Vol 232, No 3 (2018)
- Year: 2018
- Articles: 11
- URL: https://journals.rcsi.science/1072-3374/issue/view/14930
Article
V. V. Zhikov (1940–2017)
Regularity of a Boundary Point for the p(x)-Laplacian
Abstract
We study the behavior of solutions to the Dirichlet problem for the p(x)-Laplacian with a continuous boundary function. We prove the existence of a weak solution under the assumption that p is separated from 1 and ∞. We present a necessary and sufficient Wiener type condition for regularity of a boundary point provided that the exponent p has the logarithmic modulus of continuity at this point.
Regularity of Solutions to Quasilinear Parabolic Systems with Time-Nonsmooth Principal Matrix and the Neumann Boundary Condition
Abstract
We consider a quasilinear parabolic system of equations with nondiagonal principal matrix in a model parabolic cylinder with the Neumann condition on the plane part Γ of the lateral surface of the cylinder. We prove the partial regularity (the Hölder continuity) of the weak solution in a neighborhood of Γ by the method of A(t)-caloric approximations adapted to the problem with the Neumann boundary condition.
The Poisson Equation and Estimates for Distances Between Stationary Distributions of Diffusions
Abstract
We estimate distances between stationary solutions to Fokker–Planck–Kolmogorov equations with different diffusion and drift coefficients. To this end we study the Poisson equation on the whole space. We have obtained sufficient conditions for stationary solutions to satisfy the Poincaré and logarithmic Sobolev inequalities.
The Norm Resolvent Convergence for Elliptic Operators in Multi-Dimensional Domains with Small Holes
Abstract
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 \( {W}_2^1 \). For the convergence rate we obtain sharp estimates relative to the smallness order.
Equations of Magnetohydrodynamic Boundary Layer for a Modified Incompressible Viscous Medium. Boundary Layer Separation
Abstract
We study the behavior of a magnetohydrodynamic stationary boundary layer of a modified fluid in the sense of Ladyzhenskaya. We study how a magneric force affects the behavior of a continuous medium. We establish the influence of the magnetic field on the point of separation of the boundary layer from the solid streamlined surface.
Extraction of Pairs of Harmonics from Trigonometric Polynomials by Phase-Amplitude Operators
Abstract
We propose a method for extracting the sums of two harmonics from trigonometric polynomials Tn(t) by the method of amplitude-phase transformations. Such transformations send polynomials Tn(t) to similar polynomials by two simplest operations: multiplication by a real constant X and shift by a real phase λ, i.e., Tn(t) → X ・ Tn(t − λ). The harmonics are extracted by adding similar polynomials.
The Stabilization Rate of Solutions to the Cauchy Problem for Parabolic Equations
Abstract
We study sufficient conditions on lower-order coefficients of a nondivergence-form parabolic equation that guarantee the power rate of the uniform stabilization of the solution to the Cauchy problem on every compact set K of RN for any bounded initial function.
Localized Modes Due to Defects in High Contrast Periodic Media Via Two-Scale Homogenization
Abstract
The spectral problem for an infinite periodic medium perturbed by a compact defect is considered. For a high contrast small ε-size periodicity and a finite size defect we consider the critical ε2-scaling for the contrast. We employ two-scale homogenization for deriving asymptotically explicit limit equations for the localized modes and associated eigenvalues. Those are expressed in terms of the eigenvalues and eigenfunctions of a perturbed version of a two-scale limit operator introduced by V. V. Zhikov with an emergent explicit nonlinear dependence on the spectral parameter for the spectral problem at the macroscale. Using the method of asymptotic expansions supplemented by a high contrast boundary layer analysis, we establish the existence of the actual eigenvalues near the eigenvalues of the limit operator, with “ε square root” error bounds. An example for circular or spherical defects in a periodic medium with isotropic homogenized properties is given.
Approximation of Two-Dimensional Viscoelastic Flows of General Form
Abstract
We consider the initial-boundary value problem for approximations of the system of integro-differential equations generalizing the equations of motion for viscoelastic fluids. We prove the existence and convergence theorems and give some examples of non-Newtonian fluids described by the model under consideration.
A Counterexample Related to the Regularity of the p-Stokes Problem
Abstract
We construct a solenoidal vector field u belonging to \( {W}^{2,q}\left(\Omega \right)\cap {W}_0^{1,s}\left(\Omega \right),q\in \left(1,n\right),s\in \left(1,\infty \right) \), such that (1 + |Du|)p − 2, p ∈ (1, ∞), p ≠ 2, does not belong to the Muckenhoupt class A∞(Ω). Thus, one cannot use the Korn inequality in weighted Lebesgue spaces to prove the natural regularity of the p-Stokes problem.