Comparison principle for second order elliptic operators and applications

Rabah Tahraoui

doi:10.1016/j.anihpc.2005.02.005

JournalsaihpcVol. 23, No. 2pp. 159–183

Comparison principle for second order elliptic operators and applications

Rabah Tahraoui
CEREMADE, Université Paris 9, Dauphine, UMR 7534, place du Maréchal de Lattre de Tassigny, 75775 Paris cedex 16, France; I.U.F.M. Rouen 2, rue du Tronquet, 76131 Mont-St-Aignan, France

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Abstract

Given two elliptic operators $A_{1}$ and $A_{2}$ , let us consider the following problems:

{- A_{i} u_{i} = f_{i}, u_{i} ∣_{\partial_{Ω}} = 0, f_{i} ⩾ 0 in Ω, i = 1, 2.

In this work, we give some sufficient conditions ensuring the following comparison principle:

u_{1} (x) ⩾ u_{2} (x) \forall x \in Ω.

As application, we study the behaviour, as $∣ x ∣ \to + \infty$ , of the solution $u$ of the problem

⎩ ⎨ ⎧ - A u = f ∣ x ∣ \to + \infty lim u (x) = 0. in R^{N},

Résumé

Etant donnés deux opérateurs

{- A_{i} u_{i} = f_{i}, u_{i} ∣_{\partial_{Ω}} = 0, f_{i} ⩾ 0 in Ω, i = 1, 2.

Dans ce travail, on se propose de donner des conditions suffisantes portant sur $A_{i}$ , $f_{i}$ permettant de comparer les deux solutions $u_{1}$ et $u_{2}$ :

u_{1} (x) ⩾ u_{2} (x) \forall x \in Ω.

Comme application, nous étudions le comportement asymptotique, quand $∣ x ∣ \to + \infty$ , de la solution $u$ du problème suivant :

⎩ ⎨ ⎧ - A u = f ∣ x ∣ \to + \infty lim u (x) = 0. in R^{N},

Cite this article

Rabah Tahraoui, Comparison principle for second order elliptic operators and applications. Ann. Inst. H. Poincaré Anal. Non Linéaire 23 (2006), no. 2, pp. 159–183

DOI 10.1016/J.ANIHPC.2005.02.005