Vol 211, No 11 (2020)

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.

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.

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.

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.