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Vol 84, No 6 (2020)
- Year: 2020
- Articles: 8
- URL: https://ogarev-online.ru/1607-0046/issue/view/7547
Articles
Complete description of the Lyapunov spectra of continuous families of linear differential systems withunbounded coefficients
Abstract
For every positive integer $n$ and every metric space $M$ we consider the class$\widetilde{\mathcal{U}}^n(M)$ of all parametric families $\dot x = A(t, \mu)x$, where $x\in\mathbb{R}^n$, $t\ge 0$,$\mu\in M$, of linear differential systems whose coefficients are piecewise continuous and, generally speaking,unbounded on the time semi-axis for every fixed value of the parameter $\mu$ such that if a sequence$(\mu_k)$ converges to $\mu_0$ in the space of parameters, then the sequence $(A( {\cdot} ,\mu_k))$\linebreakconverges uniformly on the semi-axis to the matrix $A( {\cdot} ,\mu_0)$. For the familiesin $\widetilde{\mathcal{U}}^n(M)$, we obtain a complete description of individual Lyapunov exponents andtheir spectra as functions of the parameter.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2020;84(6):3-22
3-22
Geometric estimates of solutions of quasilinear elliptic inequalities
Abstract
Suppose that $p>1$ and $\alpha$ are real numbers with $p-1 \le \alpha \le p$. Let $\Omega$ be a non-emptyopen subset of $\mathbb{R}^n$, $n \ge 2$. We consider the inequality$$\operatorname{div} A (x, D u)+b (x) |D u|^\alpha\ge 0,$$where $D=(\partial/\partial x_1, \partial/\partial x_2, …, \partial/\partial x_n)$ is the gradient operator,$A\colon \Omega \times \mathbb{R}^n \to \mathbb{R}^n$ and $b\colon \Omega \to [0, \infty)$ are certain functions and$$C_1|\xi|^p\le\xi A(x, \xi),\quad |A (x, \xi)|\le C_2|\xi|^{p-1},\qquad C_1, C_2=\mathrm{const}>0, \quad p>1,$$for almost all $x \in \Omega$ and all $\xi \in \mathbb{R}^n$. We obtain estimates for solutions of this inequality usingthe geometry of $\Omega$. In particular, these estimates yield regularity conditions for boundary points.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2020;84(6):23-72
23-72
Threshold resonances and virtual levels in the spectrum of cylindrical and periodic waveguides
Abstract
We describe and classify the thresholds of the continuous spectrum and the resulting resonances forgeneral formally self-adjoint elliptic systems of second-order differential equations with Dirichlet or Neumannboundary conditions in domains with cylindrical and periodic outlets to infinity (in waveguides). These resonancesarise because there are “almost standing” waves, that is, non-trivial solutions of the homogeneousproblem which do not transmit energy. We consider quantum, acoustic, and elastic waveguides as examples.Our main focus is on degenerate thresholds which are characterized by the presence of standing waves withpolynomial growth at infinity and produce effects lacking for ordinary thresholds. In particular, we describe theeffect of lifting an eigenvalue from the degenerate zero threshold of the spectrum. This effect occurs for elasticwaveguides of a vector nature and is absent from the scalar problems for cylindrical acoustic and quantumwaveguides. Using the technique of self-adjoint extensions of differential operators in weighted spaces, weinterpret the almost standing waves as eigenvectors of certain operators and the threshold as the correspondingeigenvalue. Here the threshold eigenvalues and the corresponding vector-valued functions not decaying at infinitycan be obtained by approaching the threshold (the virtual level) either from below or from above. Hence their propertiesdiffer essentially from the customary ones. We state some open problems.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2020;84(6):73-130
73-130
On the group of spheromorphisms of a homogeneous non-locally finite tree
Abstract
We consider a tree $\mathbb{T}$ all whose vertices have countable valency. Its boundary is the Baire space $\mathbb{B}\simeq\mathbb{N}^\mathbb{N}$ and the set of irrational numbers $\mathbb{R}\setminus\mathbb{Q}$ is identified with $\mathbb{B}$ by continued fraction expansions. Removing $k$ edges from $\mathbb{T}$, we get a forest consisting of copies of $\mathbb{T}$. A spheromorphism (or hierarchomorphism) of $\mathbb{T}$ is an isomorphism of two such subforests regarded as a transformation of $\mathbb{T}$ or $\mathbb{B}$. We denote the group of all spheromorphisms by $\operatorname{Hier}(\mathbb{T})$. We show that the correspondence $\mathbb{R}\setminus \mathbb{Q}\simeq \mathbb{B}$ sends the Thompson group realized by piecewise $\mathrm{PSL}_2(\mathbb{Z})$-transformations to a subgroup of $\operatorname{Hier}(\mathbb{T})$. We construct some unitary representations of $\operatorname{Hier}(\mathbb{T})$, show that the group $\operatorname{Aut}(\mathbb{T})$ of automorphisms is spherical in $\operatorname{Hier}(\mathbb{T})$ and describe the train (enveloping category) of $\operatorname{Hier}(\mathbb{T})$.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2020;84(6):131-164
131-164
Bogolyubov's theorem for a controlled system related to a variational inequality
Abstract
We consider the problem of minimizing an integral functional on the solutions of a controlled systemdescribed by a non-linear differential equation in a separable Banach space and a variational inequality.The variational inequality determines a hysteresis operator whose input is a trajectory of the controlled system andwhose output occurs in the right-hand side of the differential equation, in the constraint on the control, and in the functionalto be minimized. The constraint on the control is a multivalued map with closed non-convex values and the integrandis a non-convex function of the control. Along with the original problem, we consider the problem of minimizing theintegral functional with integrand convexified with respect to the control, on the solutions of the controlled systemwith convexified constraints on the control (the relaxed problem).By a solution of the controlled system we mean a triple: the output of the hysteresis operator, the trajectory,and the control. We establish a relation between the minimization problem and the relaxed problem. This relationis an analogue of Bogolyubov's classical theorem in the calculus of variations. We also study the relation betweenthe solutions of the original controlled system and those of the system with convexified constraints on the control.This relation is usually referred to as relaxation. For a finite-dimensional space we prove the existence of an optimalsolution in the relaxed optimization problem.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2020;84(6):165-196
165-196
On the uniform approximation of functions of bounded variation by Lagrange interpolationpolynomials with a matrix ${\mathcal L}_n^{(\alpha_n,\beta_n)}$ of Jacobi nodes
Abstract
Let sequences $\{\alpha_n\}_{n=1}^{\infty}$, $\{\beta_n\}_{n=1}^{\infty}$ satisfy therelations $\alpha_n\in\mathbb{R}$, $\beta_n\in\mathbb{R}$,$\alpha_n=o(\sqrt{n/\ln n})$, $\beta_n=o(\sqrt{n/\ln n})$ as $n\to \infty $, and let $[a,b]\subset (0,\pi)$ and$f\in C[a,b]$. We redefine the function $f$ as $F$ on the interval $[0,\pi]$ bypolygonal arcs in such a way that the function remains continuous and vanishes on a neighbourhood of the ends of the interval. Also let the function $f$ andthe pair of sequences $\{\alpha_n\}_{n=1}^{\infty}$, $\{\beta_n\}_{n=1}^{\infty}$ beconnected by the equiconvergence condition. Then for the classical Lagrange–Jacobiinterpolation processes $\mathcal{L}_n^{(\alpha_n,\beta_n)}(F,\cos\theta)$ toapproximate $f$uniformly with respect to $\theta $ on $[a,b]$ it is sufficient that $f$ have bounded variation $V^{b}_{a}(f)<\infty$ on $[a,b]$. Inparticular, if the sequences $\{\alpha_n\}_{n=1}^{\infty}$ and$\{\beta_n\}_{n=1}^{\infty}$ are bounded, then for the classical Lagrange–Jacobiinterpolation processes $\mathcal{L}_n^{(\alpha_n,\beta_n)}(F,\cos\theta)$ toapproximate $f$ uniformly with respect to $\theta $on $[a,b]$ it is sufficient that the variation of $f$ be bounded on$[a,b]$, $V^{b}_{a}(f)<\infty$.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2020;84(6):197-222
197-222
223-223
224-224