0000000000520975

AUTHOR

Joonas Niinikoski

showing 4 related works from this author

Äärellisten Borel-mittojen Fourier-muunnoksista euklidisissa avaruuksissa

2016

Tutkielmassa esitellään euklidisen avaruuden äärellisille Borel-mitoille Fourier-muunnokset kompleksiarvoisina kuvauksina ja tutkitaan niiden vähenemistä mentäessä ääarettömän kauas origosta. Keskeisenä kysymyksenä on, millä reunaehdoilla annetun kompkatikantajaisen ja äärellisen Borel-mitan Fourier muunnos vähenee polynomiaalisesti (eli sitä voidaan dominoida jollain euklidisen normin negatiivisella potenssilla) kaikkialla riittävän kaukana tai ainakin ”keskimääräisesti”. Mikäli tällainen Borel-mitta on absoluuttisesti jatkuva Lebesguen mitan suhteen kompaktikantajaisella ja sileällä tiheysfunktiolla, niin sen Fourier-muunnos vähenee aina polynomiaalisesti kaikkialla. Ongelmaa tarkastellaa…

Gaussin kuvauskapasitiivinen dimensioFourier-muunnospolynomiaalinen väheneminenFrostmanin lemmaRuellen siirto-operaattorisymbolinen dynamiikkaäärellinen Borel-mittaRieszin energiaGibbsin mittaRajchman-mitta
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Volume preserving mean curvature flows near strictly stable sets in flat torus

2021

In this paper we establish a new stability result for the smooth volume preserving mean curvature flow in flat torus $\mathbb T^n$ in low dimensions $n=3,4$. The result says roughly that if the initial set is near to a strictly stable set in $\mathbb T^n$ in $H^3$-sense, then the corresponding flow has infinite lifetime and converges exponentially fast to a translate of the strictly stable (critical) set in $W^{2,5}$-sense.

osittaisdifferentiaaliyhtälötMean curvature53C44 (Primary) and 35K93 (Secondary)Applied Mathematics010102 general mathematicsMathematical analysisSense (electronics)Stability result01 natural sciences010101 applied mathematicsSet (abstract data type)differentiaaligeometriastrictly stable setsMathematics - Analysis of PDEsFlow (mathematics)Volume (thermodynamics)Independent setFOS: Mathematics0101 mathematicsFlat torusAnalysisMathematicsperiodic stabilityvolume preserving mean curvature flowAnalysis of PDEs (math.AP)
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Stationary sets of the mean curvature flow with a forcing term

2020

We consider the flat flow approach for the mean curvature equation with forcing in an Euclidean space $\mathbb R^n$ of dimension at least 2. Our main results states that tangential balls in $\mathbb R^n$ under any flat flow with a bounded forcing term will experience fattening, which generalizes the result by Fusco, Julin and Morini from the planar case to higher dimensions. Then, as in the planar case, we are able to characterize stationary sets in $\mathbb R^n$ for a constant forcing term as finite unions of equisized balls with mutually positive distance.

osittaisdifferentiaaliyhtälötMean curvature flowForcing (recursion theory)Mean curvatureEuclidean spaceApplied Mathematics010102 general mathematicsMathematical analysisstationary setscritical setsvariaatiolaskenta01 natural sciences35J93Term (time)010101 applied mathematicsMathematics - Analysis of PDEsFlow (mathematics)forced mean curvature flowBounded functionFOS: Mathematics0101 mathematicsConstant (mathematics)AnalysisAnalysis of PDEs (math.AP)MathematicsAdvances in Calculus of Variations
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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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