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Том 226, № 4 (2017)
- Год: 2017
- Статей: 11
- URL: https://ogarev-online.ru/1072-3374/issue/view/14862
Article
Operator Bruwier Series and Initial Problem for a Linear Differential–Difference Equation in a Banach Space
Аннотация
We prove the existence and uniqueness of a solution to the one-point initial problem for the linear differential–difference equation u′(z) = Au(z + h), h ϵ ℂ, in some classes of exponential type entire functions. We obtain a representation of a unique solution to this problem by using the operator Bruwier series.
335-343
On Boundedness of Bergman Projection Operators in Banach Spaces of Holomorphic Functions in Half-Plane and Harmonic Functions in Half-Space
Аннотация
We present a simple proof of the boundedness of holomorphic and harmonic Bergman projection operators on a half-plane and a half-space respectively on the Orlicz space, the variable exponent Lebesgue space, and the variable exponent generalized Morrey space. The approach is based on an idea due to V. P. Zaharyuta and V. I. Yudovich (1962) to use Calderón–Zygmund operators for proving the boundedness of the Bergman projection in Lebesgue spaces on the unit disc. We also study the rate of growth of functions near the boundary in the spaces under consideration.
344-354
Asymptotics of Spectra of Compact Pseudodifferential Operators with Nonsmooth Symbols with Respect to Spatial Variables
Аннотация
We consider compact selfadjoint pseudodifferential operators with symbols that are not smooth with respect to x on a fixed set. We obtain conditions for the validity of the Weyl formula for the spectral asymptotics and apply the results to some class of pseudodifferential operators.
355-374
Korn Inequality for a Thin Periodic Corrugated Beam
Аннотация
We obtain the asymptotically exact weighted Korn inequality for a thin (of thickness δh) periodically corrugated (with step δ) beam clamped at its ends. The constant in the inequality is independent of both small parameters δ and h, but the distribution of weighted factors under longitudinal and transverse displacements essentially differs from the optimal one in the case of a straight beam.
375-387
The Mixed Fourier–Bessel Transform of a Radial Bessel j-Function
Аннотация
We find the weighted spherical mean of the kernel of the mixed Fourier–Bessel transform and the mixed Fourier–Bessel transform of a radial compactly supported function. In a space of weighted distributions, we obtain a formula for the mixed Fourier–Bessel transform of a radial Bessel j-function in terms of weighted Kipriyanov distributions.
388-401
Asymptotics of Eigenvalues in Spectral Gaps Under Regular Perturbations of Walls of a Periodic Waveguide
Аннотация
We find asymptotic representations of eigenvalues inside gaps of the continuous spectrum of a periodic waveguide with local smooth gently sloped (of depth ε ≪ 1) perturbations of walls. These eigenvalues reach the upper or lower gap edge as ε → +0. We consider several variants of the gap edge structure and obtain conditions guaranteeing the existence or absence of points of the discrete spectrum in small neighborhoods. We calculate the total number of eigenvalues in a gap for small ε. To justify the asymptotic expansions, we use elementary tools of the theory of spectral measure.
402-444
Operator Estimates in Homogenization of Elliptic Systems of Equations
Аннотация
We study homogenization of nonselfadjoint second order elliptic systems with ε-periodic rapidly oscillating coefficients as ε → 0. We obtain the L2- and H1-estimates for the homogenization error of order ε. The estimates admit the operator form and can be written in terms of the resolvents of the original and approximate systems in the operator norm \( {\left\Vert \cdot \right\Vert}_{L^2\to {L}^2} \) or \( {\left\Vert \cdot \right\Vert}_{L^2\to {H}^1} \). The shift method is used for obtaining such estimates.
445-461
Semiclassical Asymptotics of Solutions to Hartree Type Equations Concentrated on Segments
Аннотация
We study the nonlinear eigenvalue problem for two-dimensional Hartree type equations with selfaction potentials possessing logarithmic singularity and depending on the distance between points. To find a series of asymptotic eigenvalues, we derive a counterpart of the Bohr–Sommerfeld quantization rule. The corresponding asymptotic eigenfunctions are localized near a plane segment.
462-516
Asymptotics of the Spectrum of a Two-dimensional Hartree Type Operator Near Upper Boundaries of Spectral Clusters. Asymptotic Solutions Located Near a Circle
Аннотация
We consider the eigenvalue problem for a two-dimensional perturbed resonance oscillator. The role of perturbation is played by an integral Hartree type nonlinearity, where the selfaction potential depends on the distance between points and has logarithmic singularity. We obtain asymptotic eigenvalues near the upper boundaries of spectral clusters appeared near eigenvalues of the unperturbed operator.
517-530
On the Semiclassical Analysis of the Ground State Energy of the Dirichlet Pauli Operator in Non-Simply Connected Domains
Аннотация
We consider the Dirichlet Pauli operator in bounded connected domains in the plane, with a semiclassical parameter. We show that the ground state energy of the Pauli operator is exponentially small as the semiclassical parameter tends to zero and estimate the decay rate. This extends our recent results discussing a recent paper by Ekholm–Kovařík–Portmann, including non-simply connected domains.
531-544
Erratum
Erratum to: Asymptotic Approximations of the Solution to a Boundary Value Problem in a Thin Aneurysm Type Domain
545-545