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Limit behaviors of some boundary value problems with high and or low valued parameters

De
31 pages
Niveau: Supérieur, Doctorat, Bac+8
Limit behaviors of some boundary value problems with high and/or low valued parameters Salima Hassani?, Serge Nicaise†, Abderrahman Maghnouji‡ Abstract The first aim of this paper is to give a general variational framework for bilinear forms depending on two parameters tending to zero and to infinity respectively and allowing to analyze the three limit problems. Secondly we give different illustrative applications for transmission problems involving some elasticity systems, diffusion problems and Maxwell systems where one parameter tends to infinity and/or a part of the domain squeezes to a smooth surface. These limit procedures lead to new transmission problems, like a coupling between the Lamé system and the Stokes system. 1 Introduction Some partial differential equations are characterized by the fact that their coefficients are very different in some subpart of the domain where they are set in such a way that their ratio becomes very large. As an example, we can cite the case of the diffusion problem [15]: ?div (a?u) = f in D, where a = 1 in a fixed part of the domain D and a goes to infinity in the remainder Da of the domain D. In that case, it is not difficult to obtain the limit problem, see for instance [15]. In a similar manner, it is not difficult to obtain the limit problem of the above problem if a remains fixed but the sub-domain squeezes to a smooth hypersurface of codimension 1.

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  • unique solution

  • limit procedures

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  • ?université de valenciennes et du hainaut cambrésis


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Limit
behaviors of some boundary value problems high and/or low valued parameters Salima Hassani,Serge Nicaise, Abderrahman Maghnouji
Abstract
with
The first aim of this paper is to give a general variational framework for bilinear forms depending on two parameters tending to zero and to infinity respectively and allowing to analyze the three limit problems. Secondly we give different illustrative applications for transmission problems involving some elasticity systems, diffusion problems and Maxwell systems where one parameter tends to infinity and/or a part of the domain squeezes to a smooth surface. These limit procedures lead to new transmission problems, like a coupling between the Lamé system and the Stokes system.
Introduction
Some partial differential equations are characterized by the fact that their coefficients are very different in some subpart of the domain where they are set in such a way that their ratio becomes very large. As an example, we can cite the case of the diffusion problem [15]:
div(aru) =finD,
wherea= 1in a fixed part of the domainDandagoes to infinity in the remainderDaof the domainD that case, it is not difficult to obtain the limit problem, see for instance [15].. In In a similar manner, it is not difficult to obtain the limit problem of the above problem ifa remains fixed but the sub-domain squeezes to a smooth hypersurface of codimension 1. But the situation becomes more difficult if both the parameteragoes to infinity and the domainDa becomes small. Such a situation was studied for instance in [15] for diffusion problems in its full generality. For diffusion problems or the Lamé system, asymptotic expansion of the solution when the sub-domain squeezes to a smooth hypersurface of codimension 1 can be found in [1, 2] for instance. To our knowledge a systematic analysis of general systems of partial differential equations, where one parameter tends to infinity in a sub-domain that squeezes to a smooth hypersurface of codimension 1 was not performed. Hence for a family of bilinear forms depending on two Université de Valenciennes et du Hainaut Cambrésis, LAMAV, FR CNRS 2956 F-59313 - Valenciennes Cedex 9 France, salima_hassani@yahoo.fr Université de Valenciennes et du Hainaut Cambrésis, LAMAV, FR CNRS 2956, F-59313 - Valenciennes Cedex 9 France, Serge.Nicaise@univ-valenciennes.fr Université de Valenciennes et du Hainaut Cambrésis, LAMAV, FR CNRS 2956, F-59313 - Valenciennes Cedex 9 France, Abderrahman.Maghnouji@univ-valenciennes.fr
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