Том 229, № 1 (2018)

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.

An Anscombe-type theorem for the large deviations principle for trajectories of a random process is proved. As a consequence, the moderate deviations principle for the compound renewal processes is obtained.

New sharp Kolmogorov-type inequalities for the norms of the Riesz derivatives ∥D^αf∥_∞ of functions \( f\in {L}_{\infty, E}^{\nabla}\left({\mathrm{\mathbb{R}}}^m\right) \) are obtained. Some applications of obtained inequalities are investigated.

Abstract

The problem of maximum of the functional

\( {I}_n\left(\upgamma \right)={r}^{\upgamma}\left({B}_0,0\right)\prod \limits_{k=1}^nr\left({B}_k,{a}_k\right) \)

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 I_n(γ) for γ > 1.