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Discrete analogs of pseudo-differential operators and equations in discrete Sobolev–Slobodetsky spaces are considered. Using suitable discrete boundary conditions, we prove the unique solvability of the discrete boundary-value problem. [ABSTRACT FROM AUTHOR]

The paper investigates connections and relations between different types (concepts) of hypoelliptic differential operators (symbols (characteristic polynomials) of these operators). [ABSTRACT FROM AUTHOR]

We consider the discrete Dirichlet boundary value problem for a discrete elliptic pseudodifferential equation in the quadrant and study its solvability in discrete counterparts of the Sobolev–Slobodetskii space. The study is based on a special factorization of the elliptic symbol. We compare the solutions to the discrete Dirichlet problem and its continuous counterpart. [ABSTRACT FROM AUTHOR]

We introduce integral transforms to study problems for the singular Bessel differential operators B−γ with negative parameter −γ < 0. A solution to the singular Bessel equation B−γu+u = 0 is a function u = 핁μ expressed in terms of the Bessel functions of the first kind with positive parameter μ = (γ + 1)/2. Based on the notion of a 핋-pseudoshift, we formulate the Levitan addition theorem. We construct the Bessel 핁-transform and the corresponding class of singular 핁-pseudodifferential operators. We prove a theorem on the order of 핁-pseudodifferential operators in the class of Sobolev–Kipriyanov functions. [ABSTRACT FROM AUTHOR]

For a pair of smooth transversally intersecting submanifolds in some enveloping smooth manifold, we study the algebra generated by pseudodifferential operators and (co)boundary operators corresponding to submanifolds. We establish that such an algebra has 18 types of generating elements. For operators from this algebra, we define the concept of symbol and obtain the composition formula. [ABSTRACT FROM AUTHOR]

We study the main operations (argument shift, differentiation, etc.) in the generalized spaces of type S and some classes of analytic functions and pseudodifferential operators in spaces of this kind, as well as the properties of the Fourier transforms of generalized functions, convolutions, convolvers, and multiplicators. The correct solvability of the nonlocal time problem is proved for one class of pseudodifferential equations in generalized spaces of type S. Its solution is presented in the form of convolution of the fundamental solution with an initial function, which is an element of the space of generalized functions of the ultradistribution type. [ABSTRACT FROM AUTHOR]

In this paper, we propose an abstract scheme for the study of special operators and apply this scheme to examining elliptic pseudo-differential operators and related boundary-value problems on manifolds with nonsmooth boundaries. In particular, we consider cases where boundaries may contain conical points, edges of various dimensions, and even peak points. Using the constructions proposed, we present well-posed formulations of boundary-value problems for elliptic pseudo-differential equations on manifolds discussed in Sobolev–Slobodecky spaces. [ABSTRACT FROM AUTHOR]

We consider a computation of a group of stable homotopy classes for pseudodifferential elliptic boundary problems. We study this problem in terms of topological K-groups of spaces in the following cases: boundary-value problems on manifolds with boundary, transmission problems with conditions on a closed submanifold of codimension one, and nonlocal problems with contractions. [ABSTRACT FROM AUTHOR]

We consider discrete pseudo-differential equations with elliptic symbols and the corresponding discrete boundary-value problems in special canonical domains of multidimensional spaces. The solvability of such equations and boundary-value problems in discrete analogs of Sobolev–Slobodetsky spaces is examined. [ABSTRACT FROM AUTHOR]

We prove the correct solvability of the nonlocal multipoint (in time) problem for the evolutionary equation with pseudodifferential operator in the space of generalized periodic functions and an initial function, which is an element of the space of periodic ultradistributions. [ABSTRACT FROM AUTHOR]

Contrary to the opinion that has prevailed for the last several decades about the incorrectness of the inverse–MEEG problems (see, for example, the paper of D. Sheltraw and E. Coutsias in Journal of Applied Physics, 94, No. 8, 5307–5315 (2003)), in this note it is shown that this problem is absolutely well posed: it has a unique solution, but in a special class of functions (different from those considered by biophysicists). The solution has the form q = q0 + p0δ|∂Y, where q0 is an ordinary function defined in the domain of the region Y occupied by the brain, and p0δ|∂Y is a δ-function on the boundary of the domain Y with a certain density p0. Moreover, the operator of this problem realizes an isomorphism of the corresponding function spaces. This result was obtained due to the fact that: (1) Maxwell's equations are taken as a basis; (2) a transition was made to the equations for the potentials of the magnetic and electric fields; (3) the theory of boundary value problems for elliptic pseudodifferential operators with an entire index of factorization is used. This allowed us to find the correct functional class of solutions of the corresponding integral equation of the first kind. Namely: the solution has a singular boundary layer in the form of a delta function (with some density) at the boundary of the domain. From the point of view of the MEEG problem, this means that the sought-for current dipoles are also concentrated in the cerebral cortex. [ABSTRACT FROM AUTHOR]

We construct new bases of real functions from L2(Br) and from L2(ℚp). These functions are eigenfunctions of the p-adic pseudo-differential Vladimirov operator, which is defined on a compact set Br ⊂ ℚp of the field of p-adic numbers ℚp or, respectively, on the entire field ℚp. A relation between the basis of functions from L2(ℚp) and the basis of p-adic wavelets from L2(ℚp) is found. As an application, we consider the solution of the Cauchy problem with the initial condition on a compact set for a pseudo-differential equation with a general pseudo-differential operator that is diagonal in the basis constructed. [ABSTRACT FROM AUTHOR]

We consider a simplest elliptic pseudodifferential equation in a multi-dimensional cone (multi-dimensional angle) and describe all possible structures of its solutions related to the wave factorization of the elliptic symbol. Depending on the index of wave factorization, we consider various statements of well-posed boundary-value problems. The existence of solutions is studied in Sobolev-Slobodetskii spaces. [ABSTRACT FROM AUTHOR]

We consider model pseudodifferential equations in canonical multidimensional domains with boundary singularities presented by the union of cones or a cone of lower dimension. We study the solvability of these equations by using the wave factorization concept. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We use the step-by-step method to prove the solvability of nonlocal problem for an autonomous quasilinear parabolic pseudodifferential equation with nonsmooth symbols and deviations of the argument. [ABSTRACT FROM AUTHOR]

We prove that the Cauchy problem for a pseudodifferential equation with pseudo-Bessel operator with variable symbol is solvable in the class of bounded and even functions on ℝ: [ABSTRACT FROM AUTHOR]

We consider compact selfadjoint pseudodifferential operators under the assumption that the decay order of symbols with respect to ξ depends on a point x. We show that the asymptotic Weyl formula is valid for such operators. Bibliography: 12 titles. [ABSTRACT FROM AUTHOR]

We present a new variant of H-measures defined on spectra of general algebra of test symbols and derive the localization properties of such H-measures. Applications for the compensated compactness theory are given. In particular, we present new compensated compactness results for quadratic functionals in the case of general pseudo-differential constraints. The case of inhomogeneous second order differential constraints is also studied. Bibliography: 9 titles. [ABSTRACT FROM AUTHOR]

Based on a special Fourier-Bessel transform introduced by Kipriyanov and Katrakhov, we consider singular pseudodifferential operators including linear singular differential operators. We introduce B-elliptic operators and construct regularizers for B-elliptic equations. A priori estimates are proved. Bibliography: 7 titles. [ABSTRACT FROM AUTHOR]

It is well known that the composition of two elliptic differential operators of orders ω and ω′ is again an elliptic operator of order ω+ω′. This fact is not true for the composition of two multi-order systems elliptic in the sense of Douglis and Nirenberg. We consider a new more general class of multi-order pseudodifferential operators. This class of multi-order essentially elliptic operators is closed with respect to the composition, coordinate and basis transformation. Moreover, every multi-order essentially elliptic operator has a parametrix belonging to the class and therefore is a Fredholm operator. Bibliography: 5 titles. [ABSTRACT FROM AUTHOR]

We prove the theorem of existence of a solution to the inhomogeneous equation with the one-dimensional Schrödinger operator in the space of quickly decreasing functions arising in the theory of asymptotic solutions to singularly perturbed partial differential equations. [ABSTRACT FROM AUTHOR]

A quasilinear boundary-value problem is considered for a degenerate parabolic pseudodifferential equation in a Sobolev space. By the Laplace transformation, this problem is reduced to a boundary-value problem for a degenerate elliptic equation with a parameter. The latter problem is studied with the help of Vishik-Grushin methods and the Rothe theorem. The approach proposed differs from those previously used in relation to similar problems. [ABSTRACT FROM AUTHOR]

We consider a class of pseudodifferential operators acting on test functions as a convolution with a given generalized function. For evolution families with generators from this class we construct an analog of a probabilistic representation. Bibliography: 9 titles. [ABSTRACT FROM AUTHOR]

The wave factorization of an elliptic pseudodifferential operator and the solvability of the equation with such an operator are described. [ABSTRACT FROM AUTHOR]

This paper shows that for mechanical systems, the dimension of whose base space is larger than time (there also exist spatial coordinates), the system of equations defining the evolution of the system must be a hyperbolic system of pseudodifferential equations. [ABSTRACT FROM AUTHOR]

Pseudodifferential operators occurring in real equations of continuous-medium mechanics are discussed. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Kuramoto-Sivashinskii equation on the multidimensional torus with a Riemannian metric: $$u_t = - (P(\nabla u,\nabla u) + \Delta u + \nu \Delta ^2 u),\bar u = 0,x \in T^n ,$$ where $$\bar u = \frac{1}{{volT^n }}\int\limits_{T^n } {ud\mu } ,Pu = u - \bar u,\nu > 0$$ . For this equation the theorem on energy transfer holds. It is formulated in the following way. Let $$\sum {a_k \xi _k } $$ be the Fourier expansion of the function u. Denote by P N and P the operators of rejection of the “leading” and, respectively, “lowest” terms of the Fourier expansion (harmonics), i.e., $$P_N u = \sum\limits_1^N {a_k \xi _k } ,P_N^ \bot = u - P_N u$$ . For any ρ > 0, N ∈ ℕ, s ≥ n/2+5, and λ ∈ (0, 1), there exists R such that for any function. ϕ ∈ $$\bar C^\infty (T^n )$$ lying outside the ball $$n_{C^1 } \leqslant R$$ in the ball $$Q = \{ n_s \leqslant \rho \left\| \varphi \right\|_{C^1 } \} $$ , there exists an instant of time t ∈ (0, 1) such that g ϕ=ψ and $$\left\| {P_N^ \bot \psi } \right\|_s^2 \geqslant \lambda \left\| \psi \right\|_s^2 $$ . Here, R is a constant depending on the metric ( g), n s is the sth Sobolev norm, and $$n_{C^1 } $$ is the C 1-norm. [ABSTRACT FROM AUTHOR]

Studies the ring properties of skew-Laurent series rings and of pseudodifferential operator rings. Definitions of a Laurent ring; Distributive semilocal rings; Notation and general properties of Laurent rings.

We obtain the conditions of one-valued solvability of some infinite-dimensional pseudodifferential equations and construct some classes of invertible pseudodifferential operators with symbols depending on two arguments. [ABSTRACT FROM AUTHOR]

The $C^*$-algebra $\{\backslash mathcal\; A\}$ generated by the operators of pseudodifferential boundary value problems on a manifold $\{\backslash mathcal\; M\}$ with smooth closed disjoint edges and boundary $\backslash Gamma$ is studied. The operators act in the space $L\_2(\{\backslash mathcal\; M\}\; )\backslash oplus\; L\_2(\backslash Gamma).$ The goal of this paper is to describe all (up to an equivalence) irreducible representations of the algebra $\{\backslash mathcal\; A\}.$ Bibliography: 12 titles. [ABSTRACT FROM AUTHOR]