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Vol 211, No 11 (2020)
- Year: 2020
- Articles: 7
- URL: https://journal-vniispk.ru/0368-8666/issue/view/7471
Topological classification of integrable geodesic billiards on quadrics in three-dimensional Euclidean space
Abstract
We consider geodesic billiards on quadrics in $\mathbb{R}^3$. We consider the motion of a point mass inside a billiard table, that is, inside a domain lying on a quadric bounded by finitely many quadrics confocal with the given one and having angles at corner points of the boundary equal to ${\pi}/{2}$. According to the well-known Jacobi-Chasles theorem this problem turns out to be integrable. We introduce an equivalence relation on the set of billiard tables and prove a theorem on their classification. We present a complete classification of geodesic billiards on quadrics in $\mathbb{R}^3$ up to Liouville equivalence. Bibliography: 19 titles.
Matematicheskii Sbornik. 2020;211(11):3-40
3-40
On the integral characteristic function of the Sturm-Liouville problem
Abstract
We introduce a function whose zeros, and only these zeros, are eigenvalues of the corresponding Sturm-Liouville problem. The boundary conditions of the problem depend continuously on the spectral parameter. Therefore, it makes sense to call the function thus constructed a characteristic function of the Sturm-Liouville problem (however, it is not a characteristic function in the ordinary sense). An investigation of the function thus obtained enables us to prove the solvability of the problem in question, to find the asymptotic behaviour of the eigenvalues, to obtain comparison theorems, and to introduce an indexing of the eigenvalues and the zeros of eigenfunctions in a natural way. Bibliography: 31 titles.
Matematicheskii Sbornik. 2020;211(11):41-53
41-53
Extensions of the space of continuous functions and embedding theorems
Abstract
The machinery of $s$-dimensionally continuous functions is developed for the purpose of applying it to the Dirichlet problem for elliptic equations. With this extension of the space of continuous functions, new generalized definitions of classical and generalized solutions of the Dirichlet problem are given. Relations of these spaces of $s$-dimensionally continuous functions to other known function spaces are studied. This has led to a new construction (seemingly more successful and closer to the classical one) of $s$-dimensionally continuous functions, using which new properties of such spaces have been identified. The embeddings of the space $C_{s,p}(\overline Q)$ in $C_{s',p'}(\overline Q)$ for $s'>s$ and $p'>p$, and, in particular, in $ L_q(Q)$ are proved. Previously, $W^1_2(Q)$ was shown to embed in $C_{n-1,2}(\overline Q)$, which secures the $(n-1)$-dimensional continuity of generalized solutions. In the present paper, the more general embedding of $W^1_r(Q)$ in $C_{s,p}(\overline Q)$ is verified and the corresponding exponents are shown to be sharp. Bibliography: 33 titles.
Matematicheskii Sbornik. 2020;211(11):54-71
54-71
Limits, standard complexes and $\mathbf{fr}$-codes
Abstract
For a strongly connected category $\mathscr{C}$ with pairwise coproducts, we introduce a cosimplicial object, which serves as a sort of resolution for computing higher derived functors of $\lim \colon \mathrm{Ab}^{\mathscr{C}}\to \mathrm{Ab}$. Applications involve the Künneth theorem for higher limits and $\lim$-finiteness of $\mathbf{fr}$-codes. A dictionary for the $\mathbf{fr}$-codes with words of length $\leq 3$ is given. Bibliography: 19 titles.
Matematicheskii Sbornik. 2020;211(11):72-95
72-95
Asymptotically sharp two-sided estimate for domains of univalence of holomorphic self-maps of a disc with an invariant diameter
Abstract
Holomorphic self-maps of the unit disc with two fixed diametrically opposite boundary points and an invariant diameter are investigated. Asymptotically sharp estimates for domains of univalence are obtained for functions in such classes, which depend on the product of the angular derivatives at the boundary fixed points. Bibliography: 16 titles.
Matematicheskii Sbornik. 2020;211(11):96-117
96-117
Vanishing properties of $f$-minimal hypersurfaces in a complete smooth metric measure space
Abstract
Let $(N^{n+1},g,e^{-f}dv)$ be a complete smooth metric measure space with $M^{n}$ being a complete noncompact $f$-minimal hypersurface in $N^{n+1}$. In this paper, we extend the classical vanishing theorems for $L^2$-harmonic $1$-forms on a complete minimal hypersurface to a weighted manifold. In addition, we obtain a vanishing result under the assumption that $M^n$ has sufficiently small weighted $L^n$-norm of the second fundamental form on $M^{n}$, which can be regarded as a generalization of a result by Yun and Seo. Bibliography: 26 titles.
Matematicheskii Sbornik. 2020;211(11):118-128
118-128
Asymptotic analysis of solutions of ordinary differential equations with distribution coefficients
Abstract
Ordinary differential equations of the form $$\tau(y)- \lambda ^{2m} \varrho(x) y=0, \qquad \tau(y) =\sum_{k,s=0}^m(\tau_{k,s}(x)y^{(m-k)}(x))^{(m-s)},$$on the finite interval $x\in[0,1]$ are under consideration. Here the functions $\tau_{0,0}$ and $\varrho$ are absolutely continuous and positive and the coefficients of the differential expression $\tau(y)$ are subject to the conditions $$\tau_{k,s}^{(-l)}\in L_2[0,1], \qquad 0\le k,s \le m, \quad l=\min\{k,s\},$$where $f^{(-k)}$ denotes the $k$th antiderivative of the function $f$ in the sense of distributions. Our purpose is to derive analogues of the classical asymptotic Birkhoff-type representations for the fundamental system of solutions of the above equation with respect to the spectral parameter as $\lambda \to \infty$ in certain sectors of the complex plane $\mathbb C$. We reduce this equation to a system of first-order equations of the form $$\mathbf y'=\lambda\rho(x)\mathrm B\mathbf y+\mathrm A(x)\mathbf y+\mathrm C(x,\lambda)\mathbf y,$$where $\rho$ is a positive function, $\mathrm B$ is a matrix with constant elements, the elements of the matrices $\mathrm A(x)$ and $\mathrm C(x,\lambda)$ are integrable functions, and $\|\mathrm C(x,\lambda)\|_{L_1}=o(1)$ as $\lambda \to \infty$. For systems of this kind, we obtain new results concerning the asymptotic representation of the fundamental solution matrix, which we use to make an asymptotic analysis of the above scalar equations of high order. Bibliography: 44 titles.
Matematicheskii Sbornik. 2020;211(11):129-166
129-166