Perturbations singulières pour des EDP linéaires et non linéaires en presence de discontinuités

Abstract : In my thesis, we study some singular perturbation problems (\textit{i.e} problems which are characterized by the presence of a small parameter) that develop boundary layers in some conditions more delicate than usual, namely when the limit solution is not regular. I consider here two classes of regular problems associated with the Laplacian and bilaplacian, and a nonlinear problem derived from the Plateau problem (minimal surfaces), for which the limit function displays some singularities (namely ordinary discontinuity for the forme and infinite normal derivative on some parts of the boundary of the domain for the later).\\ Th first part of this thesis is concerned with the study of two singular linear models associated with non classical singular perturbations for some PDEs with singular source function. This kind of equations appears in several applications in physics; this occurs for instance with singular vortex structures in fluid mechanics or when folding occurs in elasticity, and in automatic control. Classically, the presence of a small parameter in some partial differential equations may lead to the appearance of classical boundary layers near the boundary for the regularized solution. If we consider moreover a discontinuous source function (even possibly a distribution), we note the appearance of non classical boundary layers in the interior of the domain; the study some of these boundary layers is the main object of this first part.\\ In the second part of my thesis, I am interested in the study of the minimal surface problem on a domain formed by two concentric circles. For some classes of boundary data, this problem does not have a solution and its weak solution called ``the generalized solution'' has an infinite gradient on some parts of the boundary. To solve this difficulty, we introduce an elliptic regularisation which involve boundary layers near the boundary. The main result of this part consists in giving some representation formula of the solutions, and of some approximate solutions, and some estimatesof the order of convergence.
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Mathematics [math]. Université Paris Sud - Paris XI, 2001. French


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Contributor : Makram Hamouda <>
Submitted on : Wednesday, November 6, 2002 - 12:06:55 PM
Last modification on : Wednesday, November 6, 2002 - 12:06:55 PM

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Makram Hamouda. Perturbations singulières pour des EDP linéaires et non linéaires en presence de discontinuités. Mathematics [math]. Université Paris Sud - Paris XI, 2001. French. <tel-00001931>

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