Applications of the duality between the Homogeneous Complex Monge–Ampère Equation and the Hele-Shaw flow
Annales de l'Institut Fourier, to appear, 30 p.

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.

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.

Received : 2015-09-17
Revised : 2017-12-08
Accepted : 2017-12-15
Published online : 2019-03-08
Classification:  32W20,  76D27
Keywords: Complex Monge–Ampère equations, Hele-Shaw flows
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     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, to appear, 30 p.

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