Том 232, № 3 (2018)

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.

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.

We propose a method for extracting the sums of two harmonics from trigonometric polynomials T_n(t) by the method of amplitude-phase transformations. Such transformations send polynomials T_n(t) to similar polynomials by two simplest operations: multiplication by a real constant X and shift by a real phase λ, i.e., T_n(t) → X ・ T_n(t − λ). The harmonics are extracted by adding similar polynomials.

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.

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.