Friedman 2; Mercaldo, Anna 3; Posteraro, M.Rosaria 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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Betta, M.Francesca; Brock, Friedman; Mercaldo, Anna; Posteraro, M.Rosaria. 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. http://www.numdam.org/articles/10.1016/S1631-073X(02)02295-1/

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