# Volume 69, № 12, 2017

### On the solvability of boundary-value problems with continuous and generalized gluing conditions for the equation of mixed type with loaded term

Ukr. Mat. Zh. - 2017. - 69, № 12. - pp. 1587-1595

We study the unique solvability of the boundary-value problems with normal derivatives and continuous and generalized gluing conditions for a loaded equation of the third order.

### Hölder continuity and the Harnack inequality for the solutions of an elliptic equation containing the $p$ -Laplacian and uniformly degenerating in a part of the domain

Ukr. Mat. Zh. - 2017. - 69, № 12. - pp. 1596-1604

We consider quasilinear equations of the $p$-Laplacian type uniformly degenerating in a part of the domain. An analog of the Harnack inequality is proved for nonnegative solutions and the Holder continuity of the solutions is established by using this inequality.

### Inverse problem for the heat equation in a rectangular domain

Ukr. Mat. Zh. - 2017. - 69, № 12. - pp. 1605-1614

We establish conditions for the existence and uniqueness of a smooth solution to the inverse problem for the two-dimensional heat equation with unknown leading coefficient depending on time and the space variable.

### Point interactions on the line and Riesz bases of δ -functions

Ukr. Mat. Zh. - 2017. - 69, № 12. - pp. 1615-1624

We present the description of a relationship between the Sobolev spaces $W^1_2 (R),\; W^2_2 (R)$ and the Hilbert space $\ell_2$. Let $Y$ be a finite or countable set of points on $R$ and let $d := \mathrm{inf} \bigl\{ | y\prime y\prime \prime | , y\prime , y\prime \prime \in Y, y\prime \not = y\prime \prime \bigr\}$. By using this relationship, we prove that if d = 0, then the systems of delta-functions $\bigl\{ \delta (x y_j), y_j \in Y \bigr\} $ and their derivatives $\bigl\{ \delta \prime (x y_j), y_j \in Y \bigr\} $ do not form Riesz bases in the closures of their linear spans in the Sobolev spaces $W^1_2 (R),\; W^2_2 (R)$ but, conversely, form these bases in the case where $d > 0$. We also present the description of the Friedrichs and Krein extensions and prove their transversality. Moreover, the construction of a basis boundary triple and the description of all nonnegative selfadjoint extensions of the operator $A\prime$ are proposed.

### On the exact constants in Hardy – Littlewood inequalities

Ukr. Mat. Zh. - 2017. - 69, № 12. - pp. 1625-1632

We obtain the exact constants for the Hardy – Littlewood inequalities.

### On the interference of the weight and boundary contour for algebraic polynomials in weighted Lebesgue spaces. II

Ukr. Mat. Zh. - 2017. - 69, № 12. - pp. 1633-1651

We continue to study the estimation of the modulus of algebraic polynomials on the boundary contour with weight function, when the contour and the weight function have certain singularities with respect to the their quasinorm in the weighted Lebesgue space. In particular, the exact estimates were obtained for polynomials orthonormal on the curve with respect to the weight function with zeros on the same curve.

### The classical M. A. Buhl problem, its Pfeiffer – Sato solutions and the classical Lagrange – D’Alembert principle for the integrable heavenly type nonlinear equations

Prykarpatsky Ya. A., Samoilenko A. M.

Ukr. Mat. Zh. - 2017. - 69, № 12. - pp. 1652-1689

The survey is devoted to old and recent investigations of the classical M. A. Buhl problem of description of the compatible linear vector field equations and their general M. G. Pfeiffer and modern Lax – Sato-type special solutions. In particular, we analyze the related Lie-algebraic structures and the properties of integrability for a very interesting class of nonlinear dynamical systems called the dispersion-free heavenly type equations, which were introduced by Pleba´nski and later analyzed in a series of articles. The AKS-algebraic and related \scrR -structure schemes are used to study the orbits of the corresponding coadjoint actions, which are intimately connected with the classical Lie – Poisson structures on them. It is shown that their compatibility condition coincides with the corresponding heavenly type equations under consideration. It is also demonstrated that all these equations are originated in this way and can be represented as a Lax compatibility condition for specially constructed loop vector fields on the torus. The infinite hierarchy of conservations laws related to the heavenly equations is described and its analytic structure connected with the Casimir invariants is indicated. In addition, we present typical examples of equations of this kind demonstrating in detail their integrability via the scheme proposed in the paper. The relationship between a very interesting Lagrange – d’Alembert-type mechanical interpretation of the devised integrability scheme and the Lax – Sato equations is also discussed.

### Problem with free boundary for the systems of equations of reaction-diffusion type

Rasulov M. S., Takhirov Zh. O.

Ukr. Mat. Zh. - 2017. - 69, № 12. - pp. 1690-1700

We consider a problem with free boundary for systems of quasilinear parabolic equations. A part of the boundary conditions are given in the nonlocal form. The a priori estimates of the H¨older norms are established. These estimates are used to prove the existence and uniqueness of the solution.

### A Tauberian theorem for the power-series summability method

Ukr. Mat. Zh. - 2017. - 69, № 12. - pp. 1701-1713

We introduce a one-sided Tauberian condition in terms of the weighted general control modulo oscillatory behavior of integer order $m$ with $m \geq 1$ for the power-series summability method.

### Mykola Ivanovych Portenko (on his 75th birthday)

Dorogovtsev A. A., Kopytko B.I., Osipchuk M. M.

Ukr. Mat. Zh. - 2017. - 69, № 12. - pp. 1714-1716

### Weighted limit solution of a nonlinear differential equation at a singular point and its property

Dzhumabaev D. S., Uteshova R. E.

Ukr. Mat. Zh. - 2017. - 69, № 12. - pp. 1717-1722

On a finite interval, we consider a system of nonlinear ordinary differential equations with a singularity at the left endpoint of the interval. The definition of weighted limit solution is introduced and its attracting property is established.

### Index of volume 69 of „Ukrainian Mathematical Journal”

Ukr. Mat. Zh. - 2017. - 69, № 12. - pp. 1723-1729