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dc.contributor.authorNiinikoski, Joonas
dc.date.accessioned2021-08-19T10:23:31Z
dc.date.available2021-08-19T10:23:31Z
dc.date.issued2021
dc.identifier.isbn978-951-39-8812-8
dc.identifier.urihttps://jyx.jyu.fi/handle/123456789/77434
dc.description.abstractThe 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 n-dimensional sets. An important question is when a given mean curvature type of flow exists at all times, and thus does not form singularities. A singularity of a flow is a time where one cannot continue the flow, and usually the evolving set experiences topological changes. The work consists of three articles. In the first article [A], the focus lies on a behavior of a volume preserving mean curvature flow starting nearby a so-called strictly stable set in a three- or four-dimensional flat torus. The contribution of the first article is to show that if the previous flow starts sufficiently close to the strictly stable set in the H3-sense, then the flow exists at all times and converges, up to a small translation, to the set at an exponential rate. In particular, such a flow does not experience singularities. The second article [B] and the third article [C] concern generalizations of mean curvature type of flows, so-called flat flows, obtained via the minimizing movement method. Advantages of such a generalization are that it is defined at all times and requires less regularity for a given initial set compared to a mean curvature type of flow. In [B], it is shown that a flat flow of volume preserving mean curvature flow, starting from a bounded set of finite perimeter, has a shape of a finite union of equisized balls with mutually disjoint interiors in the asymptotical sense. The previous result relies on a new quantitative Alexandrov’s theorem, also proven in [B]. This theorem says that if a bounded C2-regular set, with a fixed upper bound on perimeter and a fixed lower bound on volume in an n-dimensional Euclidean space, has a boundary mean curvature close to a constant value in the Ln−1-sense, then the set is close to a finite union of equisized balls, with mutually disjoint interiors, in the Hausdorff-sense. In [C], it is shown that finite unions of n-dimensional tangent balls are not invariant under flat flows of any mean curvature flow with a bounded forcing. This is already proven in the two-dimensional case, so the third article generalizes this result to the higher dimensions.en
dc.format.mimetypeapplication/pdf
dc.language.isoeng
dc.publisherJyväskylän yliopisto
dc.relation.ispartofseriesJYU dissertations
dc.relation.haspart<b>Artikkeli I:</b> Niinikoski, J. (2021). Volume preserving mean curvature flows near strictly stable sets in flat torus. <i>Journal of Differential Equations, 276, 149-186.</i> DOI: <a href="https://doi.org/10.1016/j.jde.2020.12.010"target="_blank">10.1016/j.jde.2020.12.010</a>. <a href="https://arxiv.org/abs/1907.03618"target="_blank"> Preprint</a>
dc.relation.haspart<b>Artikkeli II:</b> Julin, V. and Niinikoski, J. (2020). Quantitative Alexandrov theorem and asymptotic behavior of the volume preserving mean curvature flow. <i>To appear in Anal. PDE.</i> <a href="https://arxiv.org/abs/2005.13800"target="_blank"> Preprint</a>
dc.relation.haspart<b>Artikkeli III:</b> Julin, V., & Niinikoski, J. (2021). Stationary sets of the mean curvature flow with a forcing term. <i>Advances in Calculus of Variations.</i> DOI: <a href="https://doi.org/10.1515/acv-2021-0019"target="_blank">10.1515/acv-2021-0019</a>. JYX: <a href="https://jyx.jyu.fi/handle/123456789/77405"target="_blank"> jyx.jyu.fi/handle/123456789/77405</a>
dc.rightsIn Copyright
dc.titleAsymptotical behavior of volume preserving mean curvature flow and stationary sets of forced mean curvature flow
dc.typeDiss.
dc.identifier.urnURN:ISBN:978-951-39-8812-8
dc.relation.issn2489-9003
dc.rights.copyright© The Author & University of Jyväskylä
dc.rights.accesslevelopenAccess
dc.type.publicationdoctoralThesis
dc.format.contentfulltext
dc.rights.urlhttp://rightsstatements.org/page/InC/1.0/?language=en
dc.date.digitised


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