Convergence of nonlocal geometric flows to anisotropic mean curvature motion. (English) Zbl 1471.53072
Summary: We consider nonlocal curvature functionals associated with positive interaction kernels, and we show that local anisotropic mean curvature functionals can be retrieved in a blow-up limit from them. As a consequence, we prove that the viscosity solutions to the rescaled nonlocal geometric flows locally uniformly converge to the viscosity solution to the anisotropic mean curvature motion. The result is achieved by combining a compactness argument and a set-theoretic approach related to the theory of De Giorgi’s barriers for evolution equations.
MSC:
| 53E10 | Flows related to mean curvature |
| 35D40 | Viscosity solutions to PDEs |
| 35K93 | Quasilinear parabolic equations with mean curvature operator |
| 35R11 | Fractional partial differential equations |
Keywords:
nonlocal curvature flow; anisotropic mean curvature flow; geometric equations; de Giorgi’s barriers for geometric evolutions; level-set method; viscosity solutionsReferences:
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