Search results for "large time behavior"

showing 2 items of 2 documents

Quantitative Alexandrov theorem and asymptotic behavior of the volume preserving mean curvature flow

2020

We prove a new quantitative version of the Alexandrov theorem which states that if the mean curvature of a regular set in Rn+1 is close to a constant in the Ln sense, then the set is close to a union of disjoint balls with respect to the Hausdorff distance. This result is more general than the previous quantifications of the Alexandrov theorem, and using it we are able to show that in R2 and R3 a weak solution of the volume preserving mean curvature flow starting from a set of finite perimeter asymptotically convergences to a disjoint union of equisize balls, up to possible translations. Here by a weak solution we mean a flat flow, obtained via the minimizing movements scheme. peerReviewed

Mathematics - Differential Geometrymean curvature flowMathematics - Analysis of PDEsDifferential Geometry (math.DG)FOS: Mathematicsminimizing movements35J93 53C44 53C45constant mean curvaturelarge time behaviorAnalysis of PDEs (math.AP)
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Stationary sets and asymptotic behavior of the mean curvature flow with forcing in the plane

2020

We consider the flat flow solutions of the mean curvature equation with a forcing term in the plane. We prove that for every constant forcing term the stationary sets are given by a finite union of disks with equal radii and disjoint closures. On the other hand for every bounded forcing term tangent disks are never stationary. Finally in the case of an asymptotically constant forcing term we show that the only possible long time limit sets are given by disjoint unions of disks with equal radii and possibly tangent. peerReviewed

osittaisdifferentiaaliyhtälötMathematics - Analysis of PDEsforced mean curvature flowFOS: Mathematicsstationary setscritical setsGeometry and TopologyAstrophysics::Earth and Planetary Astrophysicslarge time behaviorAnalysis of PDEs (math.AP)
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