Quantitative Alexandrov theorem and asymptotic behavior of the volume preserving mean curvature flow
Julin, V., & Niinikoski, J. (2023). Quantitative Alexandrov theorem and asymptotic behavior of the volume preserving mean curvature flow. Analysis and PDE, 16(3), 679710. https://doi.org/10.2140/apde.2023.16.679
Published in
Analysis and PDEDate
2023Discipline
MatematiikkaAnalyysin ja dynamiikan tutkimuksen huippuyksikköMathematicsAnalysis and Dynamics Research (Centre of Excellence)Copyright
© 2023 the Authors
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.
Publisher
Mathematical Sciences PublishersISSN Search the Publication Forum
21575045Keywords
Publication in research information system
https://converis.jyu.fi/converis/portal/detail/Publication/183485861
Metadata
Show full item recordCollections
Related funder(s)
Research Council of FinlandFunding program(s)
Research costs of Academy Research Fellow, AoFAdditional information about funding
This research was supported by the Academy of Finland grant 314227.License
Related items
Showing items with similar title or keywords.

Asymptotical behavior of volume preserving mean curvature flow and stationary sets of forced mean curvature flow
Niinikoski, Joonas (Jyväskylän yliopisto, 2021)The main subject of this dissertation is mean curvature type of flows, in particular the volume preserving mean curvature flow. A classical flow in this context is seen as a smooth time evolution of ndimensional sets. An ... 
Volume preserving mean curvature flows near strictly stable sets in flat torus
Niinikoski, Joonas (Elsevier BV, 2021)In this paper we establish a new stability result for smooth volume preserving mean curvature flows in flat torus Tn in dimensions n = 3, 4. The result says roughly that if an initial set is near to a strictly stable set ... 
Stationary Sets and Asymptotic Behavior of the Mean Curvature Flow with Forcing in the Plane
Fusco, Nicola; Julin, Vesa; Morini, Massimiliano (Springer, 2022)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 ... 
The asymptotics of the areapreserving mean curvature and the Mullins–Sekerka flow in two dimensions
Julin, Vesa; Morini, Massimiliano; Ponsiglione, Marcello; Spadaro, Emanuele (Springer, 2022)We provide the first general result for the asymptotics of the area preserving mean curvature flow in two dimensions showing that flat flow solutions, starting from any bounded set of finite perimeter, converge with ... 
Consistency of the Flat Flow Solution to the Volume Preserving Mean Curvature Flow
Julin, Vesa; Niinikoski, Joonas (Springer, 2024)We consider the flat flow solution, obtained via a discrete minimizing movement scheme, to the volume preserving mean curvature flow starting from C1,1regular set. We prove the consistency principle, which states that ...