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This paper is devoted to the study of the qualitative properties of solutions to boundary-value problems for strongly elliptic differential-difference equations. Some results for these equations such as existence and smoothness of generalized solutions in certain subdomains of Q were obtained earlier. Nevertheless, the smoothness of generalized solutions of such problems can be violated near the boundary of these subdomains even for infinitely differentiable right-hand side. The subdomains are defined as connected components of the set that is obtained from the domain Q by throwing out all possible shifts of the boundary Q by vectors of a certain group generated by shifts occurring in the difference operators. For the one dimensional Neumann problem for differential-difference equations there were obtained conditions on the coefficients of difference operators, under which for any continuous right-hand side there is a classical solution of the problem that coincides with the generalized solution. 2 Also there was obtained the smoothness (in Sobolev spaces W k ) of generalized solutions of the second and the third boundary-value problems for strongly elliptic differential-difference equations in subdomains excluding -neighborhoods of certain points. However, the smoothness (in Ho lder spaces) of generalized solutions of the second boundary-value problem for strongly elliptic differential-difference equations on the boundary of adjacent subdomains was not considered. In this paper, we study this question in Ho lder spaces. We establish necessary and sufficient conditions for the coefficients of difference operators that guarantee smoothness of the generalized solution on the boundary of adjacent subdomains for any right-hand side from the Ho lder space.

In this paper, we consider the Laplace operator in a flat band whose lower boundary periodically oscillates under the Dirichlet boundary condition. The period and the amplitude of oscillations are two independent small parameters. The main result obtained in the paper is the absence of internal lacunas in the lower part of the spectrum of the operator for sufficiently small period and amplitude. We obtain explicit upper estimates of the period and amplitude in the form of constraints with specific numerical constants. The length of the lower part of the spectrum, in which the absence of lacunas is guaranteed, is also expressed explicitly in terms of the period function and the amplitude.

In this paper, we formulate mathematical foundations of applications of boundary integral equations with strongly singular integrals understood in the sense of finite Hadamard value to numerical solution of certain boundary-value problems. We describe numerical schemes for solving boundary strongly singular equations based on quadrature formulas and the collocation method. Also, we make references to known results on the mathematical justification of the numerical methods described in the paper.

A dynamical system $$ {\displaystyle \begin{array}{ll}{u}_{tt}-\Delta u-\nabla 1\mathrm{n}\;\rho \cdot \nabla u=0& in\kern0.6em {\mathrm{\mathbb{R}}}_{+}^3\times \left(0,T\right),\\ {}{\left.u\right|}_{t=0}={\left.{u}_t\right|}_{t=0}=0& in\kern0.6em \overline{{\mathrm{\mathbb{R}}}_{+}^3},\\ {}{\left.{u}_z\right|}_{z=0}=f& for\kern0.36em 0\le t\le T,\end{array}} $$ is under consideration, where ρ = ρ(x, y, z) is a smooth positive function; f = f(x, y, t) is a boundary control; u = uf (x, y, z, t) is a solution. With the system one associates a response operator R : f ↦ uf|z = 0. The inverse problem is to recover the function ρ via the response operator. A short representation of the local version of the BC-method, which recovers ρ via the data given on a part of the boundary, is provided. If ρ is constant, the forward problem is solved in explicit form. In the paper, the corresponding representations for the solutions and response operator are derived. A way to use them for testing the BC-algorithm, which solves the inverse problem, is outlined. The goal of the paper is to extend the circle of the BC-method users, who are interested in numerical realization of methods for solving inverse problems.

The paper is devoted to spectral problems for strongly elliptic second-order systems in bounded Lipschitz domains. We consider the spectral Dirichlet and Neumann problems and three problems with spectral parameter in conditions at the boundary: the Poincare–Steklov problem and two transmission problems. In the style of a survey, we discuss the main properties of these problems, both self-adjoint and non-self-adjoint. As a preliminary, we explain several facts of the general theory of the main boundary value problems in Lipschitz domains. The original definitions are variational. The use of the boundary potentials is based on results on the unique solvability of the Dirichlet and Neumann problems. In the main part of the paper, we use the simplest Hilbert L2-spaces Hs, but we describe some generalizations to Banach spaces Hsp of Bessel potentials and Besov spaces Bsp at the end of the paper.

This paper studies conditions for local (in time) solvability of a qualitatively new singularllimit problem, the free (unknown) boundary problem appearing recently. In fact, there are not so many different free boundary problems, which corresponds to not so large a variety of principally different phase transitions of the first and second kinds. Therefore, the appearance of principally new problems elicits interest. This paper studies structural features of a certain problem on the basis of a certain method developed previously, precisely, the localization method [1, 3, 9].

In this paper, a simple shear flow in a half-space, which has interesting properties from the point of view of boundary regularity, is described. It is a solution with a bounded velocity field to both the homogeneous Stokes system and the Navier–Stokes equation, and satisfies the homogeneous initial and boundary conditions. The gradient of the solution may become unbounded near the boundary. The example significantly simplifies an earlier construction by K. Kang, and shows that the boundary estimates obtained in a recent paper by the first author are sharp. Bibliography: 4 titles.

The paper deals with the so-called M-transform, which maps divergence-free vector fields in Ω T := {x ∈ Ω| dist(x, ∂Ω) < T}, Ω ⊂⊂ $$ \mathbb{R} $$ 3, to the space of transversal fields. The latter space consists of vector fields in Ω T tangential to the equidistant surfaces of the boundary ∂Ω. In papers devoted to the dynamical inverse problem for the Maxwell system, in the framework of the BC-method, the operator M T was defined for T < T ω, where T ω depends on the geometry of Ω. This paper provides a generalization for arbitrary T. It is proved that M T is partially isometric, and its intertwining properties are established. Bibliography: 6 titles.

This is the first paper of a conceived series under the common title “The boundary control method in inverse problems.” The aim of the series is to expound systematically an approach to inverse problems based upon its relationship with control theory. The 1d-variant of the method is shown with the example of the classical problem of recovering the density of an inhomogeneous string, and both dynamical and spectral statements of the problem are considered. The paper is written in such a way as to serve as an introduction to the multidimensional BC-method: the basic tools and constructions are amenable to further generalization to multidimensional problems. Bibliography: 31 titles.

In the half-plane {−∞ < x < +∞} × {0 < y < +∞}, the Dirichlet problem is considered for differential-difference equations of the kind $$ {u}_{xx}+\sum_{k=1}^m{a}_k{u}_{xx}\left(x+{h}_{k,}y\right)+{u}_{yy}=0 $$ , where the amount m of nonlocal terms of the equation is arbitrary and no commensurability conditions are imposed on their coefficients a1, . . . , am and the parameters h1, . . . , hm determining the translations of the independent variable x. The only condition imposed on the coefficients and parameters of the studied equation is the nonpositivity of the real part of the symbol of the operator acting with respect to the variable x. Earlier, it was proved that the specified condition (i.e., the strong ellipticity condition for the corresponding differential-difference operator) guarantees the solvability of the considered problem in the sense of generalized functions (according to the Gel’fand–Shilov definition), a Poisson integral representation of a solution was constructed, and it was proved that the constructed solution is smooth outside the boundary line. In the present paper, the behavior of the specified solution as y → +∞ is investigated. We prove the asymptotic closeness between the investigated solution and the classical Dirichlet problem for the differential elliptic equation (with the same boundary function as in the original nonlocal problem) determined as follows: all parameters h1, . . . , hm of the original differential-difference elliptic equation are assigned to be equal to zero. As a corollary, we prove that the investigated solutions obey the classical Repnikov–Eidel’man stabilization condition: the solution stabilizes as y → +∞ if and only if the mean value of the boundary function over the interval (−R,+R) has a limit as R → +∞.

During the past several years, there has been an increase in the development of data assimilation methods and their application. The first data assimilation applications were in meteorology, and now the data assimilation technique is an important component of numerical weather forecasting. The data revolution in oceanography is bringing the daily practice of physical oceanography closer to that of dynamic meteorology. There are now large observational data sets of temperature and salinity in the Atlantic from global projects such as WOCE, SECTIONS, TOGA, COARE, PIRATA, and others. Improved assimilation techniques are needed in order to have ocean models fully exploit the new observational facilities. Data assimilation improves the estimate of the ocean and atmospheric physical state by extracting as much information as possible from both the measurements and the dynamic model, combining them in an optimal way. Assimilation may be used to improve initial and/or boundary conditions and to evaluate poorly known model parameters. Data assimilation in ocean models has been ongoing in the scientific literature for approximately 30 years (for a review, see [8]). It is necessary to understand which processes can be realistically described by the ocean models and which is the role of measurements. Two extreme opinions exist. One considers that the ocean is like a quasi-stationary medium and that it is sufficient to make only a few measurements in order to have a good description of the ocean state. From this point of view, the numerical models are useless (or simply wrong) and data assimilation techniques are restricted to a statistical correction process. The second opinion is that the ocean is a highly turbulent fluid that does not have a memory about its previous states. In this case, information generated by measurements becomes meaningless in a short time interval, and it is necessary to continuously monitor the ocean. Thus, only well-designed resolution models would be able to describe real processes in the ocean and data assimilation would be useful only to obtain correct initial (and possible boundary) conditions. Reality, however, lies somewhere in between these two extreme points of view. Thus, the scientific question is how to represent this situation numerically. The answer depends on both the model and the data assimilation techniques as well as on the local dynamic. There are two general concepts which have been discussed for data assimilation. The first is the “variational/adjoint” method, which has been the most popular scheme (see, e.g., [22, 24, 28]). An example of this technique is to assume that the model initial (and/or boundary) conditions are unknown parameters and that there is observation data distributed over some time interval. The technique seeks the optimal initial conditions by comparing the model trajectory with the measurements with respect to some criteria. This can be formulated as a constrained minimization problem. The present paper does not deal with this class of methods so there is no need to go into the details of this problem. The other class of methods is the “sequential data assimilation.” Starting from some initial condition the model solution is sequentially updated whenever measurements are available. The model solution will approach the observed state under certain conditions. This group of methods requires an updating scheme, which combines the model solution and the measurements to find the “best” state estimate. The Kalman-filter method belongs to this class of data assimilation technique. The Kalman filter is derived in a number of books on control theory (e.g., [1, 11]). In oceanography, the Kalmanfilter has been used in [7, 16]. The main idea of this method is to solve dynamic equations using the error covariance matrix equation, where an error is the difference between the model and observation values. These applications have been presented in [9, 16] and are commonly used. In this paper, another method for the definition of the error covariance matrix is presented. It continues the studies published in our paper by K. Belyaev, S. Meyers, and J. J. O’Brien.

In this paper, we study the problem of boundary control of string oscillations, which is carried out over a period of time less than the critical time. The control is performed by a displacement of one end of the string, whereas at the other end a uniform boundary condition with oblique derivative is given, and the direction of this derivative does not coincide with characteristics. The classical statement of the problem is considered. Necessary and sufficient conditions for the existence of a unique control are found, and the control itself is obtained in an explicit analytical form.

The paper deals with the uniqueness and stability of generalized solutions to the mixed boundary-value problem for the elasticity theory system in an unbounded domain, coinciding with a cone in a neighborhood of infinity. It is assumed that the boundary of the domain consists of two parts: the Dirichlet condition is prescribed on one of them, denoted by Γ1, and the Neumann condition is prescribed on the other. The paper contains sufficient conditions (in terms of metric properties of Γ1) for the validity of the Korn and Hardy inequalities, which imply the uniqueness and stability of the solution to the considered problem in appropriate function spaces. Bibliography: 8 titles.

In this paper, we consider diffeomorphic mappings of a nondegenerate parallelepiped onto another such parallelepiped. We prove that the image of any face of the first parallelepiped under such a mapping is a certain face of the second parallelepiped. Moreover, we also obtain several auxiliary results on the transformation of boundary points of a nondegenerate parallelepiped under the action of a diffeomorphism.

This paper is a survey of our last results about solutions to the Dirichlet and Robin boundary problems, the Robin transmission problem for an elliptic quasilinear second-order equation with the constant p– and variable p(x)-Laplacians, as well as to the degenerate oblique derivative problem for elliptic linear and quasilinear second-order equations in a conical bounded n-dimensional domain.

In the paper, for a Lame-type system, the inverse problem on recovering the fast wave velocity from the boundary dynamical data (the response operator) is solved. The velocity is determined in the near-boundary domain, the depth of determination being proportional to the observation time. We use the BC-method, which is an approach to inverse problems based on their connections with boundary control theory.

Certain Hardy inequalities in domains of Euclidean space contain sharp but unreachable constants. V. G. Maz’ya and other authors used this fact to improve the corresponding inequalities by adding new integral terms. In this paper, a survey of results in this direction initiated by H. Brezis and M. Marcus is presented. Also, we give some generalizations of Brezis–Marcus-type inequalities to the case of Rellich-type inequalities with weights that are powers of the distance from a point to the boundary of the domain. Generalizations to the case of conformally invariant integral inequalities in simply connected and doubly connected planar hyperbolic domains are discussed.

Bifurcation diagrams of Steiner tree topologies for four boundary points are constructed in this paper. Also some properties of such diagrams for an arbitrary number of points are considered.

In the paper, the extremal boundary of the space of weakly additive, order-preserving, normalized, positive-homogeneous, and semiadditive functionals on a compact space is studied. The dimension of the extremal boundary of the convex compact OS(n) is found.

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.

The paper is devoted to the calculation of the Morse index on geodesic lines upon smooth surfaces embedded into the 3D Euclidean space. The interest in this theme is created by the fact that the wave field composed of the surface waves slides along the boundaries guided by the geodesic lines, which, generally speaking, give birth to numerous caustics. The same circumstance takes place in problems of the short-wave diffraction by 3D bodies in the shadowed part of the surface of the body, where the creeping waves arise. Two types of geodesic flows are considered upon the surface when they are generated by a point source and by an initial wave front, for instance, by the light-shadow boundary in the short-wave diffraction by a smooth convex body. The position of the points where geodesic lines meet caustics, i.e., focal points, is found and it is proved that all focal points are simple (not multiple) irrespective of the geometric structure of the caustics arisen. The mathematical techniques in use are based on the complexification of the geometrical spreading problem for a geodesics/rays tube.

In this paper, we study the Carleman classes in Jordan domains of the complex plane. We obtain a quasianalyticity criterion for the regular Carleman classes, which is universal for all weakly uniform domains. The proof is based on the solution of the Dirichlet problem with an unbounded boundary function and a Beurling result on the estimate of the harmonic measure.

In this paper, we construct stability regions (in the linear approximation) of triangular libration points for the planar, restricted, elliptic three-body problem and examine bifurcations that occur when parameters of the system pass through the boundaries of these regions. A new scheme for the construction of stability regions is presented, which leads to approximation formulas describing these boundaries. We prove that on one part of the boundary, the main scenario of bifurcation is the appearance of nonstationary 4π-periodic solutions that are close to a triangular libration point, whereas on the other part, the main scenario is the appearance of quasiperiodic solutions.

The results of a recent paper by A. V. Vasin, S. V. Kislyakov, and the author on the relationship between the local boundary smoothness of an analytic function and local boundary smoothness of its modulus are extended to the case of generalized pointwise Holder type conditions of order between one and two.

Results of a recent paper by A. V. Vasin, S. V. Kislyakov, and the author are extended to the case of outer functions in the upper half-plane. As in the case of the disk, it can only be guaranteed that the smoothness of an outer function is at least one half of that for its modulus, but the quantitative manifestation of this effect is different; in particular, it depends on the position of the point at which the smoothness is measured. Bibliography: 8 titles.

A function f meromorphic on ℂ\ℝ is said to be in the generalized Nevanlinna class N κ (κ ϵ ℤ+), if f is symmetric with respect to ℝ and the kernel $$ {\mathbf{N}}_{\omega }(z)\coloneq \frac{f(z)-\overline{f\left(\omega \right)}}{z-\overline{\omega}} $$ has κ negative squares on ℂ+. The generalized Stieltjes class $$ {\mathbf{N}}_{\kappa}^k\left(\kappa, k\in {\mathrm{\mathbb{Z}}}_{+}\right) $$ is defined as the set of functions f ϵ N κ such that z f ϵ N k . The full indefinite Stieltjes moment problem $$ {MP}_{\kappa}^k\left(\mathbf{s}\right) $$ consists in the following: Given κ, k ϵ ℤ+, and a sequence $$ \mathbf{s}={\left\{{s}_i\right\}}_{i=0}^{\infty } $$ of real numbers, to describe the set of functions $$ f\in {\mathbf{N}}_{\kappa}^k $$ , which satisfy the asymptotic expansion $$ f(z)=-\frac{s_0}{z}-\cdots -\frac{s_2n}{z^{2n+1}}+o\left(\frac{1}{z^{2n+1}}\right)\kern1em \left(z=-y\in {\mathrm{\mathbb{R}}}_{-},y\uparrow \infty \right) $$ for all n big enough. In the present paper, we will solve the indefinite Stieltjes moment problem $$ {MP}_{\kappa}^k\left(\mathbf{s}\right) $$ within the M. G. Krein theory of u-resolvent matrices applied to a Pontryagin space symmetric operator A [0;N] generated by $$ {\mathfrak{J}}_{\left[0;N\right]} $$ . The u-resolvent matrices of the operator A [0;N] are calculated in terms of generalized Stieltjes polynomials, by using the boundary triple’s technique. Some criteria for the problem $$ {MP}_{\kappa}^k\left(\mathbf{s}\right) $$ to be solvable and indeterminate are found. Explicit formulae for Pade approximants for the generalized Stieltjes fraction in terms of generalized Stieltjes polynomials are also presented.

This article deals with short-wave diffraction by a strongly elongated body of revolution (an axially symmetric problem). In this case, the classical method of the Leontovich–Fock parabolic equation (equations of the Schrodinger type, to be more precise) appears to be nonapplicable, because the corresponding recurrent system of equations loses its asymptotic nature, and the equations themselves, including the main parabolic equation, gain singularity in their coefficients. This paper introduces a new boundary layer in a neighborhood of the light-shadow boundary, which is determined by other scales than the Fock boundary layer. In the arising main parabolic equation, the variables are not separated, so it turns out to be impossible to construct a solution in analytic form. In this case we state a nonstationary scattering problem, where the arc length along the geodesic lines (meridians) plays the part of time, and the problem itself is resolved by numerical methods. Bibliography: 11 titles.

A quaternion field is a pair p = {α, u} of a function α and a vector field u given on a 3d Riemannian manifold Ω with boundary. A field is said to be harmonic if ∇α = rot u in Ω. The linear space of harmonic fields is not an algebra with respect to quaternion multiplication. However, it may contain commutative algebras, which is the subject of the paper. Possible applications of these algebras to the impedance tomography problem are touched upon.

Local regularity up to the flat part of the boundary is proved for certain classes of distributional solutions that are L∞L3,q with q finite. The corresponding result for the interior case was recently proved by Wang and Zhang, see also Phuc’s paper. For local regularity up to the flat part of the boundary, q = 3 was established by G. A. Seregin. Our result can be viewed as an extension of it to L3,q with q finite. New scale-invariant bounds, refined pressure decay estimates near the boundary and development of a convenient new ϵ-regularity criterion, are central themes in providing this extension.

In this paper, we extend a Gonchar–Grigorjan type estimate for the norm of the holomorphic parts of meromorphic functions in finitely connected Jordan domains with C2-smooth boundary in the case where the poles are in a compact set. A uniform estimate for Cauchy type integrals is also given.

In the paper, for a Lame type system the inverse problem of recovering the fast and slow wave velocities from the boundary dynamical data (the response operator) is solved. The velocities are determined in a near-boundary domain, the depth of determination being proportional to the observation time. The BC-method, which is an approach to inverse problems based on their connections with boundary control theory, is used.

This paper considers elliptic problems with high-order derivatives multiplied by a small parameter. We found the algebraic conditions for an operator and the boundary conditions that guarantee the Fredholm property. An a priori estimate for the solution with a constant independent of the small parameter is proved. These results are known for elliptic boundary-value problems with small parameter in the half-space Rn+. We extend them to the case of bounded domains with smooth boundary. The small-parameter coercive conditions are formulated, and a two-sided estimate is proved.

Local estimates are derived for solutions to a perturbed Stokes system, which arises as a reduction of the Stokes system near a curved part of the boundary of the domain if a diffeomorphism flattening the boundary is applied. The estimates obtained in the paper play the crucial role in the investigation of the partial regularity of weak solutions to the Navier--Stokes system near the curved part of the boundary. Bibliography: 8 titles.

In this paper, the asymptotic behavior of solutions ue of the Poisson equation in the e-periodically perforated domain Ωe ⊂ \( {{\mathbb{R}}^n} \), n ≥ 3, with the third nonlinear boundary condition of the form ∂νue + e−γσ(x, ue) = e −γg(x) on a boundary of cavities, is studied. It is supposed that the diameter of cavities has the order eα with α > 1 and any γ. Here, all types of asymptotic behavior of solutions ue, corresponding to different relations between parameters α and γ, are studied.

Let Ω be a bounded open and oriented connected subset of ℝ n which has a compact topological boundary Γ, let C be the Dirac operator in ℝ n , and let ℝ0,n be the Clifford algebra constructed over the quadratic space ℝ n . An ℝ0,n -valued smooth function f : Ω → ℝ0,n in Ω is called monogenic in Ω if Df = 0 in Ω. The aim of this paper is to present the most general condition on Γ obtained so far for which a Holder continuous function f can be decomposed as F + − F − = f on Γ, where the components F ± are extendable to monogenic functions in Ω± with Ω+ := Ω, and Ω− := ℝ n \ (Ω ⋃ Γ), respectively.

In the present paper, it is shown that the dynamical boundary data (the response operator) that correspond to measurements on the boundary of a Riemannian manifold, do determine the distances (wave travel times) from the boundary points to an interior source with given semigeodesic coordinates. The procedure that finds these distances is in principle amenable to numerical realization. Bibliography: 4 titles.

The aim of this paper is to extend the results on regional stability developed by the authors [17] to the case where the target subregion is a surface. We consider a linear distributed parameter system, and we explore the problem of stabilizing this system on a subregion (internal or boundary) of the spatial domain. We give sufficient conditions ensuring the regional stabilizability and we characterize the minimum cost stabilizing control. Illustrative examples are given.

In this paper, we discuss a probabilistic approach to construction of a solution of a free boundary problem for parabolic and integro-differential equations. This problem admits an interpretation as an optimization problem for a stochastic process with diffusion and jumps satisfying a stochastic differential equation. The results are applied to calculation of American option prices in the Black–Scholes and Merton models. Bibliography: 22 titles.

The paper studies the region of values Dm,n(T) of the system {f(z1), f(z2),..., f(zm), f(r1), f(r2),..., f(rn)}, where m ≥ 1; n > 1; zj, j = 1, ... m, are arbitrary fixed points of the disk U = {z: ¦z¦ 0 for Im z ≠= 0, z ∈ U. An algebraic characterization of the set Dm,n(T) in terms of nonnegative-definite Hermitian forms is provided, and all the boundary functions are described. As an implication, the region of values of f(z1) in the subclass of functions f ∈ T with prescribed values f(rj) (j = 1, 2, 3) is determined. Bibliography: 12 titles.

This paper considers the Schwarz problem that consists in finding a J-analytic function by its real part on the boundary. The Fredholm solvability of this problem is proved. The integral representation of J-analytic functions by Cauchy-type integrals with real density is obtained.

Let $$ F \in \mathbb{C}{\left[ {X,\,Y} \right]^2} $$ be an etale map of degree deg F = d. An etale map $$ G \in \mathbb{C}{\left[ {X,Y} \right]^2} $$ is called a d-inverse approximation of F if deg G ≤ d and F ◦ G =(X + A(X, Y), Y + B(X, Y)) and G ◦ F =(X + C(X, Y), Y + D(X, Y)), where the orders of the four polynomials A, B, C, and D are greater than d. It is a well-known result that every $$ {\mathbb{C}^2} $$ -automorphism F of degree d has a d-inverse approximation, namely, F −1. In this paper, we prove that if F is a counterexample of degree d to the two-dimensional Jacobian conjecture, then F has no d-inverse approximation. We also give few consequences of this result. Bibliography: 18 titles.

In this paper, we present a simple method to prove the correctness and self-adjointness of operators B 3 corresponding to some boundary problems. We also give the unique solutions for these problems. The algorithm is easy to implement via computer algebra systems. In our examples, Derive and Mathematica were used. Bibliography: 10 titles.

The general construction for finding all “essentially different” nonlinearities in equations of mathematical physics is exemplified by the inverse problem of the Grad–Shafranov equation. We present an algorithm that allows one to recover relatively fast all essentially different sought-for nonlinear right-hand sides of the Grad–Shafranov equation. We present the first example of a domain with smooth boundary for which the inverse problem has at most one solution in the class of affine functions, and also in the class of exponential functions. We select some subset of simply connected domains that model, in some sense, the so-called doublet configurations, for which the inverse problem has at least two essentially different solutions in the class of analytic functions. In the concluding subsection of the paper, we indicate a method of recovery, from boundary data, of all essentially different nonlinearities in equations of mathematical physics of considerably general kind, which includes a system of equations that describes combustion and detonation processes.

The aim of this paper is to consider an output controllability problem. It consists in driving the state of a distributed parabolic system toward a state between two prescribed functions on a boundary subregion of the system evolution domain with minimum energy control. Two necessary conditions are given. The first is formulated in terms of the subdifferential associated with a minimized functional. The second is formulated as a system of equations for arguments of the Lagrange systems. Numerical illustrations show the efficiency of the second approach and lead to open questions.

The aim of this paper is to study differentiable functions of several variables defined on a domain G ⊂ ℝ with irregular boundary. We suggest a new integral representation, which allows us to establish estimates for L p -modules of continuity and embedding theorems for functional spaces that have defined L p -modules of continuity behavior.

In this paper, we study linear-fractional models for one-parameter semigroups of holomorphic mappings via Schroder’s and Abel’s functional equations. By using some limit schemes in the spirit of Kœnigs to solve those equation, we obtain new results on the asymptotic behavior of one-parameter semigroups having a boundary Denjoy-Wolff fixed point. In addition, we establish infinitesimal versions of the Burns-Krantz rigidity theorem for semigroups and their generators.

We generalize the class of contractive-valued functions analytic in the unit disk (Schur class) to the case of finitely connected domains with operator-valued weights on the boundary. The two fundamental decompositions (the inner-outer factorization and the decomposition into the orthogonal sum of the pure and unitary parts) are extended to our setting. This generalization allows one to use a uniform approach to functional models developed recently in papers of the author and D. Yakubovich. Moreover, it gives an alternative language to handle multi-valued functions in the multiply connected case.

In the paper, we study the inverse problem of finding the solution u and the coefficient q from the following data: $$Mu = u_t - L(x,t,D_x )u + g(x,t,u,\nabla u) + q(x,t)u(x,t) = f(x,t), (x,t) \in Q = G \times (0,T), u/s = \varphi (x,t), \left. {\frac{{\partial u}}{{\partial n}}} \right|_s = \psi (x,t), \left. u \right|_{t = 0} = u_0 (x), S = \Gamma \times (0,T),$$ where G ⊂ ℝn is a bounded domain with boundary Γ and L is a second-order elliptic operator. We prove that the problem is locally solvable in time or in the case where the norms of its data are sufficiently small.

The paper refines and supplements some classical and modern distortion theorems for functions regular in the disk |z| < 1 and satisfying the condition |f(z)| < 1 for |z| < 1. Distortion bounds at points of the disk |z| < 1 and also on its boundary |z| = 1, provided that |f(z)| = 1, are considered both for univalent and multivalent mappings. Bibliography: 22 titles.

The attainable set of a linear control system can have both smooth and nonsmooth boundary. This smoothness property is known to be used to classify such systems. One approach, suggested by A. I. Ovseevich in the case of a smooth control set, is based on connecting smoothness of the attainable set with spherical observability of the dual system. This paper generalizes these results to the case of nonsmooth control sets. The corresponding property of the spherical observability notion can be treated as observability with several bearing-only observations.