Comptes Rendus
no. 7
Partial Differential Equations
Existence via compactness for maximal monotone elliptic operators
[Existence par compacticité pour opérateurs maximaux monotone elliptiques]

Présenté par : Évariste Sanchez-Palencia

Piotr Gwiazda1 ; Anna Zatorska-Goldstein1
1 Institute of Applied Mathematics and Mechanics, Warsaw University, Banacha 2, 02-097 Warsaw, Poland
Comptes Rendus. Mathématique, Volume 340 (2005) no. 7, pp. 489-492.

Dans cette Note nous proposons une méthode nouvelle de démonstation de l'existence de solutions de −divA(x,∇u)∋f, où A(x,∇u) a un graphe maximale monotone dépendant de x. L'idée de cette méthode est d'utiliser la théorie des mésures de Young et la méthode de compacticité par compensation. Une autre approche a été proposée ailleurs. Néanmoins, notre méthode permet d'obtenir la convergence forte des solutions approchées.

In this Note we propose a new method of proving the existence of solutions to −divA(x,∇u)∋f, when A(x,∇u) has x-dependent maximal monotone graph. The idea is based on the theory of Young measures and on the method of compensated compactness. Alternative approaches were proposed elsewhere. However, our method allows us to obtain also the strong convergence of approximate solutions.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2005.02.017

Piotr Gwiazda 1 ; Anna Zatorska-Goldstein 1

1 Institute of Applied Mathematics and Mechanics, Warsaw University, Banacha 2, 02-097 Warsaw, Poland 
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Piotr Gwiazda; Anna Zatorska-Goldstein. Existence via compactness for maximal monotone elliptic operators. Comptes Rendus. Mathématique, Volume 340 (2005) no. 7, pp. 489-492. doi : 10.1016/j.crma.2005.02.017. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2005.02.017/

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  • Agnieszka Świerczewska-Gwiazda Nonlinear parabolic problems in Musielak–Orlicz spaces, Nonlinear Analysis: Theory, Methods Applications, Volume 98 (2014), p. 48 | DOI:10.1016/j.na.2013.11.026
  • Piotr Gwiazda On measure-valued solutions to a two-dimensional gravity-driven avalanche flow model, Mathematical Methods in the Applied Sciences, Volume 28 (2005) no. 18, p. 2201 | DOI:10.1002/mma.660

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