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.

Baspinar E., Citti G. (2019). Uniqueness of viscosity mean curvature flow solution in two sub-riemannian structures. SIAM JOURNAL ON MATHEMATICAL ANALYSIS, 51(3), 2633-2659 [10.1137/17M1150797].

Uniqueness of viscosity mean curvature flow solution in two sub-riemannian structures

Citti G.
2019

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.
2019
Baspinar E., Citti G. (2019). Uniqueness of viscosity mean curvature flow solution in two sub-riemannian structures. SIAM JOURNAL ON MATHEMATICAL ANALYSIS, 51(3), 2633-2659 [10.1137/17M1150797].
Baspinar E.; Citti G.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11585/701100
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