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The diffusion equation is considered in an infinite 1-periodic medium. We find large-time approximations for its fundamental solution. The approximation precision has pointwise and integral estimates of orders $$ O\left({t}^{-\frac{d+j+1}{2}}\right) $$ and $$ O\left({t}^{-\frac{j+1}{2}}\right) $$ , j = 0, 1, …, respectively. The approximations are constructed on the base of the known fundamental solution of the averaged equation with constant coefficients, its derivatives, and solutions of a family of auxiliary problems on the periodicity cell. The family of problems on the cell is generated recurrently. These results are used to construct approximations of the operator exponential of the diffusion equation with precision estimates in operator norms in Lp-spaces, 1 ≤ p ≤ ∞. For the analogous equation in an e-periodic medium, where e is a small parameter, we obtain approximations of the operator exponential in Lp-operator norms for a fixed time with precision of order O(en), n = 1, 2, …

The diffusion equation is considered in an infinite 1-periodic medium. For its fundamental solution we find approximations at large values of time t. Precision of approximations has pointwise and integral estimates of orders O(t(-d+j+1)/2) and O(t(-j+1)/2), j=0,1,…, respectively. Approximations are constructed based on the known fundamental solution of the averaged equation with constant coefficients, its derivatives, and solutions of a family of auxiliary problems on the periodicity cell. The family of problems on the cell is generated recurrently. These results are used for construction of approximations of the operator exponential of the diffusion equation with precision estimates in operator norms in Lp-spaces, 1≤p≤∞. For the analogous equation in an ε-periodic medium (here ε is a small parameter) we obtain approximations of the operator exponential in Lp-operator norms for a fixed time with precision of order O(εn), n=1,2,….

We consider the Γ-convergence of a sequence of integral functionals Fn(u), defined on the functions u from the Sobolev space W1,α(Ω) (α > 1); Ω is a bounded Lipschitz domain, where the integrand fn(x, u,∇u) depends on a function u and its gradient ∇u. As functions of ξ, the integrands fn(x, s, ξ) are convex and satisfy a two-sided power estimate on the coercivity and growth with different exponents α < β. Moreover, the integrands fn(x, s, ξ) are equi-continuous over s in some sense with respect to n. We prove that for the functions from L ∞ (Ω) ∩ W1,β(Ω) the Γ-limit functional coincides with an integral functional F(u) for which the integrand f(x, s, ξ) is of the same class as fn(x, s, ξ).

We study the convergence of continuous spectrum eigenfunctions for differential operators of divergence type with e-periodic coefficients, where e is a small parameter. Two cases are considered, the case of classical homogenization, where the coefficient matrix satisfies the ellipticity condition uniformly with respect to e, and the case of two-scale homogenization, where the coefficient matrix has two phases and is highly contrast with hard-to-soft-phase contrast ratio 1: e2.

Differential operators corresponding to elliptic equations of divergent type with 1-periodic coefficients are considered. The equations are put in Sobolev spaces with an arbitrary 1-periodic Borel measure on the entire space R d . In the study of the spectrum of operators of this kind, the Bloch principle is of fundamental importance. According to this principle, all points of the desired spectrum are obtained when studying the equation on the unit cube with quasiperiodic boundary conditions. The proof of the Bloch principle for problems in the above formulation is proved, in several versions of the principle. Examples of the application of the principle to finding the spectrum of specific operators, for example, for the Laplacian in a weighted space or on a singular structure of lattice type.

We study the Γ-convergence as e→0 of a family of integral functionals with integrand f{sub e}(x,u,∇u), where the integrand oscillates with respect to the space variable x. The integrands satisfy a two-sided power estimate on the coercivity and growth with different exponents. As a consequence, at least two different variational Dirichlet problems can be connected with the same functional. This phenomenon is called Lavrent'ev's effect. We introduce two versions of Γ-convergence corresponding to variational problems of the first and second kind. We find the Γ-limit for the aforementioned family of functionals for problems of both kinds; these may be different. We prove that the Γ-convergence of functionals goes along with the convergence of the energies and minimizers of the variational problems. Bibliography: 23 titles. (paper)

We prove Korn-type inequalities for thin periodic structures of period $\varepsilon$ and thickness $\varepsilon h(\varepsilon)$, where $h(\varepsilon)\to 0$ as $\varepsilon\to 0$, among which there are plane grids, spatial rod and box structures. These inequalities are important in homogenization of corresponding elasticity problems.

Elliptic equations of -Laplacian type are investigated. There is a?well-known logarithmic condition on the modulus of continuity of the nonlinearity exponent , which ensures that a?Laplacian with variable order of nonlinearity inherits many properties of the usual -Laplacian of constant order. One of these is the so-called improved integrability of the gradient of the solution. It is proved in this paper that this property holds also under a?slightly more general condition on the exponent , although then the improvement of integrability is logarithmic rather than power-like. The method put forward is based on a?new generalization of Gehring's lemma, which relies upon the reverse H?lder inequality with increased support and exponent on the right-hand side. A counterexample is constructed that reveals the extent to which the condition on the modulus of continuity obtained is sharp. Bibliography: 28 titles.

We consider the divergent elliptic equations whose weight function and its inverse are assumed locally integrable. The equations of this type exhibit the Lavrentiev phenomenon, the nonuniqueness of weak solutions, as well as other surprising consequences. We classify the weak solutions of degenerate elliptic equations and show the attainability of the so-called W-solutions. Investigating the homogenization of arbitrary attainable solutions, we find their different asymptotic behavior. Under the assumption of the higher integrability of the weight function we estimate the difference between the exact solution and certain special approximations.

In this paper, the proof of a Trotter-Kato type theorem in a variable Banach space is given and some special cases and examples are considered.

A parabolic equation with asymmetric elliptic operator of the second order is considered. Estimates of the Nash-Aronson type are established in the case when the skew-symmetric part of the heat matrix is unbounded. These estimates are used in the proof of the central limit theorem for diffusion in an incompressible stochastic flow.

A new approach is proposed to the study of plane problems of elasticity on rod junctures. This approach is based on so-called structural theorems and yields a fairly simple method of passing to the limit in the integral identity and thus obtaining the limit problem. In the case of nonlinear problems, the structural theorems make it possible to use variational methods like that of Γ-convergence of energy functionals.

The asymptotic behavior of the operator exponent related to the Cauchy problem for a parabolic equation with periodic coefficients is studied either under the reduction of the periodicity cell or for large times. Estimates for the closeness of the operator exponentials (the original and the limit) with respect to the L2-operator norm and the related H1-estimates are obtained under minimal assumptions concerning the smoothness of the heat matrix and of the initial data.

Korn-type inequalities for thin periodic structures of period e and width eh(e) with h(e) → 0 are presented. Periodic meshes, three-dimensional road structures, and three-dimensional box structures are considered. A particular attention is paid to structures with the so-called critical width when \(\mathop {lim}\limits_{\varepsilon \to 0} h\left( \varepsilon \right)\varepsilon ^{ - 1} > 0\).

Holder continuity of solutions is proved for a new class of degenerate divergent second-order elliptic equations with weight not satisfying the Muckenhoupt condition. There is no Harnack inequality and no Sobolev embedding theorems with higher summation exponent for these equations. As an example an equation is considered in a domain divided into two parts by a hyperplane. In each part the weight function is a power one, the powers are different, and their absolute values do not exceed the space dimension.

The concept of two-scale convergence associated with a fixed periodic Borel measure is introduced. In the case when is Lebesgue measure on the torus convergence in the sense of Nguetseng-Allaire is obtained. The main properties of two-scale convergence are revealed by the simultaneous consideration of a sequence of functions and a sequence of their gradients. An application of two-scale convergence to the homogenization of some problems in the theory of porous media (the double-porosity model) is presented. A mathematical notion of softly or weakly coupled parallel flows is worked out. A homogenized operator is constructed and the convergence result itself is interpreted as a strong two-scale resolvent convergence. Problems concerning the behaviour of the spectrum under homogenization are touched upon in this connection.

The case when smooth functions are not dense in a weighted Sobolev space is considered. New examples of the inequality (where is the closure of the space of smooth functions) are presented. We pose the problem of 'viscosity' or 'attainable' spaces (that is, spaces that are in a certain sense limits of weighted Sobolev spaces corresponding to 'well-behaved' weights, which means weights bounded above and away from zero) such that . A precise definition of this property of 'attainability' is given in terms of the convergence of the solutions of the corresponding elliptic equations. It is proved that an attainable space always exists, but does not in general coincide with the extreme spaces and . Examples of strict inclusions are presented.