https://neckpinch.com/authors/shokhrukh-yu-kholmatov/New arXiv papers by Shokhrukh Yu. Kholmatov, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT 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/2403.09902arxiv:2403.09902Thu, 14 Mar 2024 00:00:00 GMTShokhrukh Yu. Kholmatov — We study forced anisotropic curvature flow of droplets on an inhomogeneous horizontal hyperplane. As in [Bellettini, Kholmatov: J. Math. Pures Appl. (2018)] we establish the existence of smooth flow, starting from a regular droplet and satisfying the prescribed anisotropic Young's law, and also the existence of a 1/2-Hölder continuous in time minimizing movement solution starting from a set of finite perimeter. Furthermore, we investigate various properties of minimizing movements, including comparison principles, uniform boundedness and the consistency with the smooth flow.Other curvature flowsAnisotropic & nonlocal flowsWeak & singular flowsHomogeneous spaces 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/2401.06307arxiv:2401.06307Fri, 12 Jan 2024 00:00:00 GMTShokhrukh Kholmatov — In this paper we prove that in ℝ³ the minimizing movement solutions for mean curvature motion of droplets, obtained in [Bellettini, Kholmatov: J. Math. Pure Appl. (2018)], coincide with the smooth mean curvature flow of droplets with a prescribed (possibly nonconstant) contact angle.Mean curvature flowWeak & singular flowsFree boundary & capillary 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