The problem of the product of inner radii of four nonoverlapping domains, some of there are symmetric about unit circle

Considered in the paper is one quite general problem of geometric function theory on extremal decomposition of the complex plane, namely to determine the maximum of product of the inner radii of $n$ non-overlapping domains $\{B_{k}\}_{k=1}^{n},$, symmetric with respect to the unit circle, and the power $\gamma$ of the inner radius of a domain $\{B_ {0}\}$, which contains the origin. Starting point of the theory of extremal problems on non-overlapping domains is the result of Lavrent’ev \cite{Lavr} who in 1934 solved the problem of a product of conformal radii of two mutually nonoverlapping simply connected domains. It was the first result of this direction. Goluzin \cite{Goluzin} generalized this problem in the case of an arbitrary finite number of mutually disjoint domains and obtained an accurate evaluation for the case of three domains. Further, Kuzmina \cite{Kuzm} showed that the problem of the evaluation for the case of four domains is reduced to the smallest capacity problems in a certain continuum family and received the exact inequality for $n=4$. For $n\geq5$ full solution of the problem is not obtained at this time. The problem, considered in this paper, stated in \cite{Dubinin-1994} by V.N. Dubinin and earlier in different form by G.P. Bakhtina \cite{Bakhtina-1984}. Let $a_{0}=0$, $|a_{1}|=\ldots=|a_{n}|=1$, $a_{k}\in B_{k}\in \overline{\mathbb{C}}$, where $B_{0},\ldots, B_{n}$ are disjoint domains, and $B_{1},\ldots, B_{n}$ are symmetric about the unit circle. Find the exact upper bound for $r^\gamma(B_0,0)\prod\limits_{k=1}^n r(B_k,a_k)$, where $r(B_k,a_k)$ is the inner radius of $B_k$ with respect to $a_k$. For $\gamma=1$ and $n\geq2$ this problem was solved by L.V. Kovalev \cite{kovalev-2000,kovalev2-2000} and for $\gamma_{n}=0,38n^{2}$ and $n\geq2$ under the additional assumption that the maximum $\alpha_{0}$ of the angles between neighbouring line segments $[0, a_{k}]$ do not exceed $2\pi/\sqrt{2\gamma}$ it was solved in \cite{BahDenV}. In the present paper this problem is solved for three non-overlapping symmetric domains and for $0<\gamma\leq1.233$ without additional restrictions, moreover, for the first time such $1<\gamma$ are considered for this case. Was proved the lemma, by which it was obtained the estimate of the inner radius of a domain $\{B_ {0}\}$, which contains the origin. Using this lemma and the result of paper \cite{BahDenV}, it was proved that for $\alpha_{0}>2\pi/\sqrt{2\gamma}$ consided product does not exceed some expression.

Extremal decomposition of the complex plane with free poles

We consider an open extremal problem in geometric function theory of complex variables on the maximum of the functional $$r^\gamma\left(B_0,0\right)\prod\limits_{k=1}^n r\left(B_k,a_k\right),$$ where \(B_{0}\), ..., \(B_{n}\), \(n\ge 2\), are pairwise disjoint domains in \(\overline{\mathbb{C}}\), \(a_0 = 0\), \(|a_{k}| = 1\), \(k=\overline{1,n}\), and \(\gamma\in (0, n]\) (\(r(B,a)\) is the inner radius of the domain \(B\subset\overline{\mathbb{C}}\) relative to a point \(a\in B\)). For all values of the parameter \(\gamma\in (0, n]\), it is necessary to show that its maximum is attained for a configuration of domains \(B_{k}\) and points \(a_{k}\), \(k=\overline{0,n}\), possessing the \(n\)-fold symmetry. The problem was solved by V.N. Dubinin [1, 2] for \(\gamma=1\) and by G.V. Kuz’mina [4] for \(0 \lt \gamma \lt 1\). L.V. Kovalev [4] obtained its solution for \(n \ge 5\) under the additional assumption that the angles between neighbouring line segments \([0, a_{k}]\) do not exceed \(2\pi /\sqrt{\gamma}\). In particular, this problem will be solved in the present paper for \(n=2\) and \(\gamma\in(1,\,2]\).

About one extremal problem for open sets and partially non-overlapping domains

Sharp estimates of a product of the inner radii for pairwise disjoint domains are obtained. In particular, we solve an extremal problem in the case of any finite number of free poles on the rays.

Problem on extremal decomposition of the complex plane

In geometric function theory of a complex variable problems on extremal decomposition with free poles on the unit circle are well known. One of such problem is the problem on maximum of the functional $${r^\gamma }({B_0},0)\prod\limits_{k = 1}^n r ({B_k},{a_k}),$$ where B0, B1, B2,..., Bn, n ≥ 2, are pairwise disjoint domains in ¯𝔺, a0 = 0, |ak| = 1, $k = \overline {1,n}$ and γ ∈ 2 (0; n], r(B, a) is the inner radius of the domain, B ⊂ ¯𝔺, with respect to a point a ∈ B. In the paper we consider a more general problem in which restrictions on the geometry of the location of points ak, $k = \overline {1,n}$ are weakened.

Some estimates for extremal decomposition of the complex plane

In geometric function theory of complex variable extremal problems on non-overlapping domains are well-known classic direction. A lot of such problems are reduced to determination of the maximum of product of inner radii on the system of non-overlapping domains satisfying a certain conditions. In this paper, we consider the well-known problem of maximum of the functional \(r^\gamma\left(B_0,0\right)\prod\limits_{k=1}^n r\left(B_k,a_k\right)\), where \(B_{0}\),..., \(B_{n}\) are pairwise disjoint domains in \(\overline{\mathbb{C}}\), \( a_0=0 \), \(|a_{k}|=1\), \(k=\overline{1,n}\) are different points of the circle, \(\gamma\in (0, n]\), and \(r(B,a)\) is the inner radius of the domain \(B\subset\overline{\mathbb{C}}\) relative to the point \( a \). This problem was posed as an open problem in the Dubinin paper in 1994. Till now, this problem has not been solved, though some partial solutions are available. In the paper an estimate for the inner radius of the domain that contains the point zero is found. The main result of the paper generalizes the analogous results of [1, 2] to the case of an arbitrary arrangement of systems of points on \(\overline{\mathbb{C}}\).

The problem of extreme decomposition of the plane

Considered here is one quite general problem about the description of extremal configurations maximizing the product of inner radii mutually non-overlapping domains the next following form: \begin{equation} J_{n}(\gamma)=\left[r\left(B_0,0\right)r\left(B_\infty,\infty\right)\right]^{\gamma}\prod\limits_{k=1}^n r\left(B_k,a_k\right),\end{equation} where $\gamma\in\mathbb{R^{+}}$, $A_{n}=\{a_{k}\}_{k=1}^{n}$ -- $n$-radial points system, $B_0$, $B_\infty$, $\{B_{k}\}_{k=1}^{n}$ -- set of systems of mutually disjoint domains, \textup{(}$a_{0}=0\in B_0\subset{\mathbb{C}}$, $\infty\in B_\infty\subset\overline{\mathbb{C}}$, $a_k\in B_k\subset{\mathbb{C}}$\textup{)}, achieved for some configuration of domains $B_{0}$, $B_{k}$, $B_{\infty}$ and points $a_{0}$, $a_{k}$, $\infty$, $k=\overline{1, n}$. The functional (1) evaluation for the first time was obtained in 1988 by V.M. Dubinin \cite{4} for {$\gamma=\frac{1}{2}$ и $n\geqslant2$} for systems of disjoint domains using symmetrization method. A special case, when domains are univalent, was examined by G.V. Kuzmina in \cite{KYZMINA-01} After the result of V.M. Dubinin in the general formulation for the arbitrary multiply-concted domains was not until 2017. In 2017 in the work of I. Dengi, A. Targonsky \cite{iratarg} was the result for $\gamma_{n}=0.08n^{2}$, $n\geq7$. The result was obtained through a lower bound of $min_t (n-1)x_{1}+x_{2},$ where $x_{1}(t)+x_{2}(t)$ is the equation $F^\prime (x)=t$ solution, $F^\prime (x)=4x\ln(x)-2(x-1)\ln|x-1|-2(x+1)\ln(x+1)+\frac{2}{x}, $ $y_0 \leqslant t<0$, $y_{0}\approx-1,0599$. In this paper, a much better estimate of $min_t (n-1)x_{1}+x_{2}$ was obtained through a lower bound with the specified parameters. On the basis of this, the article succeeded in obtaining an estimate of the maximum of the functional (1) over a larger interval of the parameter $\gamma$, $\gamma\in(0,\gamma_{n}]$, $n\geq7$. Received the result for for any points of the circle $|a_k|=R$, $k=\overline{1,n}$, and any pairwise disjoint systems of domains $B_k$, $a_{0}=0\in B_0\subset{\mathbb{C}}$, $\infty\in B_\infty\subset\overline{\mathbb{C}}$, $a_k\in B_k\subset{\mathbb{C}}$, $k=\overline{1,n}.$

On Conformal Radii of Non-Overlapping Simply Connected Domains

The paper deals with the following open problem stated by V.N. Dubinin. Let $a_{0}=0$, $|a_{1}|=\ldots=|a_{n}|=1$, $a_{k}\in B_{k}\subset \overline{\mathbb{C}}$, where $B_{0},\ldots, B_{n}$ are disjoint domains. For all values of the parameter $\gamma\in (0, n]$ find the exact upper bound for $r^\gamma(B_0,0)\prod\limits_{k=1}^n r(B_k,a_k)$, where $r(B_k,a_k)$ is the conformal radius of $B_k$ with respect to $a_k$. For $\gamma=1$ and $n\geqslant2$ the problem was solved by V.N. Dubinin. In the paper the problem is solved for $\gamma\in (0, \sqrt{n}\,]$ and $n\geqslant2$ for simply connected domains.The paper deals with the following open problem stated by V.N. Dubinin. Let a0 = 0, ιa1ι =...= ιanι = 1, ak ∈ Bk ⊂ , where B0, ..., Bn are disjoint domains. For all values of the parameter γ∈ (0; n] find the exact upper bound nfor rγ(B0; 0) ∏ r(Bk; ak), where r(Bk; ak) is the conformal radius of Bk with respect to ak. For γ = 1 k=1 and n ≥ 2 the problem was solved by V.N. Dubinin. In the paper the problem is solved for γ ∈ (0; √n ] and n ≥ 2 for simply connected domains.

Parabolic-hyperbolic transmission problem in disjoint domains

In applications, especially in engineering, often are encountered composite or layered structures, where the properties of individual layers can vary considerably from the properties of the surrounding material. Layers can be structural, thermal, electromagnetic or optical, etc. Mathematical models of energy and mass transfer in domains with layers lead to so called transmission problems. In this paper we investigate a mixed parabolic-hyperbolic initial-boundary value problem in two nonadjacent rectangles with nonlocal integral conjugation conditions. It was considered more examples of physical and engineering tasks which are reduced to transmission problems of similar type. For the model problem the existence and uniqueness of its weak solution in appropriate Sobolev-like space is proved. A finite difference scheme approximating this problem is proposed and analyzed.