Том 233, № 4 (2018)
- Год: 2018
- Статей: 9
- URL: https://journals.rcsi.science/1072-3374/issue/view/14946
Article
On Regularity of Solutions for Initial-Boundary Value Problems for the Zakharov–Kuznetsov Equation
On Some Degenerate Elliptic Equations Arising in Geometric Problems
Аннотация
We consider some fully nonlinear degenerate elliptic operators and we investigate the validity of certain properties related to the maximum principle. In particular, we establish the equivalence between the sign propagation property and the strict positivity of a suitably defined generalized principal eigenvalue. Furthermore, we show that even in the degenerate case considered in the present paper, the well-known condition introduced by Keller–Osserman on the zero-order term is necessary and sufficient for the existence of entire weak subsolutions.
On New Structures in the Theory of Fully Nonlinear Equations
Аннотация
We describe the current state of the theory of equations with m-Hessian stationary and evolution operators. It is quite important that new algebraic and geometric notions appear in this theory. In the present work, a list of those notions is provided. Among them, the notion of m-positivity of matrices is quite important; we provide a proof of an analog of Sylvester’s criterion for such matrices. From this criterion, we easily obtain necessary and sufficient conditions for existence of classical solutions of the first initial boundary-value problem for m-Hessian evolution equations. The asymptotic behavior of m-Hessian evolutions in a semibounded cylinder is considered as well.
On Feedback-Principle Control for Systems with Aftereffect Under Incomplete Phase-Coordinate Data
Аннотация
For a nonlinear system of differential equations with aftereffect, two mutually complement game minimax (maximin) problems for the quality functional are considered. Assuming that a part of phase coordinates of the system is measured (with error) sufficiently frequently, we provide solving algorithms that are stable with respect to the information noise and computational errors. The proposed algorithms are based on the Krasovskii extremal translation principle.
The Riesz Basis Property with Brackets for Dirac Systems with Summable Potentials
Аннотация
In the space ℍ = (L2[0, π])2, we study the Dirac operator \( {\mathrm{\mathcal{L}}}_{P,U} \) generated by the differential expression ℓP(y) = By′ + Py, where
and the regular boundary conditions
The elements of the matrix P are assumed to be complex-valued functions summable over [0, π]. We show that the spectrum of the operator \( {\mathrm{\mathcal{L}}}_{P,U} \) is discrete and consists of eigenvalues {λn}n ∈ ℤ such that \( {\uplambda}_n={\uplambda}_n^0+o(1) \) as |n| → ∞, where \( {\left\{{\uplambda}_n^0\right\}}_{n\in \mathrm{\mathbb{Z}}} \) is the spectrum of the operator \( {\mathrm{\mathcal{L}}}_{0,U} \) with zero potential and the same boundary conditions. If the boundary conditions are strongly regular, then the spectrum of the operator \( {\mathrm{\mathcal{L}}}_{P,U} \) is asymptotically simple. We show that the system of eigenfunctions and associate functions of the operator \( {\mathrm{\mathcal{L}}}_{P,U} \) forms a Riesz base in the space ℍ provided that the eigenfunctions are normed. If the boundary conditions are regular, but not strongly regular, then all eigenvalues of the operator \( {\mathrm{\mathcal{L}}}_{0,U} \) are double, all eigenvalues of the operator \( {\mathrm{\mathcal{L}}}_{P,U} \) are asymptotically double, and the system formed by the corresponding two-dimensional root subspaces of the operator \( {\mathrm{\mathcal{L}}}_{P,U} \) is a Riesz base of subspaces (Riesz base with brackets) in the space ℍ.
Smoothness of Generalized Solutions of the Dirichlet Problem for Strongly Elliptic Functional Differential Equations with Orthotropic Contractions
Аннотация
In the disk, we consider the first boundary-value problem for a functional differential equation containing transformations of orthotropic contractions of independent variables of the unknown function. We study the smoothness of generalized solutions inside special-type subdomains and near their boundaries and pose strong ellipticity conditions.
Well-Posedness and Spectral Analysis of Integrodifferential Equations Arising in Viscoelasticity Theory
Аннотация
We study the well-posedness of initial-value problems for abstract integrodifferential equations with unbounded operator coefficients in Hilbert spaces and provide a spectral analysis of operator functions that are symbols of the specified equations. These equations represent an abstract form of linear partial integrodifferential equations arising in viscoelasticity theory and other important applications. For the said integrodifferential equations, we obtain well-posedness results in weighted Sobolev spaces of vector functions defined on the positive semiaxis and valued in a Hilbert space. For the symbols of the said equations, we find the localization and the structure of the spectrum.
Method of Guiding Functions for Existence Problems for Periodic Solutions of Differential Equations
Аннотация
We provide a review and systematic explanation of various generalizations of the guiding function method. The current state of the said method and its applications to various kinds of problems for nonlinear periodic systems described by differential and functional differential equations are considered.