Vol 234, No 1 (2018)

We consider one of the classical problems of the geometric theory of functions of a complex variable on a maximum of the functional

\( {\left[r\left({B}_0.0\right)r\left({B}_{\infty },\infty \right)\right]}^{\upgamma}\prod \limits_{k=1}^nr\left({B}_k,{a}_k\right), \)

where n ∈ ℕ, n ≥ 2, γ ∈ ℝ⁺, \( {A}_n={\left\{{a}_k\right\}}_{k=1}^n \) is a system of points such that |a_k| = 1, a₀ = 0, B₀, B_∞, \( {\left\{{B}_k\right\}}_{k=1}^n \) is a system of pairwise nonoverlapping domains, \( {a}_k\in {B}_k\subset \overline{\mathrm{\mathbb{C}}} \), \( k=\overline{0,n} \), \( \infty \in {B}_{\infty}\subset \overline{\mathrm{\mathbb{C}}} \), r(B, a) is the inner radius of the domain \( B\subset \overline{\mathrm{\mathbb{C}}} \) with respect to the point a ∈ B. We have analyzed this problem under some weaker restrictions on pairwise nonoverlapping domains.

We have obtained the pointwise Bernstein–Walsh type estimation for algebraic polynomials in the unbounded regions with piecewise asymptotically conformal boundary, having exterior and interior zero angles, in the weighted Lebesgue space.

We study various Stieltjes integrals as Poisson–Stieltjes, conjugate Poisson–Stieltjes, Schwartz–Stieltjes, and Cauchy–Stieltjes ones and prove theorems on the existence of their finite angular limits a.e. in terms of the Hilbert–Stieltjes integral. These results hold for arbitrary bounded integrands that are differentiable a.e. and, in particular, for integrands of the class \( \mathcal{C}\mathrm{\mathcal{B}}\mathcal{V} \) (countably bounded variation).

The class of γ-generating matrices and its subclasses of regular and singular γ-generating matrices were introduced by D. Z. Arov in [8], where it was shown that every γ-generating matrix admits an essentially unique regular–singular factorization. The class of generalized γ-generating matrices was introduced in [14, 20]. In the present paper, subclasses of singular and regular generalized –generating matrices are introduced and studied. As the main result, a theorem of existence of the regular–singular factorization for a rational generalized γ-generating matrix is proved.

The paper deals with energy (weak) solutions u (t; x) of the class of equations with the model representative

\( \left(\left|u\right|{p}^{-1}u\right)t-\Delta p(u)=0,\kern0.5em \left(t,x\right)\in \left(0,T\right)\times \varOmega, \varOmega \in {\mathrm{\mathbb{R}}}^n,n\ge 1,p>0, \)

and with the following blow-up condition for the energy:

\( \varepsilon (t):= {\int}_{\Omega}{\left|u\left(t,x\right)\right|}^{p+1} dx+{\int}_0^t{\int}_{\Omega}{\left|{\nabla}_xu\left(\tau, x\right)\right|}^{p+1} dx d\tau \to \infty \mathrm{as}\;t\to T, \)

where Ω is a smooth bounded domain. In the case of flat peaking, namely, under the condition

\( {\displaystyle \begin{array}{cc}\varepsilon (t)\le F\upalpha (t){\upomega}_0{\left(T-t\right)}^{-\upalpha}& \forall t0,\upalpha >\frac{1}{p+1}, \)

a sharp estimate of the profile of a solution has been obtained in a neighborhood of the blow-up time t = T.