# Volume 53, № 6, 2001

### On Some Noncoercive Variational Inequalities

Gallo А., Piccirillo A. M., Toscano L.

Ukr. Mat. Zh. - 2001. - 53, № 6. - pp. 723-741

We study existence and regularity of solutions of noncoercive variational inequalities.

### Asymptotics of Solutions of the Sturm–Liouville Equation with Respect to a Parameter

Gomilko A. M., Pivovarchik V. N.

Ukr. Mat. Zh. - 2001. - 53, № 6. - pp. 742-757

On a finite segment [0, *l*], we consider the differential equation $$\left( {a\left( x \right)y\prime \left( x \right)} \right)\prime + \left[ {{\mu \rho }_{\text{1}} \left( x \right) + {\rho }_{2} \left( x \right)} \right]y\left( x \right) = 0$$ with a parameter μ ∈ *C*. In the case where *a*(*x*), ρ(*x*) ∈ *L* _{∞}[0, *l*], ρ_{ j }(*x*) ∈ *L* _{1}[0, *l*], *j* = 1, 2, *a*(*x*) ≥ *m* _{0} > 0 and ρ(*x*) ≥ *m* _{1} > 0 almost everywhere, and *a*(*x*)ρ(*x*) is a function absolutely continuous on the segment [0, *l*], we obtain exponential-type asymptotic formulas as \(\left| {\mu } \right| \to \infty\) for a fundamental system of solutions of this equation.

### Extremal Problems in the Theory of Capacities of Condensers in Locally Compact Spaces. III

Ukr. Mat. Zh. - 2001. - 53, № 6. - pp. 758-782

We complete the construction of the theory of interior capacities of condensers in locally compact spaces begun in the previous two parts of the work. A condenser is understood as an ordered finite collection of sets each of which is marked with the sign + or − so that the closures of sets with opposite signs are mutually disjoint. The theory developed here is rich in content for arbitrary (not necessarily compact or closed) condensers. We obtain sufficient and (or) necessary conditions for the solvability of the main minimum problem of the theory of capacities of condensers and show that, under fairly general assumptions, these conditions form a criterion. For the main minimum problem (generally speaking, unsolvable even for a closed condenser), we pose and solve dual problems that are always solvable (even in the case of a nonclosed condenser). For all extremal problems indicated, we describe the potentials of minimal measures and investigate properties of extremals. As an auxiliary result, we solve the well-known problem of the existence of a condenser measure. The theory developed here includes (as special cases) the main results of the theory of capacities of condensers in \(\mathbb{R}^n\) , *n* ≥ 2, with respect to the classical kernels.

### On Exact Estimates for the Pointwise Approximation of the Classes $W^rH^ω$ by Algebraic Polynomials

Ukr. Mat. Zh. - 2001. - 53, № 6. - pp. 783-799

We obtain estimates for the approximation of functions of the class *W* ^{r} *H* ^{ω}, where ω(*t*) is a convex modulus of continuity such that *t*ω′(*t*) does not decrease, by algebraic polynomials with regard for the position of a point on the segment [−1, 1]. The estimates obtained cannot be improved for all moduli of continuity simultaneously.

### On Integral Representations of an Axisymmetric Potential and the Stokes Flow Function in Domains of the Meridian Plane. II

Ukr. Mat. Zh. - 2001. - 53, № 6. - pp. 800-809

We obtain new integral representations for an axisymmetric potential and the Stokes flow function in an arbitrary simply-connected domain of the meridian plane. The boundary properties of these integral representations are studied for domains with closed rectifiable Jordan boundary.

### On Some Integral Transformations and Their Application to the Solution of Boundary-Value Problems in Mathematical Physics

Ukr. Mat. Zh. - 2001. - 53, № 6. - pp. 810-819

We obtain a formula for the expansion of an arbitrary function in a series in the eigenfunctions of the Sturm–Liouville boundary-value problem for the differential equation of cone functions. On the basis of this result, we derive a series of integral transformations (including well-known ones) and inversion formulas for them. We apply these formulas to the solution of initial boundary-value problems in the theory of heat conduction for circular hollow cones truncated by spherical surfaces.

### Linear Widths of the Besov Classes of Periodic Functions of Many Variables. II

Ukr. Mat. Zh. - 2001. - 53, № 6. - pp. 820-829

We obtain order estimates for linear widths of the Besov classes \(B_{p,{\theta }}^r\) of periodic functions of many variables in the space *L* _{q} for certain values of parameters *p* and *q* different from those considered in the first part of the work.

### Entire Dirichlet Series of Rapid Growth and New Estimates for the Measure of Exceptional Sets in Theorems of the Wiman–Valiron Type

Ukr. Mat. Zh. - 2001. - 53, № 6. - pp. 830-839

For entire Dirichlet series of the form \(F\left( z \right) = \sum\nolimits_{n = 0}^{ + \infty } {a_n e^{z{\lambda }_n } ,0 \leqslant {\lambda }_n \uparrow + \infty ,\;n \to + \infty }\) , we establish conditions under which the relation $$F\left( {{\sigma } + iy} \right) = \left( {1 + o\left( 1 \right)} \right)a_{{\nu }\left( {\sigma } \right)} e^{\left( {{\sigma + }iy} \right){\lambda }_{{\nu }\left( {\sigma } \right)} }$$ holds uniformly in \(y \in \mathbb{R}\;{as}\;{\sigma } \to + \infty\) outside a certain set *E* for which $$DE = \mathop {\lim \sup }\limits_{{\sigma } \to + \infty } h\left( {\sigma } \right)\;{meas}\;\left( {E \cap \left[ {{\sigma ,} + \infty } \right)} \right) = 0$$ where *h*(σ) is a positive continuous function increasing to +∞ on [0, +∞).

### On π-Solvable and Locally π-Solvable Groups with Factorization

Chernikov N. S., Putilov S. V.

Ukr. Mat. Zh. - 2001. - 53, № 6. - pp. 840-846

We prove that, in a locally π-solvable group *G* = *AB* with locally normal subgroups *A* and *B*, there exist pairwise-permutable Sylow π′- and *p*-subgroups *A* _{π′}, *A* _{ p } and *B* _{π′}, *B* _{ p }, *p* ∈ π, of the subgroups *A* and *B*, respectively, such that *A* _{π′} *B* _{π′} is a Sylow π′-subgroup of the group *G* and, for an arbitrary nonempty set σ \( \subseteq \) π, $$\left( {\prod\nolimits_{p \in {\sigma }} {A_p } } \right)\left( {\prod\nolimits_{p \in {\sigma }} {B_p } } \right)\quad {and}\quad \left( {A_{{\pi }\prime } \prod\nolimits_{p \in {\sigma }} {A_p } } \right)\left( {B_{{\pi }\prime } \prod\nolimits_{p \in {\sigma }} {B_p } } \right)$$ are Sylow σ- and π′ ∪ σ-subgroups, respectively, of the group *G*.

### On Optimization of “Interval” and “Pointwise” Quadrature Formulas for Classes of Monotone Functions

Ukr. Mat. Zh. - 2001. - 53, № 6. - pp. 847-956

We consider the problem of optimization of “interval” quadrature formulas (in different statements) on the class of monotone functions defined on an interval and the problem of optimization of cubature formulas with fixed knots on classes of functions defined on a *d*-dimensional cube, *d* = 2, 3,..., and monotonically nondecreasing with respect to each variable.

### On the Approximation of Functions of the Hölder Class by Triharmonic Poisson Integrals

Kharkevych Yu. I., Zhyhallo K. M.

Ukr. Mat. Zh. - 2001. - 53, № 6. - pp. 855-859

We determine the exact value of the upper bound for the deviation of the triharmonic Poisson integral from functions of the Hölder class.

### Unimprovable Integral Estimate for Solutions of the Dirichlet Problem for Quasilinear Elliptic Equations of the Second Order in a Neighborhood of an Edge

Ukr. Mat. Zh. - 2001. - 53, № 6. - pp. 860-865

We obtain an exact estimate for the second derivatives of solutions (in the weight Sobolev norm) of the Dirichlet problem for quasilinear second-order elliptic equations of nondivergence type in a neighborhood of an edge of a domain.