On the basis property of eigenelements of ordinary linear differential operators in a space of the type <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>, <Emphasis Type="Italic">p</Emphasis> ≥ 2

For a linear differential expression with matrix coefficients in the class L _p , p ≥ 2, and with a parameter λ, we consider a boundary value problem with boundary conditions at the endpoints of the interval [a, b]. Under the condition that the problem is regular, we obtain a formula for the Fourier series expansion of an arbitrary vector function of the class L _p in the root functions of the problem.     

Applications of the Differential Calculus to Nonlinear Elliptic Operators with Discontinuous Coefficients

The paper concerns Dirichlet’s problem for second order quasilinear non-divergence form elliptic equations with discontinuous coefficients. We start with suitable structure, growth, and regularity conditions ensuring solvability of the problem under consideration. Fixing then a solution u ₀ such that the linearized at u ₀ problem is non-degenerate, we apply the Implicit Function Theorem. As a result we get that for all small perturbations of the coefficients there exists exactly one solution u ≈ u ₀ which depends smoothly (in W ^2,p with p larger than the space dimension) on the data. For that, no structure and growth conditions are needed and the perturbations of the coefficients can be general L ^∞-functions of the space variable x. Moreover, we show that the Newton Iteration Procedure can be applied in order to obtain a sequence of approximate (in W ^2,p ) solutions for u ₀.     

Two inverse problems of finding a coefficient in a parabolic equation

For a parabolic equation, we consider inverse problems of reconstructing a coefficient that depends on the space variables alone. The first problem is to find a lower-order coefficient c(x) multiplying u(x, t), and the second problem is to find the coefficient a(x) multiplying Δu. As additional information, the integral of the solution with respect to time with some weight function is given. The coefficients of the equation depend both on time and on the space variables. We obtain sufficient conditions for the existence of generalized solutions of our problems; moreover, for the first problem, we also prove uniqueness and construct an iterative sequence that converges to the desired coefficient almost everywhere in the domain. We present examples of input data of these problems for which the assumptions of our theorems are necessarily true.     

Asymptotic analysis of some classes of ordinary differential equations with large high-frequency terms

A complete asymptotic expansion is constructed for solutions of the Cauchy problem for nth order linear ordinary differential equations with rapidly oscillating coefficients, some of which may be proportional to ω ^n/2, where ω is oscillation frequency. A similar problem is solved for a class of systems of n linear first-order ordinary differential equations with coefficients of the same type. Attention is also given to some classes of first-order nonlinear equations with rapidly oscillating terms proportional to powers ω ^d . For such equations with d ∈ (1/2, 1], conditions are found that allow for the construction (and strict justification) of the leading asymptotic term and, in some cases, a complete asymptotic expansion of the solution of the Cauchy problem.     

A dynamic boundary-value problem without initial conditions for almost linear parabolic equations

We study a dynamic boundary-value problem without initial conditions for linear and almost linear parabolic equations. First, we establish conditions for the existence of a unique solution of a problem without initial conditions for a certain abstract implicit evolution equation in the class of functions with exponential behavior as t → −∞. Then, using these results, we prove the existence of a unique solution of the original problem in the class of functions with exponential behavior at infinity.     

Instability,complexity, and evolution

In this paper, we consider a new class of random dynamical systems that contains, in particular, neural networks and complicated circuits. For these systems, we consider the viability problem: we suppose that the system survives only the system state is in a prescribed domain Π of the phase space. The approach developed here is based on some fundamental ideas proposed by A. Kolmogorov, R. Thom, M. Gromov, L. Valiant, L. Van Valen, and others. Under some conditions it is shown that almost all systems from this class with fixed parameters are unstable in the following sense: the probability P _t to leave Π within the time interval [0, t] tends to 1 as t → ∞. However, it is allowed to change these parameters sometimes (“evolutionary” case), then it may happen that P _t  < 1 − δ  < 1 for all t (“stable evolution”). Furthermore, we study the properties of such a stable evolution assuming that the system parameters are encoded by a dicsrete code. This allows us to apply complexity theory, coding, algorithms, etc. Evolution is a Markov process of modification of this code. Under some conditions we show that the stable evolution of unstable systems possesses the following general fundamental property: the relative Kolmogorov complexity of the code cannot be bounded by a constant as t → ∞. For circuit models, we define complexity characteristics of these circuits. We find that these complexities also have a tendency to increase during stable evolution. We give concrete examples of stable evolution. Bibliography: 80 titles. To the memory of A. N. Livshitz Published in Zapiski Nauchnykh Seminarov POMI, Vol. 360, 2008, pp. 31–69.     

On the existence and stability of periodic and almost periodic solutions of quasilinear equations with maxima

We study the problem of existence of periodic and almost periodic solutions of the scalar equation x′ (t) = − δx(t) + pmax _{u∈[t − h, t]} x(u) + f(t) where δ, p ∈ R, with a periodic (almost periodic) perturbation f(t). For these solutions, we establish conditions of global exponential stability and prove uniqueness theorems. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 6, pp. 747–754, June, 1998.     

Suites de longueur minimale associees a un ensemble normal donne

For a given subsetA of the set of real numbers, we search a sequence Λ=(λ _n) of real numbers such that bothA is the normal setB(Λ) associated to Λ, and Λ takes its values in a bounded interval, with a minimal lengthM. A lower bound ofM is obtained, which gives some necessary conditions of existency of such a bounded sequence Λ. More details are given whenA is a subset of the set of integers. In this case, the problem is to find a polynomialQ of lowest degree such that the productP.Q has non-negative coefficients, for some special given polynomialP.      

Asymptotic behavior of positive solutions to emden–fowler type equations

We study the asymptotic behavior of positive solutions to nonlinear elliptic equations of Emden–Fowler type with absorption term. For operators with variable coefficients we obtain conditions on coefficients under which the solutions have the same asymptotics as solutions to the model equation Δu = −x| ^p |u| ^σ−1 u. For positive solutions we obtain lower order terms of the asymptotic expansion at infinity. Bibliography: 10 titles.     

Electron scattering at the domain wall

We study the transmission and reflection coefficients for the Hamiltonian of a spin-polarized electron passing through the domain wall in a ferromagnetic quantum wire. We prove that total reflection occurs for energies λ ∈ (−α, α) (−α is the boundary of the essential spectrum) for both sufficiently small and sufficiently large λ, which agrees with the ballistic magnetoresistance effect in ferromagnetic nanocontacts. For energies λ > α, almost total reflection becomes almost total transmission, and both effects occur without a spin flip.     

Solvability of inhomogeneous boundary-value problems for fourth-order differential equations

We consider a Cauchy-type boundary-value problem, a problem with three boundary conditions, and the Dirichlet problem for a general typeless fourth-order differential equation with constant complex coefficients and nonzero right-hand side in a bounded domain Ω ⊂ R ² with smooth boundary. By the method of the Green formula, the theory of extensions of differential operators, and the theory of L-traces (i.e., traces associated with the differential operation L), we establish necessary and sufficient (for elliptic operators) conditions of the solvability of each of these problems in the space H ^m (Ω), m ≥ 4.     

Sums and Differences of Two Cubic Polynomials

 When f(x) is a cubic polynomial with integral coefficients, we show that almost all integers represented as the sum or difference of two values of f(x), with , are thus represented essentially uniquely.     

Homogenization of Nonlinear Degenerate Non-monotone Elliptic Operators in Domains Perforated with Tiny Holes

The paper deals with the homogenization problem beyond the periodic setting, for a degenerated nonlinear non-monotone elliptic type operator on a perforated domain Ω ^ε in ℝ ^N with isolated holes. While the space variable in the coefficients a ₀ and a is scaled with size ε (ε>0 a small parameter), the system of holes is scaled with ε ² size, so that the problem under consideration is a reiterated homogenization problem in perforated domains. The homogenization problem is formulated in terms of the general, so-called deterministic homogenization theory combining real homogenization algebras with the Σ-convergence method. We present a new approach based on the Besicovitch type spaces to solve deterministic homogenization problems, and we obtain a very general abstract homogenization results. We then illustrate this abstract setting by providing some concrete applications of these results to, e.g., the periodic homogenization, the almost periodic homogenization, and others.     

Holomorphic functional calculus of Hodge-Dirac operators in L p

We study the boundedness of the H ^∞ functional calculus for differential operators acting in L ^p (R ⁿ ; C ^N ). For constant coefficients, we give simple conditions on the symbols implying such boundedness. For non-constant coefficients, we extend our recent results for the L ^p theory of the Kato square root problem to the more general framework of Hodge-Dirac operators with variable coefficients Π _B as treated in L ²(R ⁿ ; C ^N ) by Axelsson, Keith, and McIntosh. We obtain a characterization of the property that Π _B has a bounded H ^∞ functional calculus, in terms of randomized boundedness conditions of its resolvent. This allows us to deduce stability under small perturbations of this functional calculus.     

Integrable Submanifolds in Almost Complex Manifolds

Let (M,J) be a germ of an almost complex manifold of real dimension 2m and let n (n<m) be an integer. We study a necessary and sufficient condition for M to admit an integrable submanifold N of complex dimension n. If n=m−1, we find defining functions of N explicitly from the coefficients of the torsion tensor. For J obtained by small perturbation of the standard complex structure of ℂ ^m this condition is given as an overdetermined system of second order PDEs on the perturbation. The proof is based on the rank conditions of the Nijenhuis tensor and application of the Newlander-Nirenberg theorem. We give examples of almost complex structures on ℂ³: the ones with a single complex submanifold of dimension 2 and the ones with 1-parameter or 2-parameter families of complex submanifolds of dimension 2.     

Can quantum effects due to a massless conformally coupled field avoid gravitational singularities?

Using quantum corrections from massless fields conformally coupled to gravity, we study the possibility of avoiding singularities that appear in the flat Friedmann-Robertson-Walker model. We assume that the universe contains a barotropic perfect fluid with the state equation p = ωρ, where p is the pressure and ρ is the energy density. We study the dynamics of the model for all values of the parameter ω and also for all values of the conformal anomaly coefficients α and β. We show that singularities can be avoided only in the case where α > 0 and β < 0. To obtain an expanding Friedmann universe at late times with ω > −1 (only a one-parameter family of solutions, but no a general solution, has this behavior at late times), the initial conditions of the nonsingular solutions at early times must be chosen very exactly. These nonsingular solutions consist of a general solution (a two-parameter family) exiting the contracting de Sitter phase and a one-parameter family exiting the contracting Friedmann phase. On the other hand, for ω < −1 (a phantom field), the problem of avoiding singularities is more involved because if we consider an expanding Friedmann phase at early times, then in addition to fine-tuning the initial conditions, we must also fine-tune the parameters α and β to obtain a behavior without future singularities: only a oneparameter family of solutions follows a contracting Friedmann phase at late times, and only a particular solution behaves like a contracting de Sitter universe. The other solutions have future singularities.     

Stefan problem

We prove the existence of a global classical solution of the multidimensional two-phase Stefan problem. The problem is reduced to a quasilinear parabolic equation with discontinuous coefficients in a fixed domain. With the help of a small parameter ε, we smooth coefficients and investigate the resulting approximate solution. An analytical method that enables one to obtain the uniform estimates of an approximate solution in the cross-sections t = const is developed. Given the uniform estimates, we make the limiting transition as ε → 0. The limit of the approximate solution is a classical solution of the Stefan problem, and the free boundary is a surface of the class H ^2+α,1+α/2.     

Problem without initial conditions for linear and almost linear degenerate operator differential equations

We study the problem without initial conditions for linear and almost linear degenerate operator differential equations in Banach spaces. The uniqueness of a solution of this problem is proved in the classes of bounded functions and functions with exponential behavior as t → –∞. We also establish sufficient conditions for initial data under which there exists a solution of the considered problem in the class of functions with exponential behavior at infinity.     

On an abstract integro-differential problem

In an arbitrary Banach space we consider the Cauchy problem for an integro-differential equation with unbounded operator coefficients. We establish the solvability of the problem in the weight spaces of integrable functions under certain conditions on the data. To prove this, the solution of the problem is written in explicit form with the help of analytic semigroups generated by fractional powers of the operator coefficients. Here an important role is played by the conditions on the coefficients ensuring the coercive estimates of the corresponding integral operators. Translated fromMatematicheskie Zametki, Vol. 66, No. 6, pp. 887–896, December, 1999.