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卷 83, 编号 2 (2019)
Articles
Congratulations to Armen Glebovich Sergeev
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2019;83(2):3-4
3-4
IInvestigation of the sets of real solutions of non-linear equations
摘要
We study the problem of constructing a smooth curve lying on the level setof a smooth map and issuing from an abnormal point. We find sufficientconditions for the existence of such curves and obtain bounds for thedistances to level sets of the smooth map near an abnormal point.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2019;83(2):5-20
5-20
On the existence of a linear differential system with given values of the Perron exponent
摘要
We consider linear differential systems with infinitely differentiable and, in general, unbounded coefficients on the semi-axis. We prove that the map assigning to a non-zero solution of such a system the value of the Perron exponent on thesolution can turn out to be an arbitrary continuousfunction which is constant along every line passing through zero in the solution space.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2019;83(2):21-39
21-39
Representing systems of exponentials in projective limits of weighted subspaces of $H(D)$
摘要
We consider uniformly weighted spaces of analytic functions on a boundedconvex domain in the complex plane with convex weights. For every uniformlyweighted normed space $H(D,\varphi)$ we define a special inductive limit$\mathcal H_i(D,\varphi)$ of normed spaces and a special projective limit$\mathcal H_p(D,\varphi)$ of normed spaces. We prove that$\mathcal H_i(D,\varphi)$ is the smallest locally convex space whichcontains $H(D,\varphi)$ and is invariant under differentiation, and$\mathcal H_p(D,\varphi)$ is the largest such space which is contained in $H(D,\varphi)$. We construct a representing system of exponentials in the projective limit$\mathcal H_p(D, \varphi)$ and estimate the redundancy of this system.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2019;83(2):40-60
40-60
Bounds for a class of quasilinear integral operators on the set of non-negativeand non-negative monotone functions
摘要
We consider weighted bounds for quasilinear integral operators of the form
from $L_{p,v}$ to $L_{q,u}$ on the set on non-negative and non-negative monotone functions $f$, where $u$, $v$ and $w$ are weight functions. Under the assumption that $0< r< \infty$, we obtain necessary and sufficient conditions for the validity of these bounds on the set of non-negative functions for the values of the parameters satisfying the conditions $1\leq p\leq q< \infty$ and $0< q< p< \infty$, $p\geq 1$, and also on the cones of non-negative non-increasing and non-negative non-decreasing functions for $0< q< \infty$ and $1\leq p< \infty$. Here it is assumed only that $K{( \cdot ,\cdot )}\geq 0$. However, the criteria we obtain involve the norm of a linear integral operator from $L_{p,v}$ to $L_{r,w}$ with kernel $K{( \cdot ,\cdot )}$.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2019;83(2):61-82
61-82
An exponential estimate for the cubic partial sums of multiple Fourier series
摘要
We prove an exponential integral estimate for the cubic partial sumsof multiple Fourier series on sets of large measure. This estimate yieldssome new properties of Fourier series.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2019;83(2):83-96
83-96
On the asymptotics of solutions of elliptic equations at the endsof non-compact Riemannian manifolds with metrics of a special form
摘要
We consider a linear elliptic differential equation $\Delta u+c(x)u=0$ defined on a Riemannian manifold $\mathcal{M}$ that hasan end $\mathcal{X}$ on which the metric takes the form$dl^2=h^2(r) dr^2+q^2(r) d\theta^2$ in appropriate coordinates.Here $r\in [r_0,+\infty)$, $\theta\in S$, and $S$ is a smooth compact Riemannianmanifold with metric $d\theta^2$. At the end $\mathcal{X}$, the coefficient$c(x)$ takes the form $c(x)=c(r)$. For ends of parabolic type with suchmetrics, we describe the property of asymptotic distinguishabilityof solutions of this equation. For ends of hyperbolic type, we prove a theoremon the admissible rate of convergence to zero for a difference of solutionsof this equation. For both types of ends, we formulate versions of thegeneralized Cauchy problem with initial data $(\varphi(\theta),\psi(\theta))$at the infinitely remote point and study its solubility. The results obtainedare new and, in the case of ends of parabolic type, somewhat unexpected.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2019;83(2):97-125
97-125
$\mathbb R$-factorizability of $G$-spaces in the category G-Tych
摘要
We introduce and characterize the notion of $\mathbb R$-factorizability of $G$-spaces in the category G-Tych. For $G$-spaces with $d$-openly acting groups, we establish the equivalence of $\mathbb R$-factorizability and$\mathbb R$-factorizability in G-Tych. We prove the$\mathbb R$-factorizability in G-Tych of every$\mathbb R$-factorizable $G$-space with transitive action whose phase spacepossesses the Baire property. The Dieudonne completion of an$\mathbb R$-factorizable group is shown to be the phase spaceof a $G$-space $\mathbb R$-factorizable in G-Tych. We characterize$\mathbb R$-factorizability in G-Tych under passageto the $G$-compactification.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2019;83(2):126-141
126-141
On homogenized equations of filtration in two domains with common boundary
摘要
We consider an initial-boundary value problem describing the processof filtration of a weakly viscous fluid in two distinct porous mediawith common boundary. We prove, at the microscopic level, the existenceand uniqueness of a generalized solution of the problem on the joint motionof two incompressible elastic porous (poroelastic) bodies with distinctLame constants and different microstructures, and of a viscousincompressible porous fluid. Under various assumptions on the dataof the problem, we derive homogenized models of filtration of an incompressibleweakly viscous fluid in two distinct elastic or absolutely rigid porous mediawith common boundary.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2019;83(2):142-173
142-173
Asymptotics of solutions of a modified Whitham equation with surface tension
摘要
We study the large-time behaviour of solutions of the Cauchy problemfor a modified Whitham equation,$$\begin{cases}u_{t}+i\mathbf{\Lambda}u-\partial_{x}u^3=0, &(t,x) \in\mathbb{R}^2,u(0,x)=u_0(x), &x\in \mathbb{R},\end{cases}$$where the pseudodifferential operator $\mathbf{\Lambda}\equiv \Lambda(-i\partial_{x})=\mathcal{F}^{-1}[\Lambda (\xi) \mathcal{F}]$ is givenby the symbol$$\Lambda (\xi)=a^{-{1}/{2}}\xi(\sqrt{(1+a^2\xi^2) \frac{\operatorname{tanh}a\xi}{a\xi} }-1)$$with a parameter $a>0$. This symbol corresponds to the total dispersionrelation for water waves taking surface tension into account. Assuming that the total mass of the initial data is equal to zero($\int_{\mathbb{R}}u_0(x) dx=0$) and the initial data $u_0$ are small in the norm of $\mathbf{H}^{\nu}(\mathbb{R}) \cap\mathbf{H}^{0,1}(\mathbb{R})$, $\nu \geq 22$,we prove the existence of a global-in-time solution and describe itslarge-time asymptotic behaviour.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2019;83(2):174-203
174-203
Sobolev-orthogonal systems of functions and the Cauchy problem for ODEs
摘要
We consider systems of functions ${\varphi}_{r,n}(x)$ ($r=1,2,…$,$n=0,1,…$) that are Sobolev-orthonormal with respect to a scalarproduct of the form $\langle f,g\rangle=\sum_{\nu=0}^{r-1}f^{(\nu)}(a)g^{(\nu)}(a)+\int_{a}^{b}f^{(r)}(x)g^{(r)}(x)\rho(x) dx$ and are generated by a given orthonormal system of functions$\varphi_{n}(x)$ ($n=0,1,…$). The Fourier series and sums with respectto the system $\varphi_{r,n}(x)$ ($r=1,2,…$, $n=0,1,…$) are shownto be a convenient and efficient tool for the approximate solution of theCauchy problem for ordinary differential equations (ODEs).
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2019;83(2):204-226
204-226