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Vol 213, No 10 (2022)
- Year: 2022
- Articles: 7
- URL: https://journals.rcsi.science/0368-8666/issue/view/7494
Asymptotics for problems in perforated domains with Robin nonlinear condition on the boundaries of cavities
Abstract
A boundary-value problem for a second-order elliptic equation with variable coefficients is considered in a multidimensional domain with periodic perforation by small cavities arranged along a fixed hypersurface at small distances one from another. The distances are proportional to a small parameter $\varepsilon$, and the linear sizes of the cavities are proportional to $\varepsilon\eta(\varepsilon)$, where $\eta(\varepsilon)$ is a function taking values in the interval $[0,1]$. The main result is a complete asymptotic expansion for the solution of the perturbed problem. The asymptotic expansion is a combination of an outer and an inner expansion; it is constructed using the method of matched asymptotic expansions. Both outer and inner expansions are power expansions in $\varepsilon$ with coefficients depending on $\eta$. These coefficients are shown to be infinitely differentiable with respect to $\eta\in(0,1]$ and uniformly bounded in $\eta\in[0,1]$.Bibliography: 38 titles.
Matematicheskii Sbornik. 2022;213(10):3-59
3-59
Derivative of the Minkowski function: optimal estimates
Abstract
It is well known that the derivative of the Minkowski function $?(x)$, if it exists, can take only two values, $0$ and $+\infty$. It is also known that the value of $?'(x)$ at a point $x=[0;a_1,a_2,…,a_t,…]$ is related to the limiting behaviour of the arithmetic mean $(a_1+a_2+…+a_t)/t$. In particular, as shown by Moshchevitin and Dushistova, if $a_1+a_2+…+a_t>(\kappa_2+\varepsilon)t$, where $\varepsilon>0$ and $\kappa_2\approx 4.4010487$ is some explicitly given constant, then $?'(x)=0$. They also showed that $\kappa_2$ cannot be replaced by a smaller constant. We consider the dual problem: how small can the quantity $\kappa_2t-a_1+a_2+…+a_t$ be if it is known that $?'(x)=0$? We obtain optimal estimates in this problem.Bibliography: 9 titles.
Matematicheskii Sbornik. 2022;213(10):60-89
60-89
Isometric embeddings of bounded metric spaces in the Gromov-Hausdorff class
Abstract
We show that any bounded metric space can be embedded isometrically in the Gromov-Hausdorff metric class $\operatorname{\mathcal{GH}}$. This is a consequence of the description of the local geometry of $\operatorname{\mathcal{GH}}$ in a sufficiently small neighbourhood of a generic metric space, which is of independent interest. We use the techniques of optimal correspondences and their distortions.Bibliography: 22 titles.
Matematicheskii Sbornik. 2022;213(10):90-107
90-107
Some applications of growth in $\mathrm{SL}_2(\pmb{\mathbb{F}}_p)$ to the proof of modular versions of Zaremba's conjecture
Abstract
Using growth in $\mathrm{SL}_2(\mathbb{F}_p)$ we prove that for every prime number $p$ and any positive integer $u$ there are positive integers $q=O(p^{2+\varepsilon})$, $\varepsilon > 0$, $q \equiv u \pmod{p}$, and $a < p$, $(a, p)=1$, such that the partial quotients of the continued fraction of $a/q$ are bounded by an absolute constant.Bibliography: 21 titles.
Matematicheskii Sbornik. 2022;213(10):108-129
108-129
130-138
Uniformly and locally convex asymmetric spaces
Abstract
The nonemptyness of the intersections of nested systems of convex bounded closed subsets of uniformly convex asymmetric spaces is studied. The density properties of the points of existence and points of approximative uniqueness are examined for nonempty closed subsets of uniformly convex asymmetric spaces. Problems of the existence and stability of Chebyshev centres are considered; therelationships between $\gamma$-suns, suns and the existence of best approximants are investigated. Sufficient conditions for radial $\delta$-solarity are obtained.Bibliography: 27 titles.
Matematicheskii Sbornik. 2022;213(10):139-166
139-166
The convex hull and the Caratheodory number of a set in terms of the metric projection operator
Abstract
We prove that each point of the convex hull of a compact set $M$ in a smooth Banach space $X$ can be approximated arbitrarily well by convex combinations of best approximants from $M$ to $x$ (values of the metric projection operator $P_M(x)$), where $x \in X$. As a corollary, we show that the Caratheodory number of a compact set $M \subset X$ with at most $k$-valued metric projection $P_M$ is majorized by $k$, that is, each point in the convex hull of $M$ lies in the convex hull of at most $k$ points of $M$.Bibliography: 26 titles.
Matematicheskii Sbornik. 2022;213(10):167-184
167-184