arXiv subjects

Olimjon Eshkobilov

Publications and source records attributed to Olimjon Eshkobilov.

6 recordsLinked to original sources

arXiv2026

Inverse Robin Spectral Problem for the p-Laplace Operator

We study an inverse Robin spectral problem for the $p$-Laplacian with mixed Dirichlet--Robin boundary conditions. The unknown coefficient lies on an inaccessible Robin portion, and the data are the principal eigenvalue and nonlinear conormal flux on an accessible Dirichlet patch. For a coating of thickness $\varepsilonρ$ and conductivity $\varepsilon^{p-1}$, we prove convergence of the principal eigenvalue and strong $W^{1,p}$ convergence of the eigenfunctions to an effective Robin problem with coefficient $h=ρ^{-(p-1)}$. The Rayleigh characterization gives a two-sided comparison inequality, ordered uniqueness and weighted $L^1$ stability, and Fréchet differentiability of the eigenvalue map for $1 2$; together with a quantitative linearized estimate, it yields conditional local stability. On finite-dimensional classes, branch differentiability and linearized injectivity yield local Lipschitz stability (with branch differentiability automatic when $p=2$), while monotone one-parameter families are identifiable from the principal eigenvalue.

math.AP↗

arXiv2026

Limit-Cycle Replication via Chebyshev Pullbacks and a Quadratic Ceiling for Separable Schemes

Let $H(n)$ denote the Hilbert number, i.e.\ the maximal number of limit cycles of planar polynomial vector fields of degree $\le n$. A classical lower-bound mechanism for $H(n)$ is \emph{replication}: one pulls back a vector field by a polynomial map and lifts each existing limit cycle to several disjoint copies while controlling the resulting degree. In this paper we give a fully self-contained replication theorem based on the separable Chebyshev covering \[ Φ(u,v)=(T_m(u),T_m(v)). \] Using the $m$ monotone full branches of $T_m$ on $(-1,1)$, we prove that every degree-$\le n$ polynomial vector field with $k$ limit cycles gives rise to a degree-$\le nm+m-1$ polynomial vector field with at least $m^2k$ limit cycles. Consequently, \[ H(nm+m-1)\ge m^2H(n)\qquad (m\ge 2). \] We then extend the construction to general separable pullbacks $(u,v)\mapsto (p(u),p(v))$, show that Chebyshev attains the maximal possible branch count among degree-$m$ separable pullbacks, and prove a quadratic ceiling for replication-only schemes: if one iterates separable pullbacks and no additional limit cycles are created beyond those forced by lifting, then the number of resulting limit cycles is at most quadratic in the final degree. This shows that superquadratic lower bounds, such as the known $n^2\log n$-type bounds, necessarily require mechanisms beyond pure separable replication. Finally, combining our replication theorem with the strongest currently published seed bounds, we obtain new explicit lower estimates in several degrees, including \begin{gather*} H(14)\ge 252,\qquad H(29)\ge 1080,\\ H(31)\ge 1380,\qquad H(39)\ge 2012. \end{gather*}

math.DS↗

arXiv2019

On the restricted conformal group of the (1+n)-Einstein static universe

Explicit models for the restricted conformal group of the Einstein static universe of dimension greater than two and for its universal covering group are constructed. Based on these models, as an application we determine all oriented and time-oriented conformal Lorentz manifolds whose restricted conformal group has maximal dimension. They amount to the Einstein static universe itself and two countably infinite series of its compact quotients.

math.DG↗

arXiv2019

Critical Robertson-Walker universes

The integral of the energy density function $\mathfrak m$ of a closed Robertson-Walker (RW) spacetime with source a perfect fluid and cosmological constant $Λ$ gives rise to an action functional on the space of scale functions of RW spacetime metrics. This paper studies closed RW spacetimes which are critical for this functional, subject to volume-preserving variations (critical RW spacetimes). A complete classification of critical RW spacetimes is given and explicit solutions in terms of Weierstrass elliptic functions and their degenerate forms are computed. The standard energy conditions (weak, dominant, and strong) as well as the cyclic property of critical RW spacetimes are discussed.

math.DG↗

arXiv2017

Contact manifolds, Lagrangian Grassmannians and PDEs

In this paper we review a geometric approach to PDEs. We mainly focus on scalar PDEs in n independent variables and one dependent variable of order one and two, by insisting on the underlying (2n+1)-dimensional contact manifold and the so-called Lagrangian Grassmannian bundle over the latter. This work is based on a 30-hours Ph.D course given by two of the authors (GM and GM). As such, it was mainly designed as a quick introduction to the subject for graduate students. But also the more demanding reader will be gratified, thanks to the frequent references to current research topics and glimpses of higher-level mathematics, found mostly in the last sections.

math.DG↗

arXiv2016

A conformally invariant variational problem for time-like curves

We study the conformally invariant variational problem for time-like curves in the $n$-dimensional Einstein universe defined by the conformal strain functional. We prove that the stationary curves are trapped into an Einsetin universe of dimension $2$, $3$ or $4$. We study the linearly-full stationary curves in a four-dimensional Einstein universe and we show that they can be integrated by quadratures in terms of elliptic functions, elliptic integrals and Jacobi's theta functions.

math.DG↗