arXiv subjects

Shokhrukh Yu. Kholmatov

Publications and source records attributed to Shokhrukh Yu. Kholmatov.

13 recordsLinked to original sources

arXiv2025

Crystalline elastic flow of polygonal curves: long time behaviour and convergence to stationary solutions

Given a planar crystalline anisotropy, we study the crystalline elastic flow of immersed polygonal curves, possibly also unbounded. Assuming that the segments evolve by parallel translation (as it happens in the standard crystalline curvature flow), we prove that a unique regular flow exists until a maximal time when some segments having zero crystalline curvature disappear. Furthermore, for closed polygonal curves, we analyze the behaviour at the maximal time, and show that it is possible to restart the flow finitely many times, yielding a globally in time evolution, that preserves the index of the curve. Next, we investigate the long-time properties of the flow using a Lojasiewicz-Simon-type inequality, and show that, as time tends to infinity, the flow fully converges to a stationary curve. We also provide a complete classification of the stationary solutions and a partial classification of the translating solutions in the case of the square anisotropy.

math.AP↗

arXiv2025

A De Giorgi conjecture on the regularity of minimizers of Cartesian area in 1D

We prove a $C^{1,1}$-regularity of minimizers of the functional $$ \int_I \sqrt{1+|Du|^2} + \int_I |u-g|ds,\quad u\in BV(I), $$ provided $I\subset\mathbb{R}$ is a bounded open interval and $\|g\|_\infty$ is sufficiently small, thus partially establishing a De Giorgi conjecture in dimension one and codimension one. We also extend our result to a suitable anisotropic setting.

math.AP↗

arXiv2025

Bound states of 2+1 fermionic trimers on lattice at strong couplings

In this paper, we investigate the bound states of $2+1$ fermionic trimers on a three-dimensional lattice at strong coupling. Specifically, we analyze the discrete spectrum of the associated three-body discrete Schrödinger operator $H_{γ,λ}(K),$ focusing on energies below the continuum and within its gap. Depending on the quasi-momentum $K,$ we show that if the mass ratio $γ>0$ between the identical fermions and the third particle is below a certain threshold, the operator lacks a discrete spectrum below the essential spectrum for sufficiently large coupling $λ>0.$ Conversely, if $γ$ exceeds this threshold, $H_{γ,λ}(K)$ admits at least one eigenvalue below the essential spectrum. Similar phenomena are observed in the neighborhood of the two-particle branch of the essential spectrum, which resides within the gap and grows sublinearly as $λ\to+\infty.$ For $K=0,$ the mass ratio thresholds are explicitly calculated and it turns out that, for certain intermediate mass ratios and large couplings, bound states emerge within the gap, although ground states are absent.

math.SP↗

arXiv2024

Minimizing movements for the generalized power mean curvature flow

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 \[ \int_{EΔF} f\Big(\frac{ {\rm d}_F }τ\Big)~dx \] for $f$ ranging in a large class of strictly increasing continuous functions. In particular, our analysis covers the case \[ f(r) = r^α, \qquad r \geq 0, \quad α>0, \] considered by De Giorgi. We show that the generalized minimizing movement scheme converges to the geometric evolution equation \[ f(v) = - κ\quad \text{on $\partial E(t)$,} \] where $\{E(t)\}$ are evolving subsets of $\mathbb{R}^n,$ $v$ is the normal velocity of $\partial E(t),$ and $κ$ is the mean curvature of $\partial E(t)$. 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.

math.AP↗

arXiv2024

Minimizing movements for forced anisotropic curvature flow of droplets

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.

math.AP↗

arXiv2024

On the minimality of the Winterbottom shape

In this short note we prove that the Winterbottom shape [Winterbottom: Acta Metallurgica (1967)] is a volume-constraint minimizer of the corresponding anisotropic capillary functional.

math.AP↗

arXiv2023

Existence of minimizers for the SDRI model in 2d: wetting and dewetting regime with mismatch strain

The model introduced in [Kholmatov-Piovano 2020] in the framework of the theory on Stress-Driven Rearrangement Instabilities (SDRI) [Asaro-Tiller 1972; Grinfeld 1993} for the morphology of crystalline materials under stress is considered. As in [Kholmatov-Piovano 2020] and in agreement with the models in [Lowengrub et al. 2009; Spencer 1999], a mismatch strain, rather than a Dirichlet condition as in [Crismale-Friedrich 2020], is considered to include into the analysis the lattice mismatch between the crystal and possible adjacent (supporting) materials. The existence of solutions is established in dimension two in the absence of graph-like assumptions and of the restriction to a finite number $m$ of connected components for the free boundary of the region occupied by the crystalline material, thus extending previous results for epitaxially strained thin films and material cavities. Due to the lack of compactness and lower semicontinuity for the sequences of $m$-minimizers, i.e., minimizers among configurations with at most $m$ connected boundary components, a minimizing candidate is directly constructed, and then shown to be a minimizer by means of uniform density estimates and the convergence of $m$-minimizers' energies to the energy infimum as $m\to\infty$. Finally, regularity properties for the morphology satisfied by every minimizer are established.

math.AP↗

arXiv2023

Some aspects of anisotropic curvature flow of planar partitions

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.

math.DG↗

arXiv2022

On the spectrum of Schrödinger-type operators on two dimensional lattices

We consider a family $$ \widehat H_{a,b}(μ)=\widehat H_0 +μ\widehat V_{a,b}\quad μ>0, $$ of Schrödinger-type operators on the two dimensional lattice $\mathbb{Z}^2,$ where $\widehat H_0$ is a Laurent-Toeplitz-type convolution operator with a given Hopping matrix $\hat{e}$ and $\widehat V_{a,b}$ is a potential taking into account only the zero-range and one-range interactions, i.e., a multiplication operator by a function $\hat v$ such that $\hat v(0)=a,$ $\hat v(x)=b$ for $|x|=1$ and $\hat v(x)=0$ for $|x|\ge2,$ where $a,b\in\mathbb{R}\setminus\{0\}.$ Under certain conditions on the regularity of $\hat{e}$ we completely describe the discrete spectrum of $\hat H_{a,b}(μ)$ lying above the essential spectrum and study the dependence of eigenvalues on parameters $μ,$ $a$ and $b.$ Moreover, we characterize the threshold eigenfunctions and resonances.

math.SP↗

arXiv2020

A unified model for stress-driven rearrangement instabilities

A variational model to simultaneously treat Stress-Driven Rearrangement Instabilities, such as boundary discontinuities, internal cracks, external filaments, edge delamination, wetting, and brittle fractures, is introduced. The model is characterized by an energy displaying both elastic and surface terms, and allows for a unified treatment of a wide range of settings, from epitaxially-strained thin films to crystalline cavities, and from capillarity problems to fracture models. Existence of minimizing configurations is established by adopting the direct method of the Calculus of Variations. Compactness of energy-equibounded sequences and energy lower semicontinuity are shown with respect to a proper selected topology in a class of admissible configurations that extends the classes previously considered in the literature. In particular, graph-like constraints previously considered for the setting of thin films and crystalline cavities are substituted by the more general assumption that the free crystalline interface is the boundary, consisting of an at most fixed finite number $m$ of connected components, of sets of finite perimeter. Finally, it is shown that, as $m\to\infty$, the energy of minimal admissible configurations tends to the minimum energy in the general class of configurations without the bound on the number of connected components for the free interface.

math.AP↗

arXiv2020

Minimizing movements for forced anisotropic mean curvature flow of partitions with mobilities

Under suitable assumptions on the family of anisotropies, we prove the existence of a weak global $\frac{1}{n+1}$-Hölder continuous in time mean curvature flow with mobilities of a bounded anisotropic partition in any dimension using the method of minimizing movements. The result is extended to the case when suitable driving forces are present. We improve the Hölder exponent to $\frac12$ in the case of partitions with the same anisotropy and the same mobility and provide a weak comparision result in this setting for a weak anisotropic mean curvature flow of a partition and an anisotropic mean curvature two-phase flow.

math.AP↗

arXiv2019

Asymptotics of eigenvalues of the zero-range perturbation of the discrete bilaplacian

We consider the family $$ \hat {\bf h}_μ:=\hat\varDelta\hat \varDelta - μ\hat {\bf v},\qquadμ\in\mathbb{R}, $$ of discrete Schrödinger-type operators in one-dimensional lattice $\mathbb{Z}$, where $\hat \varDelta$ is the discrete Laplacian and $\hat{\bf v}$ is of zero-range. We prove that for any $μ\ne0$ the discrete spectrum of $\hat {\bf h}_μ$ is a singleton $\{e(μ)\},$ and $e(μ)<0$ for $μ>0$ and $e(μ)>4$ for $μ<0.$ Moreover, we study the properties of $e(μ)$ as a function of $μ,$ in particular, we find the asymptotics of $e(μ)$ as $μ\searrow0$ and $μ\nearrow0.$

math.SP↗

arXiv2019

Expansion of eigenvalues of rank-one perturbations of the discrete bilaplacian

We consider the family $\hat h_μ:=\hat\varDelta\hat \varDelta - μ\hat v,$ $μ\in\mathbb{R}, $ of discrete Schrödinger-type operators in $d$-dimensional lattice $\mathbb{Z}^d$, where $\hat \varDelta$ is the discrete Laplacian and $\hat v$ is of rank-one. We prove that there exist coupling constant thresholds $μ_o,μ^o\ge0$ such that for any $μ\in[-μ^o,μ_o]$ the discrete spectrum of $\hat h_μ$ is empty and for any $μ\in \mathbb{R}\setminus[-μ^o,μ_o]$ the discrete spectrum of $\hat h_μ$ is a singleton $\{e(μ)\},$ and $e(μ)<0$ for $μ>μ_o$ and $e(μ)>4d^2$ for $μ<-μ^o.$ Moreover, we study the asymptotics of $e(μ)$ as $μ\toμ_o$ and $μ\to -μ^o$ as well as $μ\to\pm\infty.$ The asymptotics highly depend on $d$ and $\hat v.$

math.SP↗