Vol 67, No 1 (2021): Partial Differential Equations


Nonexistence of Nontrivial Weak Solutions of Some Nonlinear Inequalities with Gradient Nonlinearity

Admasu V.E., Galakhov E.I., Salieva O.A.


In this article, we modify the results obtained by Mitidieri and Pohozaev on sufficient conditions for the absence of nontrivial weak solutions to nonlinear inequalities and systems with integer powers of|the Laplace operator and with a nonlinear term of the form a(x)|∇(Δmu)|q+ b(x)|∇u|s. We obtainoptimal a priori estimates by applying the nonlinear capacity method with an appropriate choice of testfunctions. As a result, we prove the absence of nontrivial weak solutions to nonlinear inequalities and systems by contradiction.

Contemporary Mathematics. Fundamental Directions. 2021;67(1):1-13
pages 1-13 views

Asymptotic Analysis of Boundary-Value Problems for the Laplace Operator with Frequently Alternating Type of Boundary Conditions

Borisov D.I.


This work, which can be considered as a small monograph, is devoted to the study of twoand three-dimensional boundary-value problems for eigenvalues of the Laplace operator with frequently alternating type of boundary conditions. The main goal is to construct asymptotic expansions of the eigenvalues and eigenfunctions of the considered problems. Asymptotic expansions are constructed on the basis of original combinations of asymptotic analysis methods: the method of matching asymptotic expansions, the boundary layer method and the multi-scale method. We perform the analysis of the coefficients of the formally constructed asymptotic series. For strictly periodic and locally periodic alternation of the boundary conditions, the described approach allows one to construct complete asymptotic expansions of the eigenvalues and eigenfunctions. In the case of nonperiodic alternation and the averaged third boundary condition, sufficiently weak conditions on the alternation structure are obtained, under which it is possible to construct the first corrections in the asymptotics for the eigenvalues and eigenfunctions. These conditions include in consideration a wide class of different cases of nonperiodic alternation. With further, very serious weakening of the conditions on the structure of alternation, it is possible to obtain two-sided estimates for the rate of convergence of the eigenvalues of the perturbed problem. It is shown that these estimates are unimprovable in order. For the corresponding eigenfunctions, we also obtain unimprovable in order estimates for the rate of convergence.

Contemporary Mathematics. Fundamental Directions. 2021;67(1):14-129
pages 14-129 views

Averaging of Higher-Order Parabolic Equations with Periodic Coefficients

Miloslova A.A., Suslina T.A.


In L2(Rd;Cn), we consider a wide class of matrix elliptic operators Aε 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 exponent e-Aετ for τ > 0 and small ε. We show that the operator e-Aετ converges as ε → 0 in the operator norm in L2(Rd;Cn) to the exponent e-A0τ of the effective operator A0. Also we obtain an approximation of the operator exponent e-Aετ in the norm of operators acting from L2(Rd;Cn) to the Sobolev space Hp(Rd; Cn). We derive estimates of errors of these approximations depending on two parameters: ε and τ. For a fixed τ > 0 the errors have the exact order O(ε). We use the results to study the behavior of a solution of the Cauchy problem for the parabolic equation τuε(x,τ)= -(Aε uε)(x,τ)+F(x,τ) in Rd.

Contemporary Mathematics. Fundamental Directions. 2021;67(1):130-191
pages 130-191 views

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