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Vol 229, No 1 (2018)
- Year: 2018
- Articles: 8
- URL: https://journals.rcsi.science/1072-3374/issue/view/14886
Article
1-6
On quasiconformal maps and semilinear equations in the plane
Abstract
Assume that Ω is a domain in the complex plane ℂ and A(z) is a symmetric 2×2 matrix function with measurable entries, detA = 1; and such that 1/K|ξ|2 ≤ 〈A(z)ξ, ξ〉 ≤ K|ξ|2, ξ ∈ ℝ2, 1 ≤ K < ∞ . In particular, for semilinear elliptic equations of the form div (A(z)∇u(z)) = f(u(z)) in Ω; we prove a factorization theorem that asserts that every weak solution u to the above equation can be expressed as the composition u = To????; where ???? : Ω → G stands for a K−quasiconformal homeomorphism generated by the matrix function A(z); and T(w) is a weak solution of the semilinear equation ∇T(w) = J(w)f(T(w)) in G: Here, the weight J(w) is the Jacobian of the inverse mapping ????−1: Similar results hold for the corresponding nonlinear parabolic and hyperbolic equations. Some applications of these results to anisotropic media are given.
7-29
Filtration of stationary Gaussian statistical experiments
Abstract
The filtration of stationary Gaussian statistical experiments is determined by a solution of the equation of optimum filtration, which is characterized by the two-dimensional matrix of covariances. The parameters of a filtered signal are set by empiric covariances.
30-35
36-50
Pseudospectral functions of various dimensions for symmetric systems with the maximal deficiency index
Abstract
We consider the first-order symmetric system Jy′ − A(t)y = λΔ(t)y with n × n-matrix coefficients defined on an interval [a; b) with the regular endpoint a. It is assumed that the deficiency indices N± of the system satisfy the equality N_ ≤ N+ = n. The main result is the parametrization of all pseudospectral functions σ(·) of any possible dimension n????≤ n in terms of a Nevanlinna parameter τ = {C0(λ), C1(λ)}. Such parametrization is given by the linear-fractional transform
and the Stieltjes inversion formula for m???? (λ). We show that the matrix \( W\left(\uplambda \right)={\left({w}_{ij}\left(\uplambda \right)\right)}_{i,j=1}^2 \) has the properties similar to those of the resolvent matrix in the extension theory of symmetric operators. The obtained results develop the results by A. Sakhnovich; Arov and Dym; and Langer and Textorius.
51-84
Kolmogorov inequalities for the norms of the Riesz derivatives of functions of many variables
Abstract
New sharp Kolmogorov-type inequalities for the norms of the Riesz derivatives ∥Dαf∥∞ of functions \( f\in {L}_{\infty, E}^{\nabla}\left({\mathrm{\mathbb{R}}}^m\right) \) are obtained. Some applications of obtained inequalities are investigated.
85-95
Convolution equations and mean-value theorems for solutions of linear elliptic equations with constant coefficients in the complex plane
Abstract
In terms of the Bessel functions, we characterize smooth solutions of some convolution equations in the complex plane and prove a two-radius theorem for solutions of homogeneous linear elliptic equations with constant coefficients whose left-hand sides are representable in the form of a product of some non-negative integer powers of the complex differentiation operators ∂ and \( \overline{\partial} \).
96-107
On the problem of V. N. Dubinin for symmetric multiply connected domains
Abstract
Abstract
The problem of maximum of the functional
is considered. Here, \( \upgamma \in \left(0,n\right],{a}_0=0,\kern0.5em \left|{a}_k\right|=1,k=\overline{1,n},{a}_k\in {B}_k\subset \overline{\mathrm{\mathbb{C}}},\kern0.5em k=\overline{0,n},\kern0.5em {\left\{{B}_k\right\}}_{k=1}^n \) are pairwise non-overlapping domains, \( {\left\{{B}_k\right\}}_{k=0}^n \) are symmetric domains with respect to the unit circle, and r(B; a) is the inner radius of the domain \( B\subset \overline{\mathrm{\mathbb{C}}} \) with respect to the point a ∈ B. For γ = 1 and n ≥ 2, the problem was formulated as an open problem by V. N. Dubinin in 1994. L. V. Kovalev solved the Dubinin problem in 2000. The article deals with finding the maximum of the functional In(γ) for γ > 1.
108-113