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                <term xml:lang="en">Concentration phenomena</term>
                <term xml:lang="en">Topology constraints</term>
                <term xml:lang="en">Geometric flows</term>
                <term xml:lang="en">Phase-field approximation</term>
                <term xml:lang="fr">Analyse non-linéaire</term>
                <term xml:lang="fr">Phénomène de concentration</term>
                <term xml:lang="fr">Energie de Willmore</term>
                <term xml:lang="fr">Contraintes topologiques</term>
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              <p>The first part of this thesis is devoted to the theoretical and numerical study of the phase field approximation of two geometric flows, the mean curvature flow and the Willmore flow. The analysis of a particular model of approximation of the Willmore flow leads us to propose a new reaction term which charges the singularities of the normal field associated to an evolving shape. We derive a new model of approximation of the mean curvature flow which prevents topology changes. This model is in particular well adapted to the numerical approximation of 3D solutions of the Steiner problem and the Plateau problem. In the second part of the thesis, we study the asymptotic behavior of small embedded Willmore spheres in a Riemannian manifold of dimension 3. Using the formulation of Willmore equation derived by Rivière in terms of a triple system of elliptic PDEs, we show that, in the case where only two spheres appear in the asymptotic decomposition, small embedded Willmore spheres necessarily concentrate at a critical point of the scalar curvature of the ambient manifold.</p>
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              <p>La première partie de cette thèse est consacrée à l’étude théorique et numérique de l’approximation par méthode de champ de phase de deux flots géométriques, le flot de courbure moyenne et le flot de Willmore. L’analyse d’un modèle particulier d’approximation du flot de Willmore nous amène à proposer un nouveau terme de réaction qui charge les singularités du champ normal associé à une forme en évolution. On en déduit un nouveau modèle d’approximation du flot de courbure moyenne qui empêche les changements de topologie. Ce modèle est en particulier bien adapté à l’approximation numérique de solutions en 3D du problème de Steiner et du problème de Plateau. Dans la deuxième partie de la thèse, on étudie le comportement asymptotique de petites sphères de Willmore plongées dans une variété riemannienne de dimension 3. En utilisant la formulation de l’équation de Willmore donnée par Rivière en un système triple d’EDPs elliptiques, on montre que, dans le cas où seules deux sphères apparaissent dans la décomposition asymptotique, les petites sphères de Willmore se concentrent nécessairement en un point critique de la courbure scalaire de la variété ambiante.</p>
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