Applications of the duality between the Homogeneous Complex Monge–Ampère Equation and the Hele-Shaw flow  [ Applications de la dualité entre l’équation de Monge–Ampère complexe homogène et le flot de Hele-Shaw. ]
Annales de l'Institut Fourier, à paraître, p. 1-30
Nous donnons deux applications de la dualité entre l’équation de Monge–Ampère complexe homogène (HCMA) et le flot de Hele-Shaw. D’abord nous prouvons l’existence de données lisses au bord pour lesquelles la solution faible au problème de Dirichlet pour l’équation HCMA sur 1 ×𝔻 ¯ n’est pas deux fois différentiable en certains points fixés a priori ainsi que des exemples qui ne sont pas différentiables le long d’un ensemble de codimension 1 de 1 ×𝔻 ¯. Puis nous expliquons comment obtenir explicitement des familles de rayons géodésiques lisses dans l’espace des métriques Kähler sur 1 et sur le disque unité 𝔻. Ils sont construits à partir d’une famille à la fois exhaustive et croissante de domaines simplement connexes variant de manière lisse.
We give two applications of the duality between the Homogeneous Complex Monge–Ampère Equation (HCMA) and the Hele-Shaw flow. First, we prove existence of smooth boundary data for which the weak solution to the Dirichlet problem for the HCMA over 1 ×𝔻 ¯ is not twice differentiable at a given collection of points, and also examples that are not twice differentiable along a set of codimension one in 1 ×𝔻. Second, we discuss how to obtain explicit families of smooth geodesic rays in the space of Kähler metrics on 1 and on the unit disc 𝔻 that are constructed from an exhausting family of increasing smoothly varying simply connected domains.
Reçu le : 2015-09-17
Révisé le : 2017-12-08
Accepté le : 2017-12-15
Classification:  32W20,  76D27
Mots clés: l’équation de Monge–Ampère complexe, flot de Hele-Shaw
@unpublished{AIF_0__0_0_A1_0,
     author = {Ross, Julius and Nystr\"om, David Witt},
     title = {Applications of the duality between the Homogeneous Complex Monge--Amp\`ere Equation and the Hele-Shaw flow},
     note = {to appear in \emph{Annales de l'Institut Fourier}},
}
Ross, Julius; Nyström, David Witt. Applications of the duality between the Homogeneous Complex Monge–Ampère Equation and the Hele-Shaw flow. Annales de l'Institut Fourier, à paraître, pp. 1-30.

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