28 citations · 41 across the 17 of their papers we have counts for
20 papers
Crystalline elastic flow of polygonal curves: long time behaviour and convergence to stationary solutions
Giovanni Bellettini, Shokhrukh Yu. Kholmatov, Matteo Novaga
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 tr…
A De Giorgi conjecture on the regularity of minimizers of Cartesian area in 1D
Giovanni Bellettini, Shokhrukh Yu. Kholmatov
We prove a -regularity of minimizers of the functional provided is a bounded open inte…
Bound states of 2+1 fermionic trimers on lattice at strong couplings
Janikul Abdullaev, Ahmad Khalkhuzhaev, Shokhrukh Yu. Kholmatov
In this paper, we investigate the bound states of fermionic trimers on a three-dimensional lattice at strong coupling. Specifically, we analyze the discrete spectrum of the a…
Minimizing movements for the generalized power mean curvature flow
Giovanni 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 \[ \int_…
Minimizing movements for forced anisotropic curvature flow of droplets
Shokhrukh 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 existe…
On the minimality of the Winterbottom shape
Shokhrukh Yu. Kholmatov
In this short note we prove that the Winterbottom shape [Winterbottom: Acta Metallurgica (1967)] is a volume-constraint minimizer of the corresponding anisotropic capillary functio…
Crystalline hexagonal curvature flow of networks: short-time, long-time and self-similar evolutions
Giovanni 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…
Existence of minimizers for the SDRI model in : Wetting and dewetting regimes with mismatch strain
Shokhrukh Kholmatov, Paolo Piovano
The existence and the regularity results obtained in [37] for the variational model introduced in [36] to study the optimal shape of crystalline materials in the setting of stress-…
Some aspects of anisotropic curvature flow of planar partitions
Giovanni 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 smoot…
On the spectrum of Schrödinger-type operators on two dimensional lattices
Shokhrukh Yu. Kholmatov, Saidakhmat N. Lakaev, Firdavsjon M. Almuratov
We consider a family of Schrödinger-type operators on the two dimensional lattice where $\wideha…
Bose-Hubbard models with on-site and nearest-neighbor interactions: Exactly solvable case
Saidakhmat Lakaev, Shokhrukh Kholmatov, Shakhobiddin Khamidov
We study the discrete spectrum of the two-particle Schrödinger operator associated to the Bose-Hubbard Hamiltonian of…
Bound states of discrete Schrödinger operators on one and two dimensional lattices
Shokhrukh Kholmatov, Saidakhmat Lakaev, Firdavs Almuratov
We study the spectral properties of discrete Schrödinger operator associated to a one-particle system in -dimensiona…
Existence of minimizers for the SDRI model in 2d: wetting and dewetting regime with mismatch strain
Shokhrukh Yu. Kholmatov, Paolo Piovano
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 morph…
Minimizing movements for forced anisotropic mean curvature flow of partitions with mobilities
Giovanni Bellettini, Antonin Chambolle, Shokhrukh Yu. Kholmatov
Under suitable assumptions on the family of anisotropies, we prove the existence of a weak global -Hölder continuous in time mean curvature flow with mobilities of a…
Expansion of eigenvalues of rank-one perturbations of the discrete bilaplacian
Ahmad Khalkhuzhaev, Shokhrukh Yu. Kholmatov, Mardon Pardabaev
We consider the family of discrete Schrödinger-type operators in -dimensional lattice , where…
Asymptotics of eigenvalues of the zero-range perturbation of the discrete bilaplacian
Shokhrukh Yu. Kholmatov, Mardon Pardabaev
We consider the family of discrete Schrödinger-type operators in one-dimensional lattice $\m…
A unified model for stress-driven rearrangement instabilities
Shokhrukh Yu. Kholmatov, Paolo Piovano
A variational model to simultaneously treat Stress-Driven Rearrangement Instabilities, such as boundary discontinuities, internal cracks, external filaments, edge delamination, wet…
Minimizing movements for mean curvature flow of partitions
Giovanni Bellettini, Shokhrukh Yusufovich Kholmatov
We prove the existence of a weak global in time mean curvature flow of a bounded partition of space using the method of minimizing movements. The result is extended to the case whe…
Minimizers of anisotropic perimeters with cylindrical norms
G. Bellettini, M. Novaga, Sh. Yu. Kholmatov
We study various regularity properties of minimizers of the --perimeter, where is a norm. Under suitable assumptions on and on the dimension of the ambient space, we pro…
Existence of bound states of -body problem in an optical lattice
Shokhrukh Kholmatov, Zahriddin Muminov
We provide sufficient conditions to have at least one -particle bound state below the essential spectrum of a large class of -particle discrete Schrödinger operators …