Uniqueness of Viscosity Mean Curvature Flow Solution in Two Sub-Riemannian Structures

Abstract : We provide a uniqueness result for a class of viscosity solutions to sub-Riemannian mean curvature flows. In a sub-Riemannian setting, uniqueness cannot be deduced by the comparison principle, which is known only for graphs and for radially symmetry surfaces. Here we use a definition of continuous viscosity solutions of sub-Riemannian mean curvature flows motivated from a regularized Riemannian approximation of the flow. With this definition, we prove that any continuous viscosity solution of the equation is a limit of a sequence of solutions of Riemannian flow and obtain as a consequence uniqueness and the comparison principle. The results are provided in the settings of both 3-dimensional rototranslation group ${SE}(2)$ and Carnot groups of step 2, which are particularly important due to their relation to the surface completion problem of a model of the visual cortex.
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Contributor : Emre Baspinar <>
Submitted on : Friday, October 18, 2019 - 8:20:41 AM
Last modification on : Wednesday, November 27, 2019 - 1:27:11 AM

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Emre Baspinar, Giovanna Citti. Uniqueness of Viscosity Mean Curvature Flow Solution in Two Sub-Riemannian Structures. SIAM Journal on Mathematical Analysis, Society for Industrial and Applied Mathematics, 2019, 51 (3), pp.2633-2659. ⟨10.1137/17M1150797⟩. ⟨hal-02319482⟩

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