Vol 232, No 4 (2018)
- Year: 2018
- Articles: 11
- URL: https://journals.rcsi.science/1072-3374/issue/view/14934
Article
C1+α-Regularity of Viscosity Solutions of General Nonlinear Parabolic Equations
Abstract
We establish the existence of C1+α-regular viscosity solutions of fully nonlinear secondorder parabolic equations like ∂tu+H(v,Dv,D2v, t, x) = 0 allowing the main part of H to have small discontinuities with respect to x and be just measurable in t.
Meso-Scale Approximations of Fields Around Clusters of Defects
Abstract
We present a review of recent results on asymptotic approximations of solutions to boundary value problems in domains with large clusters of small defects. There are no assumptions which require periodicity within the cluster. The asymptotic approximations we discuss are uniform and include the boundary layers occurring in neighborhoods of singularly perturbed boundaries of the domains concerned. The term “meso-scale” is used to describe these approximations since they go beyond the conventional constraints of the homogenization theory.
Strong Precompactness of Bounded Sequences Under Nonlinear Ultraparabolic Differential Constraints
Abstract
It is shown that bounded sequences satisfying nonlinear differential constraints, strongly precompact under an exact condition of nondegeneration of these conditions. The proof is based on new localization principles for ultraparabolic H-measures with continuous indices.
On Sobolev Inequalities on Singular and Combined Structures
Characteristically Closed Domains for First Order Strictly Hyperbolic Systems in the Plane
Abstract
We consider a first order strictly hyperbolic system of n equations with constant coefficients in a bounded domain. It is assumed that the domain is strictly convex relative to characteristics, so that the projection along each characteristic is an involution having two fixed singular points. The natural statement of boundary value problems for such systems requires that singular points go to singular points under such transformations. We present a necessary and sufficient condition for the existence of such domains, called characteristically closed.
Operator-Norm Convergence Estimates for Elliptic Homogenization Problems on Periodic Singular Structures
Abstract
For an arbitrary periodic Borel measure μ we prove order O(ε) operator-norm resolvent estimates for the solutions to scalar elliptic problems in L2(ℝd, dμε) with ε-periodic coefficients, ε > 0. Here, με is the measure obtained by ε-scaling of μ. Our analysis includes the case of a measure absolutely continuous with respect to the standard Lebesgue measure, as well as the case of “singular” periodic structures (or “multistructures”), when μ is supported by lower-dimensional manifolds.
Homogenization of Variational Inequality for the Laplace Operator with Nonlinear Constraint on the Flow in a Domain Perforated by Arbitrary Shaped Sets. Critical Case
Abstract
We construct and justify a homogenized model of the variational inequality with the Laplace operator and a nonlinear boundary constraint on the flow on arbitrary shaped cavities generating perforation of the domain with critical values of parameters.
Multicontinuum Wave Propagation in a Laminated Beam with Contrasting Stiffness and Density of Layers
Abstract
The wave equation in a thin laminated beam with contrasting stiffness and density of layers is considered. The problem contains two parameters: ε is a geometric small parameter (the ratio of the diameter and its characteristic longitudinal size) and ω is a physical large parameter (the ratio of stiffness and densities of alternating layers). The asymptotic behavior of the solution depends on the combination of parameters ε2ω. If this value is small, then the limit model is the standard homogenized one-dimensional wave equation. On the contrary, if ε2ω is not small, then the limit model is presented by the so-called multicontinuum model, i.e., multiple one-dimensional wave equations, coupled or noncoupled and “co-existing” at every point. The proof of these results uses the milticomponent homogenization method.
Finite-Dimensional Approximations of the Steklov–Poincaré Operator for the Helmholtz Equation in Periodic Waveguides
Abstract
We consider the Dirichlet and Neumann problems for the Laplace operator in periodic waveguides. Integro-differential connections between the solution and its normal derivative, interpreted as a finite-dimensional version of the Steklov–Poincaré operator, are imposed on the artificial face of the truncated waveguide. These connections are obtained from the orthogonality and normalization conditions for the Floquet waves which are oscillating incoming/outgoing, as well as exponentially decaying/growing in the periodic waveguide. Under certain conditions, we establish the unique solvability of the problem and obtain error estimates for the solution itself, as well as for scattering coefficients in the solution. We give examples of trapped waves in periodic waveguides.