Том 100, № 5-6 (2016)

Sufficient conditions for the existence and uniqueness of a positive radially symmetric solution of the Dirichlet problem for a nonlinear elliptic second-order system with p-Laplacian are obtained. In addition, it also proved that these conditions guarantee the nonexistence of a global positive radially symmetric solution.

For the linear positive Korovkin operator \(f\left( x \right) \to {t_n}\left( {f;x} \right) = \frac{1}{\pi }\int_{ - \pi }^\pi {f\left( {x + t} \right)E\left( t \right)dt} \), where E(x) is the Egervary–Szász polynomial and the corresponding interpolation mean \({t_{n,N}}\left( {f;x} \right) = \frac{1}{N}\sum\limits_{k = - N}^{N - 1} {{E_n}\left( {x - \frac{{\pi k}}{N}} \right)f\left( {\frac{{\pi k}}{N}} \right)} \) , the Jackson-type inequalities \(\left\| {{t_{n,N}}\left( {f;x} \right) - f\left( x \right)} \right\| \leqslant \left( {1 + \pi } \right){\omega _f}\left( {\frac{1}{n}} \right),\left\| {{t_{n,N}}\left( {f;x} \right) - f\left( x \right)} \right\| \leqslant 2{\omega _f}\left( {\frac{\pi }{{n + 1}}} \right)\), where ωf (x) denotes the modulus of continuity, are proved for N > n/2. For ωf (x) ≤ Mx, the inequality \(\left\| {{t_{n,N}}\left( {f;x} \right) - f\left( x \right)} \right\| \leqslant \frac{{\pi M}}{{n + 1}}\). is established. As a consequence, an elementary derivation of an asymptotically sharp estimate of the Kolmogorov width of a compact set of functions satisfying the Lipschitz condition is obtained.

The Neumann problem for a second-order parabolic equation with integro-differential operator in the boundary condition is considered. A well-posedness theorem is proved, in particular, the integral representation of the solution is obtained, estimates for the derivatives of the solution are established, and the kernel of the inverse operator of the problem is explicitly expressed.

Erdős’ well-known problem on the maximum absolute value of the derivative of a polynomial on a connected lemniscate is extended to the case of a rational function. Moreover, under the assumption that certain lemniscates are connected, a sharp upper bound for the absolute value of the derivative of a rational function at any point in the plane different from the poles is found. The role of the extremal function is played by an appropriate Zolotarev fraction.

We consider resonances for a h-pseudo-differential operator H(x, hD_x; h) induced by a periodic orbit of hyperbolic type. We generalize the framework of Gérard and Sjöstrand, in the sense that we allow hyperbolic and elliptic eigenvalues of the Poincarémap, and look for so-called semi-excited resonances with imaginary part of magnitude −h log h, or h^δ, with 0 < δ < 1.

We consider a J-self-adjoint 2 × 2 block operator matrix L in the Feshbach spectral case, that is, in the case where the spectrum of one main-diagonal entry of L is embedded into the absolutely continuous spectrum of the other main-diagonal entry. We work with the analytic continuation of the Schur complement of amain-diagonal entry in L−z to the unphysical sheets of the spectral parameter z plane. We present conditions under which the continued Schur complement has operator roots in the sense of Markus–Matsaev. The operator roots reproduce (parts of) the spectrum of the Schur complement, including the resonances. We, then discuss the case where there are no resonances and the associated Riccati equations have bounded solutions allowing the graph representations for the corresponding J-orthogonal invariant subspaces of L. The presentation ends with an explicitly solvable example.

It is shown that a well-known expression for the capacity of the preimage of a compact set under a polynomial map remains valid in the case of a rational map, provided that the standard capacity of the preimage is replaced by its capacity in the external field determined by the poles in C of the rational function determining the map.

Asymptotic solutions based on the characteristics and the modified Maslov canonical operator of the two-dimensional wave equation with variable coefficients and right-hand side corresponding to: (a) an instantaneous source; (b) a rapidly acting, but “time spread,” source, are compared. An algorithm for approximating a (more complicated) solution of problem (b) by linear combinations of the derivatives of the (simpler) solution of problem (a) is proposed. Numerical calculations showing the accuracy of this approximation are presented. The replacement of the solutions of problem (b) by those of problem (a) becomes especially important in the case where the wave equation is considered in the domain with boundary on which the velocity of the wave equation vanishes. Then the characteristics of the problem become singular (nonstandard) and solutions of type (a) generalize to the case referred to above in a much simpler and effective way than solutions of type (b). Such a situation arises in problems where long waves (for example, tsunami waves) are incident on a sloping seashore.

Using the Karatsuba method, we obtain new estimates for Kloosterman sums modulo a prime, which, under certain constraints on the number of summands, are sharper than similar estimates found earlier.

The Moutard transformation for the two-dimensional Dirac operator with complexvalued potential is constructed. It is shown that this transformation binds the potentials of Weierstrass representations of the surfaces related by the composition of inversion and reflection with respect to the axis. An explicit analytic example of a transformation leading to the appearance of double points on the spectral curve of the Dirac operator is described analytically.

We study the wave part of a localized solution of the linear systemof shallow water equations. Given a relationship between initial conditions, the relationship between the corresponding solutions is found.

Consider an operator which is defined in Banach or Hilbert space X = X₁ × X₂ by the matrix \(L = \left( {\begin{array}{*{20}{c}}A&B \\ C&D \end{array}} \right)\), where the linear operators A: X₁ → X₁, B: X₂ → X₁, C: X₁ → X₂, and D: X₂ → X₂ are assumed to be unbounded. In the case when the operators C and B are relatively bounded with respect to the operators A and D, respectively, new conditions of closedness or closability are obtained for the operator L. For the operator L acting in a Hilbert space, analogs of Rellich–Kato theorems on the stability of self-adjointness are obtained.