https://neckpinch.com/authors/vesa-julin/New arXiv papers by Vesa Julin, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2603.24091arxiv:2603.24091Wed, 25 Mar 2026 00:00:00 GMTVedansh Arya, Seongmin Jeon, Vesa Julin — The recent work of Morini-Oronzio-Spadaro and the third author shows that, in three dimensions, a flat-flow solution of the volume-preserving mean curvature flow that converges to a single ball, which is the case for instance when the initial perimeter is less than that of two disjoint balls, converges exponentially fast in Hausdorff distance. In this paper we strengthen this result by proving that after a finite time the flow becomes smooth, satisfies the equation in the classical sense and converges exponentially fast to the limiting ball in every C^k-norm. In the proof we develop a version of Brakke's epsilon regularity theorem adapted to our setting and derive the necessary nonlinear PDE estimates directly at the level of the discrete minimizing-movement scheme. The same result holds in the planar case.Mean curvature flowVolume-preserving & constrained flowsSingularitiesWeak & singular flowsLong-time existence & convergence https://arxiv.org/abs/2307.03912arxiv:2307.03912Sat, 08 Jul 2023 00:00:00 GMTVesa Julin, Domenico Angelo La Manna — We prove that the volume preserving fractional mean curvature flow starting from a convex set does not develop singularities along the flow. By the recent result of Cesaroni-Novaga this then implies that the flow converges to a ball exponentially fast. In the proof we show that the apriori estimates due to Cinti-Sinestrari-Valdinoci imply the C^(1+α)-regularity of the flow and then provide a regularity argument which improves this into C^(2+α)-regularity of the flow. The regularity step from C^(1+α) into C^(2+α) does not rely on convexity and can probably be adopted to more general setting.Mean curvature flowAnisotropic & nonlocal flowsSingularitiesLong-time existence & convergence https://arxiv.org/abs/2301.10089arxiv:2301.10089Tue, 24 Jan 2023 00:00:00 GMTVesa Julin — In this paper I will revisit the construction of a global weak solution to the volume preserving mean curvature flow via discrete minimizing movement scheme by Mugnai-Seis-Spadaro (2016). This method is based on the gradient flow approach due to Almgren-Taylor-Wang (1993) and Luckhaus-Strurzenhecker (1995) and my aim is to replace the volume penalization by implementing the volume constraint directly in the discrete scheme, which from practical point of view is perhaps more natural. A technical novelty is the proof of the density estimate which is based on the second variation condition of the energy.Mean curvature flowVolume-preserving & constrained flowsWeak & singular flows https://arxiv.org/abs/2206.05002arxiv:2206.05002Fri, 10 Jun 2022 00:00:00 GMTVesa Julin, Joonas Niinikoski — We consider the flat flow solution, obtained via discrete minimizing movement scheme, to the volume preserving mean curvature flow starting from C^1,1-regular set. We prove the consistency principle which states that (any) such flat flow agrees with the classical solution as long as the latter exists. In particular, the flat flow is unique and smooth up to the first singular time. We obtain the result by proving the full regularity for the discrete time approximation of the flat flow such that the regularity estimates are stable with respect to the time discretization. Our method can also be applied in the case of the mean curvature flow and thus it provides an alternative proof, not relying on comparison principle, for the consistency between the flat flow solution and the classical solution for C^1,1-regular initial sets.Mean curvature flowVolume-preserving & constrained flowsWeak & singular flows https://arxiv.org/abs/2112.13936arxiv:2112.13936Mon, 27 Dec 2021 00:00:00 GMTVesa Julin, Massimiliano Morini, Marcello Ponsiglione, Emanuele Spadaro — 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 exponential rate to a finite union of equally sized disjoint disks. A similar result is established also for the periodic two-phase Mullins-Sekerka flow.Mean curvature flowVolume-preserving & constrained flows