%0 Journal Article %J Journal de Mathématiques Pures et Appliquées %D 2018 %T Minimizing movements for mean curvature flow of droplets with prescribed contact angle %A Giovanni Bellettini %A Matteo Novaga %A Shokhrukh Kholmatov %K Capillary functional %K Mean curvature flow with prescribed contact angle %K Minimizing movements %K Sets of finite perimeter %X

We study the mean curvature motion of a droplet flowing by mean curvature on a horizontal hyperplane with a possibly nonconstant prescribed contact angle. Using the solutions constructed as a limit of an approximation algorithm of Almgren–Taylor–Wang and Luckhaus–Sturzenhecker, we show the existence of a weak evolution, and its compatibility with a distributional solution. We also prove various comparison results. Résumé Nous étudions le mouvement par courbure moyenne d'une goutte qui glisse par courbure moyenne sur un hyperplan horizontal avec un angle de contact prescrit éventuellement non constant. En utilisant les solutions construites comme limites d'un algorithme d'approximation dû à Almgren, Taylor et Wang et Luckhaus et Sturzenhecker, nous montrons l'existence d'une évolution faible, et sa compatibilité avec une solution au sens des distributions. Nous démontrons également plusieurs résultats de comparaison.

%B Journal de Mathématiques Pures et Appliquées %V 117 %P 1 - 58 %G eng %U http://www.sciencedirect.com/science/article/pii/S0021782418300825 %R https://doi.org/10.1016/j.matpur.2018.06.003 %0 Journal Article %J Communications on Pure & Applied Analysis %D 2017 %T Minimizers of anisotropic perimeters with cylindrical norms %A Giovanni Bellettini %A Matteo Novaga %A Shokhrukh Kholmatov %K anisotropic Bernstein problem; %K minimal cones %K Non parametric minimal surfaces %K Sets of finite perimeter %X

We study various regularity properties of minimizers of the Φ–perimeter, where Φ is a norm. Under suitable assumptions on Φ and on the dimension of the ambient space, we prove that the boundary of a cartesian minimizer is locally a Lipschitz graph out of a closed singular set of small Hausdorff dimension. Moreover, we show the following anisotropic Bernstein-type result: any entire cartesian minimizer is the subgraph of a monotone function depending only on one variable.

%B Communications on Pure & Applied Analysis %V 16 %P 1427 %G eng %U http://aimsciences.org//article/id/47054f15-00c7-40b7-9da1-4c0b1d0a103d %R 10.3934/cpaa.2017068