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, Tome 69 (2019) no. 1, pp. 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.

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DOI : https://doi.org/10.5802/aif.3237
Classification : 32W20,  76D27
Mots clés : l’équation de Monge–Ampère complexe, flot de Hele-Shaw
@article{AIF_2019__69_1_1_0,
     author = {Ross, Julius and Nystr\"om, David Witt},
     title = {Applications of the duality between the Homogeneous Complex Monge{\textendash}Amp\`ere Equation and the Hele-Shaw flow},
     journal = {Annales de l'Institut Fourier},
     pages = {1--30},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {69},
     number = {1},
     year = {2019},
     doi = {10.5802/aif.3237},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.3237/}
}
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, Tome 69 (2019) no. 1, pp. 1-30. doi : 10.5802/aif.3237. http://archive.numdam.org/articles/10.5802/aif.3237/

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