Capacitary estimates of solutions of a class of nonlinear elliptic equations

Moshe Marcus; Laurent Véron

doi:10.1016/S1631-073X(03)00217-6

Partial Differential Equations

Capacitary estimates of solutions of a class of nonlinear elliptic equations
[Estimations capacitaires des solutions d'une classe d'équations elliptiques non linéaires]

Présenté par : Haïm Brezis

Moshe Marcus ¹ ; Laurent Véron ²

¹ Department of Mathematics, Israel Institute of Technology-Technion, 32000 Haifa, Israel
² Département de mathématiques, Faculté des sciences et techniques, Université de Tours, 37200 Tours, France

Comptes Rendus. Mathématique, Volume 336 (2003) no. 11, pp. 913-918.

Résumé
Abstract

Soit $Ω$ un domaine borné régulier de $ℝ^{N}$ and K un sous-ensemble compact de $\partial Ω$ . Supposons q⩾(N+1)/(N−1) et soit U_K la solution maximale de $(ℰ) - Δ u + u^{q} = 0$ dans $Ω$ qui s'annulle sur $\partial Ω ⧹ K$ . Nous obtenons des majorations et minorations précises de U_K au moyen de la capacité de Bessel C_2/q,q′ et montrons que U_K est σ-modérée. En outre nous corrélons les points d'explosion forte de U_K et les points épais de K pour la topologie fine associée à C_2/q,q′ et caractérisons ces points par une condition d'intégrale de chemin portant sur U_K.

Let $Ω$ be a smooth bounded domain in $ℝ^{N}$ and K a compact subset of $\partial Ω$ . Assume that q⩾(N+1)/(N−1) and denote by U_K the maximal solution of −Δu+u^q=0 in $Ω$ which vanishes on $\partial Ω ⧹ K$ . We obtain sharp upper and lower estimates for U_K in terms of the Bessel capacity C_2/q,q′ and prove that U_K is σ-moderate. In addition we relate the strong ‘blow-up’ points of U_K on $\partial Ω$ to the ‘thick’ points of K in the fine topology associated with C_2/q,q′ and characterize these points by a path integral condition on U_K.

Reçu le : 2003-04-02
Accepté le : 2003-04-02
Publié le : 2003-06-01

DOI : 10.1016/S1631-073X(03)00217-6

Affiliations des auteurs :

Moshe Marcus ¹ ; Laurent Véron ²

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     title = {Capacitary estimates of solutions of a class of nonlinear elliptic equations},
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Moshe Marcus; Laurent Véron. Capacitary estimates of solutions of a class of nonlinear elliptic equations. Comptes Rendus. Mathématique, Volume 336 (2003) no. 11, pp. 913-918. doi : 10.1016/S1631-073X(03)00217-6. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(03)00217-6/

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Cité par Sources :

^☆ This research was supported by RTN contract No. HPRN-CT-2002-00274.

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