Boundary homogenization in domains with randomly oscillating boundary

We consider a model homogenization problem for the Poisson equation in a domain with a rapidly oscillating boundary which is a small random perturbation of a fixed hypersurface. A Fourier boundary condition with random coefficients is imposed on the oscillating boundary. We derive the effective boundary condition, prove a convergence result, and establish error estimates.     

Alternating boundary layer type solutions of some singularly perturbed periodic parabolic equations with Dirichlet and Robin boundary conditions

In this paper, we continue the analysis of alternating boundary layer type solutions to certain singularly perturbed parabolic equations for which the degenerate equations (obtained by setting small parameter multiplying derivatives equal to zero) are algebraic equations that have three roots. Here, we consider spatially one-dimensional equations. We address special cases where the following are true: (a) boundary conditions are of the Dirichlet type with different values of unknown functions specified at different endpoints of the interval of interest; (b) boundary conditions are of the Robin type with an appropriate power of a small parameter multiplying the derivative in the conditions. We emphasize a number of new features of alternating boundary layer type solutions that appear in these cases. One of the important applications of such equations is related to modeling certain types of bioswitches. Special choices of Dirichlet and Robin type boundary conditions can be used to tune up such bioswitches. This article was submitted by the authors in English.     

Asymptotics of simple eigenvalues and eigenfunctions for the Laplace operator in a domain with an oscillating boundary

The asymptotic behavior of solutions to spectral problems for the Laplace operator in a domain with a rapidly oscillating boundary is analyzed. The leading terms of the asymptotic expansions for eigenelements are constructed, and the asymptotics are substantiated for simple eigenvalues. The text was submitted by the authors in English.     

On the rate of convergence of solutions in free boundary problems via penalization

The rate of convergence of approximate solutions via penalization for free boundary problems are concerned. A key observation is to obtain global bounds of penalized terms which give necessary estimates on integrations by the nonlinear adjoint method by L.C. Evans.     

Korn's inequality for periodic solids and convergence rate of homogenization

In a three-dimensional solid with arbitrary periodic Lipschitz perforation the Korn inequality is proved with a constant independent of the perforation size. The convergence rate of homogenization as a function of the Sobolev–Slobodetskii smoothness of data is also estimated. We improve foregoing results in elasticity dropping customary restrictions on the shape of the periodicity cell and superfluous smoothness and smallness assumptions on the external forces and traction.     

Homogenization of the variational inequality corresponding to a problem with rapidly varying boundary conditions

In the present paper, we investigate the asymptotic behavior of the solution of a variational inequality with one-sided constraints on ?-periodically located subsets G _ε belonging to the boundary ?Ω of the domain Ω ? ?³. We construct a limit (homogenized) problem and prove the strong (in H ₁(Ω)) convergence of the solutions of the original inequality to the solution the limit nonlinear boundary-value problem as ? → 0 in the so-called critical case.     

Pseudo almost periodic solutions for a model of hematopoiesis with an oscillating circulation loss rate

In this paper, a model of hematopoiesis with an oscillating circulation loss rate is investigated. By applying the exponential dichotomy theory, contraction mapping fixed‐point theorem, and differential inequality techniques, a set of sufficient conditions are obtained for the existence and exponential stability of positive pseudo almost periodic solutions of the model. Some numerical simulations are carried out to support the theoretical findings. Our results improve and generalize those of the previous studies. Copyright © 2015 John Wiley & Sons, Ltd.     

Existence and convergence of solutions to periodic boundary value problems for Kirchhoff equations via coincidence degree method

This paper aims to investigate the existence and convergence of solutions to periodic boundary value problems for one-dimensional Kirchhoff equation. By employing analytical skills and the coincidence degree method, some new results are obtained, which enrich and generalize the previous results.     

Asymptotic analysis of Neumann periodic optimal boundary control problem

An optimal boundary control problem in a domain with oscillating boundary has been investigated in this paper. The controls are acting periodically on the oscillating boundary. The controls are applied with suitable scaling parameters. One of the major contribution is the representation of the optimal control using the unfolding operator. We then study the limiting analysis (homogenization) and obtain two limit problems according to the scaling parameters. Another notable observation is that the limit optimal control problem has three controls, namely, a distributed control, a boundary control, and an interface control. Copyright © 2016 John Wiley & Sons, Ltd.     

On Boundary-Value Problems for the Laplacian in Bounded Domains with Micro Inhomogeneous Structure of the Boundaries

We consider boundary-value problems with rapidly alternating types of boundary conditions. We present the classification of homogenized （limit） problems depending on the ratio of small parameters, which characterize the diameter of parts of the boundary with different types of boundary conditions. Also we study the respective spectral problem of this type.     

On approximation of solutions of multidimensional SDE's with reflecting boundary conditions

Let D be either a convex domain in ^d or a domain satisfying the conditions (A) and (B) considered by Lions and Sznitman [7] and Saisho [11]. We estimate the rate of L^p convergence for Euler and Euler–Peano schemes for stochastic differential equations in D with normal reflection at the boundary of the form

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, where W is a d-dimensional Wiener process. As a consequence we give the rate of almost sure convergence for these schemes.     

A boundary element method for homogenization of periodic structures

Homogenized coefficients of periodic structures are calculated via an auxiliary partial differential equation in the periodic cell. Typically, a volume finite element discretization is employed for the numerical solution. In this paper, we reformulate the problem as a boundary integral equation using Steklov–Poincaré operators. The resulting boundary element method only discretizes the boundary of the periodic cell and the interface between the materials within the cell. We prove that the homogenized coefficients converge super-linearly with the mesh size, and we support the theory with examples in two and three dimensions.     

On decay rate of solutions for the stationary Navier–Stokes equation in an exterior domain

In this paper, we consider the asymptotic behavior of an incompressible fluid around a bounded obstacle. By adapting the Schauder's estimate for stationary Navier–Stokes equation to improve the regularity, the problem is solved by using appropriate Carleman estimates. It should be noted that the minimal decaying rate for a general scalar equation is $\exp ?(?C|x{|}^{2+})$. However, the structure of the Navier–Stokes is special. Under the assumption for any nontrivial solution to be uniform bounded which is weaker than those in [10], we got the minimal decaying rate is $\exp ?(?C|x{|}^{\frac{3}{2}+})$ which is better than the results in general scalar cases.     

Triple positive periodic solutions of nonlinear singular second-order boundary value problems

This paper studies the positive solutions of the nonlinear second-order periodic boundary value problem u″(t) + λ(t)u(t) = f(t,u(t)),a.e.t ∈ [0,2π],u(0) = u(2π),u′(0) = u′(2π),where f(t,u) is a local Carath′eodory function.This shows that the problem is singular with respect to both the time variable t and space variable u.By applying the Leggett–Williams and Krasnosel'skii fixed point theorems on cones,an existence theorem of triple positive solutions is established.In order to use these theorems,the exact a priori estimations for the bound of solution are given,and some proper height functions are introduced by the estimations.     

On the convergence rate of a parallel nonoverlapping domain decomposition method

In recent years,a nonoverlapping domain decomposition iterative procedure,which is based on using Robin-type boundary conditions as information transmission conditions on the subdomain interfaces,has been developed and analyzed.It is known that the convergence rate of this method is 1-O(h),where h is mesh size.In this paper,the convergence rate is improved to be 1-O(h1/2 H-1/2)sometime by choosing suitable parameter,where H is the subdomain size.Counter examples are constructed to show that our convergence estimates are sharp,which means that the convergence rate cannot be better than 1-O(h1/2H-1/2)in a certain case no matter how parameter is chosen.     

On the convergence of 2-dimensional markov chains in quadrants with boundary reflection

The aim of this paper is to prove a limit theorem on the weak convergence of a family of rescaled Markov chains in a quadrant with boundary reflection. The limiting process is specified in terms of solutions of a certain submartingale problem in the style used by Varadhan and Williams. The obtained result is then applied to the problem of approximating an arbitrary Brownian motion with oblique reflection in a wedge by a family of Markov chains.     

Constant-sign solutions of systems of higher order boundary value problems with integrable singularities

We consider some systems of boundary value problems where the nonlinearities may be singular in the independent variable and may also be singular in the dependent arguments. Using the Schauder fixed point theorem, we establish criteria such that the systems of boundary value problems have at least one constant-sign solution.