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Comptes Rendus Mathématique
Volume 336, n° 9
pages 703-708 (mai 2003)
Doi : 10.1016/S1631-073X(03)00184-5
Received : 10 Mars 2003 ;  accepted : 1 April 2003
Analysis of a Poisson system with boundary conditions

François  Castella a ,  Philippe  Chartier b ,  Erwan  Faou b
aIRMAR, Université de Rennes 1, 35042 Rennes cedex, France 
bINRIA Rennes, Campus Beaulieu, 35042 Rennes cedex, France 

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We consider a class of problems originating from a Raman laser amplification model, for which the equations can be written as a Poisson system with boundary conditions. Once reformulated, this system becomes an integro-differential equation that we study here in some detail. In particular, we show the existence of a smooth solution under general assumptions, and prove its uniqueness for boundary values that are not too far apart. Eventually, we completely solve the question of uniqueness for systems of dimensions one and two. To cite this article: F. Castella et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003).

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Nous étudions une classe de problèmes dont l'origine provient d'un modèle décrivant l'effet d'amplification Raman dans une fibre optique. Les équations s'écrivent sous la forme d'un système de Poisson avec conditions aux deux bouts. Après réduction, ce système s'écrit sous la forme d'une équation intégro-différentielle. Nous étudions ici cette classe de systèmes. Nous montrons l'existence d'une solution dans le cas général et l'unicité pour des données petites du problème. Pour des données quelconques, nous prouvons l'unicité en dimensions un et deux. Pour citer cet article : F. Castella et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003).




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