https://neckpinch.com/authors/giovanni-bellettini/New arXiv papers by Giovanni Bellettini, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2601.04385arxiv:2601.04385Wed, 07 Jan 2026 00:00:00 GMTGiovanni Bellettini, Virginia Lorenzini, Matteo Novaga, Riccardo Scala — We consider a fourth-order regularization of the curvature flow for an immersed plane curve with fixed boundary, using an elastica-type functional depending on a small positive parameter ε. We show that the approximating flow smoothly converges, as ε→ 0⁺, to the curvature flow of the curve with Dirichlet boundary conditions for all times before the first singularity of the limit flow.Other curvature flowsSingularitiesBoundaries & non-compact https://arxiv.org/abs/2403.20244arxiv:2403.20244Fri, 29 Mar 2024 00:00:00 GMTGiovanni Bellettini, Shokhrukh Yu. Kholmatov — Motivated by a conjecture of De Giorgi, we consider the Almgren-Taylor-Wang scheme for mean curvature flow, where the volume penalization is replaced by a term of the form ∫_EΔF f(frac rm d_F τ) dx for f ranging in a large class of strictly increasing continuous functions. In particular, our analysis covers the case f(r) = r^α, r ≥ 0, α>0, considered by De Giorgi. We show that the generalized minimizing movement scheme converges to the geometric evolution equation f(v) = − κ on $∂E(t)$, where E(t)\ are evolving subsets of ℝⁿ, v is the normal velocity of ∂E(t), and κ is the mean curvature of ∂E(t). We extend our analysis to the anisotropic setting, and in the presence of a driving force. We also show that minimizing movements coincide with the smooth classical solution as long as the latter exists. Finally, we prove that in the absence of forcing, mean convexity and convexity are preserved by the weak flow.Mean curvature flowWeak & singular flows https://arxiv.org/abs/2401.15358arxiv:2401.15358Sat, 27 Jan 2024 00:00:00 GMTGiovanni Bellettini, Shokhrukh Kholmatov, Firdavsjon Almuratov — We study the crystalline curvature flow of planar networks with a single hexagonal anisotropy. After proving the local existence of a classical solution for a rather large class of initial conditions, we classify the homothetically shrinking solutions having one bounded component. We also provide an example of network shrinking to a segment with multiplicity two.Other curvature flowsAnisotropic & nonlocal flowsPhase fields & multiphase flows https://arxiv.org/abs/2304.13454arxiv:2304.13454Wed, 26 Apr 2023 00:00:00 GMTGiovanni Bellettini, Shokhrukh Yu. Kholmatov — We consider the geometric evolution of a network in the plane, flowing by anisotropic curvature. We discuss local existence of a classical solution in the presence of several smooth anisotropies. Next, we discuss some aspects of the polycrystalline case.Other curvature flowsAnisotropic & nonlocal flows