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In this paper, we study the eta-invariant of elliptic parameter-dependent boundary-value problems and its main properties. Using Melrose's approach, we define the eta-invariant as a regularization of the winding number of the family. In this case, the regularization of the trace requires obtaining the asymptotics of the trace of compositions of invertible parameter-dependent boundaryvalue problems for large values of the parameter. Obtaining the asymptotics uses the apparatus of pseudodifferential boundary-value problems and is based on the reduction of parameter-dependent boundary-value problems to boundary-value problems with no parameter. [ABSTRACT FROM AUTHOR]

We study spectral stability estimates for the Dirichlet eigenvalues of the Laplace operator in bounded two-connected planar domains. We propose a method based on the conformal analysis of elliptic operators. Owing to this method, it is possible to obtain spectral stability estimates in domains with nonrectifiable boundaries. [ABSTRACT FROM AUTHOR]

In the unit disc, we consider the Dirichlet problem for second order elliptic system with only higher order constant coefficients. We establish the unique solvability of this problem under the assumption that the problem is Fredholm, and obtain an explicit formula for the solution. [ABSTRACT FROM AUTHOR]

We prove the validity of a regularizing property on the boundary of the double layer potential associated with the fundamental solution of a nonhomogeneous second order elliptic differential operator with constant coefficients in Schauder spaces of exponent greater or equal to two that sharpens classical results of N.M. Günter, S. Mikhlin, V.D. Kupradze, T.G. Gegelia, M.O. Basheleishvili and T.V. Burchuladze, U. Heinemann and extends the work of A. Kirsch who has considered the case of the Helmholtz operator. [ABSTRACT FROM AUTHOR]

We consider the generalized Neumann problem for a 2lth-order elliptic equation with constant real higher-order coefficients in an infinite domain containing the exterior of some circle and bounded by a sufficiently smooth contour. It consists in specifying of the (kj − 1)th-order normal derivatives where 1 ≤ k1 <... < kl ≤ 2l; for kj = j it turns into the Dirichlet problem, and for kj = j + 1 into the Neumann problem. Under certain assumptions about the coefficients of the equation at infinity, a necessary and sufficient condition for the Fredholm property of this problem is obtained and a formula for its index in Hölder spaces is given. [ABSTRACT FROM AUTHOR]

In L2(R d; C n), we consider a wide class of matrix elliptic operators A ε of order 2p (where p ≥ 2) with periodic rapidly oscillating coefficients (depending on x/ε). Here ε > 0 is a small parameter. We study the behavior of the operator exponential e - A ε τ for τ > 0 and small ε. It is shown that the operator e - A ε τ converges as ε → 0 in the operator norm in L2(R d; C n) to the exponential e - A 0 τ of the effective operator A 0. Also we obtain an approximation of the operator exponential e - A ε τ in the norm of operators acting from L2(R d; C n) to the Sobolev space Hp(R d; C n). We derive error estimates for these approximations depending on two parameters: ε and τ. For a fixed τ > 0, the errors are of the sharp order O(ε). The results are applied to study the behavior of the solution of the Cauchy problem for the parabolic equation ∂τuε(x, τ) = −(A εuε)(x, τ) + F(x, τ) in R d. [ABSTRACT FROM AUTHOR]

We consider a Schrödinger operator on a model graph with small loops assuming the violation of the typical nonresonance condition which guarantees the holomorphy property of the resolvents of elliptic operators on graphs with small edges in the general case. We study the behavior of parts of the resolvent corresponding to finite edges and small loops in a certain sense. We show that for the parts of the resolvent the holomorphy property is preserved with respect to a small parameter characterizing the length of small loops. Unlike the nonresonance case, for the part of the resolvent corresponding to small loops an additional term appears in the leading term of the Taylor series, which leads to a certain localization of the resolvent on these loops. [ABSTRACT FROM AUTHOR]

We consider q-nonlinear elliptic systems with polynomial growth q > 2 and establish the local regularity of (possibly) unbounded weak solutions under a one-sided constraint on strongly nonlinear terms. [ABSTRACT FROM AUTHOR]

We consider special classes of operators acting in functional spaces on manifolds. We can say that our approach is an operator-geometric treatment of the well-known locality principle. We described the Fredholm property in an abstract form and show how these results can be applied to the study of elliptic pseudodifferential operators on manifolds with nonsmooth boundaries. [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 present some results concerning the solvability of linear elliptic equations in bounded domains with the main coefficients almost in VMO, the drift and the free terms in Morrey classes containing Ld, and bounded zeroth order coefficient. We prove that the second-order derivatives of solutions are in a local Morrey class contained in Lp,loc. Actually, the exposition is given for fully nonlinear equations and encompasses the above mentioned results, which are new even if the main part of the equation is just the Laplacian. [ABSTRACT FROM AUTHOR]

We construct two examples of domains where the improved integrability of the gradient of a solution to the Zaremba problem for a second order elliptic operator can be estimated. [ABSTRACT FROM AUTHOR]

We obtain estimates for Dirichlet eigenvalues of divergence form elliptic operators −div [A(z)∇f(z)] in bounded non-Lipschitz domains. We propose a method based on the quasiconformal composition operators on Sobolev spaces with application to weighted Poincaré–Sobolev inequalities. [ABSTRACT FROM AUTHOR]

We consider second order elliptic operator with ε-periodic measurable coefficients in divergence form, acting in the space ℝd. For the resolvent of this operator we construct approximations in the operator norm ⋅ H 1 → L 2 with the remainder of order ε3 as ε → 0. [ABSTRACT FROM AUTHOR]

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 fail near the boundary of these subdomains even for an 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. Also there was obtained the smoothness (in Sobolev spaces W 2 k ) of generalized solutions of the second and the third boundary-value problems for strongly elliptic differential-difference equations in subdomains excluding ε-neighborhoods of certain points. However, the smoothness (in Hölder 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 Hölder 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 Hölder space. [ABSTRACT FROM AUTHOR]

We study homogenization of a second-order elliptic differential operator Aε = −div a(x/ε) ×∇ acting in an ε-periodically perforated space, where ε is a small parameter. Coefficients of the operator Aε are measurable ε-periodic functions. The simplest case where coefficients of the operator are constant is also interesting for us. We find an approximation for the resolvent (Aε + 1)−1 with remainder term of order ε2 as ε → 0 in operator L2-norm on the perforated space. This approximation turns to be the sum of the resolvent (A0 + 1)−1 of the homogenized operator A0 = −div a0∇, a0> 0 being a constant matrix, and some correcting operator εCε. The proof of this result is given by the modified method of the first approximation with the usage of the Steklov smoothing operator. [ABSTRACT FROM AUTHOR]

We consider a general nonselfadjoint second order scalar operator in a multi-dimensional domain with thin spike of finite length. The Dirichlet or Neumann boundary condition is imposed on the boundary. We prove the norm resolvent convergence of such operators to limit operators and estimate the convergence rate in two different operator norms. In the case of the Neumann boundary condition, we show that a one-dimensional operator acting along the spike appears in the limit. Based on the norm resolvent convergence, we establish the convergence of the spectra of perturbed operators to the spectra of the limit ones. In the selfadjoint case, we prove the convergence of spectral projections. [ABSTRACT FROM AUTHOR]

We consider a second-order differential-difference equation in a bounded domain Q ⊂ ℝn. We assume that the differential-difference operator contains difference operators with degenerations corresponding to the differential operators. Also, it is assumed that the considered differential-difference operator cannot be expressed by a composition of a difference operator and a strongly elliptic differential operator. The presence of degenerate difference operators does not allow us to obtain the Gårding inequality. We prove a priori estimates implying that the considered differential-difference operator is sectorial and its Friedrichs extension exists. These estimates can be applied to study the spectrum of the Friedrichs extension as well. It is well known that elliptic differential-difference equations may have solutions that do not belong even to the Sobolev space W 2 1 Q . However, using the obtained estimates, we prove smoothness of solutions at least in subdomains Qr generated by translations of the boundary, where ∪ r Q r ¯ = Q ¯ . [ABSTRACT FROM AUTHOR]

We discuss the existence of entire (i.e. defined on the whole space) subsolutions of fully nonlinear degenerate elliptic equations, giving necessary and sufficient conditions on the coefficients of the lower order terms which extend the classical Keller–Osserman conditions for semilinear elliptic equations. Our analysis shows that, when the conditions of existence of entire subsolutions fail, a priori upper bounds for local subsolutions can be obtained. [ABSTRACT FROM AUTHOR]

In this paper, we study the unique solvability of linear inverse coefficient problems with a time-independent unknown coefficient for evolution equations in Banach spaces with degenerate operators acting on the Gerasimov–Caputo fractional derivative. We apply abstract results obtained in the paper to the study of inverse problems with undetermined coefficients depending only on spatial variables for equations with polynomials on a self-adjoint, elliptic differential operator with respect to spatial variables. Also, we apply general results to the study of the unique solvability of inverse problems for time-fractional Sobolev systems. [ABSTRACT FROM AUTHOR]

The questions of the existence and uniqueness of solution to the Cauchy problem for an equation in a Banach space of distributed order at most one are investigated. Necessary and sufficient conditions for the existence of a resolving family of operators of this equation analytic in the sector are obtained. An explicit form of these operators is found. Two versions of the theorem on unique solvability of the Cauchy problem for the corresponding inhomogeneous equation are obtained: with condition of increased smoothness in spatial variables (the condition of continuity in the norm of the graph of the generator of the resolving family) and with condition of increased smoothness in the time variable (the Hölder condition). Abstract results are obtained using theory of the Laplace transform and generalization some results from the theory of analytic operator semigroups and its extensions to the case of integral equations and fractional differential equations. The conditions for unique solvability of an equation in a Banach space are used to study a class of initial boundary value problems for equations with polynomials in an elliptic differential operator with respect to spatial variables. [ABSTRACT FROM AUTHOR]

In the half-plane {−∞ < x < +∞} × {0 < y < +∞}, the Dirichlet problem is considered for differential-difference equations of the kind u xx + ∑ k = 1 m a k u xx x + h k , y + 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 → +∞. [ABSTRACT FROM AUTHOR]

For elliptic operators in divergence form with ε-periodic coefficients of an arbitrary even order 2m ≥ 4 we obtain ε2-order approximations of resolvents in the energy operator norm as ε → 0. [ABSTRACT FROM AUTHOR]

A two-dimensional homogeneous diffusion process with break is considered. The distribution of the first exit point of such a process from an arbitrary neighborhood of zero as a function of the initial point of the process is determined by an elliptic second order partial differential equation with constant coefficients and corresponds to the solution of the Dirichlet problem for this equation. A connection of this Dirichlet problem with the distribution density of the first exit point of the process from a small circular neighborhood of zero is established. In terms of this asymptotic, we find necessary and sufficient conditions under which a function of the initial point of the process satisfies a particular second order elliptical partial differential equation that corresponds to a standard Wiener process with drift and break. [ABSTRACT FROM AUTHOR]

We prove the existence, uniqueness, and stability of a solution to the Dirichlet problem in a domain Ω ⊂ ℝn, n ≥ 1, for a nonlinear elliptic operator with potentials in the space of multiplicators, where Ω is an unbounded domain or coincides with ℝn. [ABSTRACT FROM AUTHOR]

We study homogenization of fourth order elliptic operators Aε in divergence form with ε-periodic coefficients in ℝd and obtain an ε2-order approximation of the resolvents (Aε + 1)−1 in the energy operator (L2→H2)-norm as ε → 0. [ABSTRACT FROM AUTHOR]

Existence of solutions of reaction-diffusion systems of equations in unbounded domains is studied by the Leray–Schauder (LS) method based on the topological degree for elliptic operators in unbounded domains and on a priori estimates of solutions in weighted spaces. We identify some reaction-diffusion systems for which there exist two subclasses of solutions separated in the function space, monotone and nonmonotone solutions. A priori estimates and existence of solutions are obtained for monotone solutions allowing to prove their existence by the LS method. Various applications of this method are given. [ABSTRACT FROM AUTHOR]

We study homogenization of a divergence form elliptic operator Aε of order 2m ≥ 4 with ε-periodic coefficients, where ε is a small parameter. The homogenized operator  is of the same type as Aε, but with constant coefficients. For selfadjoint operators Aε without lower order terms we obtain an estimate of order ε2 for the difference of resolvents (Aε + 1)−1 and ( + 1)−1 in the operator (L2→L2)–norm. In the non-selfadjoint case, we find an approximation (Aε +1)−1 = (Â+1)−1 + ε𝒦1 + O (ε2) with corrector 𝒦1. To prove the operator estimate, we use the shift method. [ABSTRACT FROM AUTHOR]

We study homogenization of a singularly perturbed second order elliptic operator Aε acting in the whole space ℝd with ε-periodic coefficients as ε → 0. Perturbation is generated by a fourth order elliptic operator, for example, by the bi-Laplacian Δ2 with a small parameter ε2. We find approximations of the resolvent (Lε + 1)−1 of order ε and ε2 in different operator norms. Bibliography: 28 titles. [ABSTRACT FROM AUTHOR]

We consider the scalar second order elliptic differential operator in a multidimensional cylinder with the Dirichlet boundary condition. The coefficients of the operator are periodic on both outlets of the cylinder and are arbitrary in its finite part. We study a local analytic continuation of the bordered resolvent of the operator from the upper half-plane to the lower one with respect to the spectral parameter in a neighborhood of an interior point of the essential spectrum. It is shown that the size of the neighborhood depends only on geometric properties of the cylinder and the behavior of periodic components of coefficients of the operator. We introduce the notion of a resonance and formulate the corresponding boundary value problems. We describe the behavior of the bordered resolvent with respect to the spectral parameter in the neighborhood of the point of the essential spectrum. [ABSTRACT FROM AUTHOR]

We consider a class of nonlinear parabolic systems for elliptic operators of variational structure with nondiagonal principal matrices. Additional terms in the systems can have quadratic growth with respect to the gradient and arbitrary polynomial growth with respect to solutions. We obtain sufficient conditions for the time-global weak solvability of the Cauchy–Dirichlet problem and study the regularity of the solution. The case of two spatial variables is considered. [ABSTRACT FROM AUTHOR]

The monograph consists of three parts. Part I is presented here. In this monograph, we develop a new approach (mainly based on papers of the author). Many results are published for the first time here. Chapter 1 is introductory. It provides the necessary background from functional analysis (for completeness). In this monograph, we mostly use weighted HÖlder spaces; they are considered in Chap. 2. Chapter 3 plays the key role: in weighted HÖlder spaces, we consider estimates of integral operators with homogeneous difference kernels, covering potential-type integrals and singular integrals as well as Cauchy-type integrals and double layer potentials. In Chap. 4, similar estimates in weighted Lebesgue spaces are proved. Integrals with homogeneous difference kernels will play an important role in Part III of the monograph, which will be devoted to elliptic boundary-value problems. They naturally arise in integral representations of solutions of first-order elliptic systems in terms of fundamental matrices or their parametrices. The investigation of boundary-value problems for second-order and higher-order elliptic equations or systems is reduced to first-order elliptic systems. [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 study homogenization of second order elliptic differential operators Aε whose coefficients are ε-periodic and rapidly oscillate as ε→0. For the difference between the resolvent (Aε + 1)−1 and its approximations we prove the operator estimate of order ε2 in the operator L2-norm by using the first approximation method and the Steklov smoothing operator. [ABSTRACT FROM AUTHOR]

We consider a selfadjoint elliptic operator in an infinite multidimensional cylinder with the Dirichlet boundary condition which is replaced by the Robin condition on small sets located along a given line on the boundary. The shape and distribution of these sets are arbitrary. The characteristic linear size of these sets is a small parameter of the problem. It is shown that the resolvent of such an operator converges to the resolvent of the homogenized operator, and an estimate for the convergence rate is obtained. The homogenized operator is the same operator, but without alternating boundary conditions. The difference of resolvents is estimated in the norm of bounded operators acting from L2 to W 2 1. If the small sets are periodically distributed and have the same shape and the Robin condition is replaced by the Neumann condition, we derive two-sided asymptotic estimates for the lower band functions, which provides the possibility to estimate from below the length of the first band of the spectrum. [ABSTRACT FROM AUTHOR]

In two dimensions, we present a new approach to the study of the semilinear equations of the form div[A(z)∇u] = f(u), the diffusion term of which is the divergence uniform elliptic operator with measurable matrix functions A(z), whereas its reaction term f(u) is a continuous non-linear function. Assuming that f(t)/t → 0 as t → ∞, we establish a theorem on existence of weak C D ¯ ∩ W loc 1 , 2 D solutions of the Dirichlet problem with arbitrary continuous boundary data in any bounded domains D without degenerate boundary components. As consequences, we give applications to some concrete model semilinear equations of mathematical physics, arising from modeling processes in anisotropic and inhomogeneous media. With a view to the further development of the theory of boundary-value problems for the semilinear equations, we prove a theorem on the solvability of the Dirichlet problem for the Poisson equation in Jordan domains with arbitrary boundary data that are measurable with respect to the logarithmic capacity. [ABSTRACT FROM AUTHOR]

This paper is devoted to differential-difference equations with degeneration in a bounded domain Q ⊂ ℝn. We consider differential-difference operators that cannot be expressed as a composition of a strongly elliptic differential operator and a degenerated difference operator. Instead of this, the operators under consideration contain several degenerate difference operators corresponding to differential operators. Generalized solutions of such equations may not belong even to the Sobolev space W 2 1 Q . Earlier, under certain conditions on the difference and differential operators, we obtained a priori estimates and proved that, instead of the whole domain, the orthogonal projection of the generalized solution to the image of the difference operator preserves certain smoothness inside some subdomains Q r ⊂ Q U r Q ¯ r = Q ¯ . In this paper, we prove necessary and sufficient conditions in algebraic form for the existence of traces on parts of boundaries of subdomains Qr. [ABSTRACT FROM AUTHOR]

We study the connection between the stabilization of solutions of a mixed hyperbolic problem and spectral properties of the corresponding elliptic boundary value problem. We consider the first mixed problem for the wave equation in bounded and unbounded domains in ℝ, determine the class of its energy solutions, and represent the solutions in terms of the Bochner-Stieltjes integral. We study how the spectrum of the elliptic operator affects the behavior of local energy of a solution and describe a method which allows us to study the stabilization of solutions with the help of estimates in the spectral parameter for solutions of the stationary problem on the upper half-plane. [ABSTRACT FROM AUTHOR]

In the present work we study elliptic operators on manifolds with singularities in the situation where the manifold is endowed with an action of a discrete group G. As usual in elliptic theory, the Fredholm property of an operator is governed by the properties of its principal symbol. We show that the principal symbol in our situation is a pair consisting of the symbol on the main stratum (interior symbol) and the symbol at the conical point (conormal symbol). The Fredholm property of elliptic elements is obtained. [ABSTRACT FROM AUTHOR]

We consider a second order elliptic operator with variable coefficients in a multidimensional domain with a small hole and some classical boundary condition on the hole boundary. We show that the resolvent of this operator converges to the resolvent of the limit operator in the domain without holes in the sense of the norm of bounded operators acting from L2 to W21. For the convergence rate we obtain sharp estimates relative to the smallness order. [ABSTRACT FROM AUTHOR]

We establish necessary conditions for the stabilization to zero of solutions to the Dirichlet problem for a parabolic equation with elliptic operator of divergence form and coefficients depending on x and t. [ABSTRACT FROM AUTHOR]

We consider an elliptic operator in a multidimensional domain with frequent alternation of the Dirichlet condition and the Robin boundary condition in the case where the homogenized operator contains only the original Robin boundary condition. We prove the uniform resolvent convergence of the perturbed operator to the homogenized operator and obtain order sharp estimates for the rate of convergence. We construct a complete asymptotic expansion for the resolvent in the case where the resolvent acts on sufficiently smooth functions and the alternation of boundary conditions is strictly periodic and is given on a multidimensional hyperplane. Bibliography: 23 titles. [ABSTRACT FROM AUTHOR]

We establish the solvability of time-nonlocal boundary value problems for degenerate Sobolev type equations with an elliptic-parabolic second order operator at the timederivative. We prove the uniqueness of a regular solution. Bibliography: 10 titles. [ABSTRACT FROM AUTHOR]

We prove an interpolation theorem for the tensor products of approximation spaces associated with regular elliptic operators and show their applicability to the problem of approximation of elements of the tensor products of L -spaces. [ABSTRACT FROM AUTHOR]

We give an elementary solution to the problem of the index of elliptic operators associated with shift operator along the trajectories of an isometric diffeomorphism of a smooth closed manifold. This solution is based on index-preserving reduction of the operator under consideration to some elliptic pseudo-differential operator on a higher-dimension manifold and on the application of the Atiyah-Singer formula. The final formula of the index is given in terms of the symbol of the operator on the original manifold. [ABSTRACT FROM AUTHOR]

We study the spectral properties of a boundary-value problem for an elliptic second-order operator with singularly perturbed coefficients. The problem simulates resonance vibrations of an elastic system with finitely many stiff and, at the same time, light-weight inclusions of any shape. The ratio of the stiffness coefficients of the inclusions and the main body has an order of ε as ε → 0, whereas the ratio of their mass densities has an order of ε, where κ > 0. The complete asymptotic expansions of the eigenvalues and eigenfunctions of this problem are constructed and justified. [ABSTRACT FROM AUTHOR]

Basic properties of weak and strong two-scale convergence are established for Lebesgue and Sobolev spaces, in particular, those with periodic measures depending on a small parameter. These results are applied to the homogenization of elliptic operators and spectral problems. [ABSTRACT FROM AUTHOR]

We obtain a necessary and sufficient condition for the Hölder regularity of a boundary point O with respect to an elliptic second order differential operator. Sufficient conditions for the Hölder regularity of O are discussed. [ABSTRACT FROM AUTHOR]

We consider a linear second order divergence elliptic equation in a neighborhood of the plane part of the boundary. The boundary condition involves the second order derivatives with respect to tangent variables and the conormal derivative (a Venttsel type problem). We study the regularity of the solution and its derivatives in a neighborhood of the boundary in Campanato spaces and establish regularity in Hölder spaces. The results remain valid for linear systems with strongly elliptic matrices. Bibliography: 10 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]