Homogenization of the Signorini boundary‐value problem in a thick junction and boundary integral equations for the homogenized problem

We consider a mixed boundary‐value problem for the Poisson equation in a thick junction Ω_ε which is the union of a domain Ω₀ and a large number of ε—periodically situated thin cylinders. The non‐uniform Signorini conditions are given on the lateral surfaces of the cylinders. The asymptotic analysis of this problem is done as ε→0, i.e. when the number of the thin cylinders infinitely increases and their thickness tends to zero. We prove a convergence theorem and show that the non‐uniform Signorini boundary conditions are transformed in the limiting variational inequalities in the region that is filled up by the thin cylinders as ε→0. The convergence of the energy integrals is proved as well. The existence and uniqueness of the solution to this non‐standard limit problem is established. This solution can be constructed by using a penalty formulation and successive iteration. For some subclass, these problems can be reduced to an obstacle problem in Ω₀ and an appropriate postprocessing. The equations in Ω₀ finally are also treated with boundary integral equations. Copyright © 2010 John Wiley & Sons, Ltd.     

On a degenerate variational inequality with Neumann boundary conditions

A stopping time problem for degenerate reflected diffusions is studied in this paper. We give a characterization of the optimal cost as the maximum solution of a degenerate elliptic variational inequality with Neumann boundary conditions.The author would like to thank Professor L. C. Evans for very helpful suggestions on this topic.     

Contact problems with friction for hemitropic solids: boundary variational inequality approach

We study the interior and exterior contact problems for hemitropic elastic solids. We treat the cases when the friction effects, described by Tresca friction (given friction model), are taken into consideration either on some part of the boundary of the body or on the whole boundary. We equivalently reduce these problems to a boundary variational inequality with the help of the Steklov–Poincaré type operator. Based on our boundary variational inequality approach we prove existence and uniqueness theorems for weak solutions. We prove that the solutions continuously depend on the data of the original problem and on the friction coefficient. For the interior problem, necessary and sufficient conditions of solvability are established when friction is taken into consideration on the whole boundary.     

Two‐grid variational multiscale algorithms for the stationary incompressible Navier‐Stokes equations with friction boundary conditions

Two‐grid variational multiscale (VMS) algorithms for the incompressible Navier‐Stokes equations with friction boundary conditions are presented in this article. First, one‐grid VMS algorithm is used to solve this problem and some error estimates are derived. Then, two‐grid VMS algorithms are proposed and analyzed. The algorithms consist of nonlinear problem on coarse grid and linearized problem (Stokes problem or Oseen problem) on fine grid. Moreover, the stability and convergence of the present algorithms are established. Finally, Numerical results are shown to confirm the theoretical analysis. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 546–569, 2017     

Mathematical analysis of absorbing boundary conditions for the wave equation: the corner problem

Our goal in this work is to establish the existence and the uniqueness of a smooth solution to what we call in this paper the corner problem, that is to say, the wave equation together with absorbing conditions at two orthogonal boundaries. First we set the existence of a very smooth solution to this initial boundary value problem. Then we show the decay in time of energies of high order--higher than the order of the boundary conditions. This result shows that the corner problem is strongly well-posed in spaces smaller than in the half-plane case. Finally, specific corner conditions are derived to select the smooth solution among less regular solutions. These conditions are required to derive complete numerical schemes.

     

The dual variational method for n-th order ODEs with multipoint boundary conditions

We discuss the eigenvalue problem for n-th order equations with multipoint boundary conditions. We present the variational method in spite of the fact that the nonlinearity is dependent on the derivative of n/2 order. The results can be applied applied to both sublinear and superlinear cases.     

Homogenization of a boundary-value problem with a nonlinear boundary condition in a thick junction of type 3:2:1

We consider a boundary-value problem for the Poisson equation in a thick junction Ω_ε, which is the union of a domain Ω₀ and a large number of ε-periodically situated thin curvilinear cylinders. The following nonlinear Robin boundary condition ∂_νu_ε + εκ(u_ε)=0 is given on the lateral surfaces of the thin cylinders. The asymptotic analysis of this problem is performed as ε → 0, i.e. when the number of the thin cylinders infinitely increases and their thickness tends to zero. We prove the convergence theorem and show that the nonlinear Robin boundary condition is transformed (as ε → 0) in the blow-up term of the corresponding ordinary differential equation in the region that is filled up by the thin cylinders in the limit passage. The convergence of the energy integral is proved as well. Using the method of matched asymptotic expansions, the approximation for the solution is constructed and the corresponding asymptotic error estimate in the Sobolev space H¹(Ω_ε) is proved. Copyright © 2007 John Wiley & Sons, Ltd.     

关于求解变分不等式问题的2-次梯度外梯度算法收敛性的一个补注

Yair Censor,Aviv Gibali和Simeon Reich为求解变分不等式问题提出了 2-次梯度外梯度算法.关于此算法的收敛性,作者给出了部分证明,有一个问题:由算法产生的迭代点列能否收敛到变分不等式问题的一个解上,没有得到解决.此问题作为一个公开问题在文章"Extensions of Korpelevi...     

On Korn's first inequality for tangential or normal boundary conditions with explicit constants

We will prove that for piecewise C²‐concave domains in Korn's first inequality holds for vector fields satisfying homogeneous normal or tangential boundary conditions with explicit Korn constant . Copyright © 2016 John Wiley & Sons, Ltd.     

Homogenization of a composite medium with a thermal barrier

In this work, we consider a heat transfer problem between two periodic connected media exchanging a heat flux throughout their common interface. The interfacial exchange coefficient λ is assumed to tend to zero or to infinity following a rate λ=λ(ε) when the size εof the basic cell tends to zero. Three homogenized problems are determined according to the value of δ=lim_ε→0λ/ε. Copyright © 2004 John Wiley & Sons, Ltd.     

On the rate of convergence of solutions in domain with periodic multilevel oscillating boundary

In this paper we deal with the homogenization problem for the Poisson equation in a singularly perturbed domain with multilevel periodically oscillating boundary. This domain consists of the body, a large number of thin cylinders joining to the body through the thin transmission zone with rapidly oscillating boundary. Inhomogeneous Fourier boundary conditions with perturbed coefficients are set on the boundaries of the thin cylinders and on the boundary of the transmission zone. We prove the homogenization theorems and derive the estimates for the convergence of the solutions. Copyright © 2010 John Wiley & Sons, Ltd.     

非齐次边界条件泛定方程的代换选择

对具有非齐次边界条件的泛定方程齐次化过程中代换的选择进行研究和探讨.基于一些相关结论和齐次化的定义得出新的研究成果,即给出对三类非齐次边界条件齐次化都适用的代换W(x,t)=A(t)x<'3>+B(t).     

Continuity of the dynamics in a localized large diffusion problem with nonlinear boundary conditions

This paper is concerned with singular perturbations in parabolic problems subjected to nonlinear Neumann boundary conditions. We consider the case for which the diffusion coefficient blows up in a subregion Ω₀ which is interior to the physical domain Ω⊂Rn. We prove, under natural assumptions, that the associated attractors behave continuously as the diffusion coefficient blows up locally uniformly in Ω₀ and converges uniformly to a continuous and positive function in .     

Finite termination of a smoothing-type algorithm for the monotone affine variational inequality problem

In this paper, we consider the monotone affine variational inequality problem (AVIP for short). Based on a smooth reformulation of the AVIP, we propose a Newton-type method to solve the monotone AVIP, where a testing procedure is embedded into our algorithm. Under mild assumptions, we show that the proposed algorithm may find a maximally complementary solution to the monotone AVIP in a finite number of iterations. Preliminary numerical results are reported.     

Holder continuity of solutions to a parametric variational inequality

We prove a Hölder continuity property of the locally unique solution to a parametric variational inequality without assuming differentiability of the given data.This research was supported by the World Laboratory (Lausanne).     

美式利率期权定价的抛物型变分不等式

应用PDE方法对美式利率期权定价问题进行理论分析.在CIR利率模型下美式利率期权定价问题可归结为一个退化的一维抛物型变分不等式.通过引入惩罚函数证明了该变分不等式的解的存在唯一性,然后研究了自由边界的一些性质,如单调性,光滑性和自由边界在终止期的位置.     

A variational approach to stochastic nonlinear diffusion problems with dynamical boundary conditions

We study a nonlinear partial differential equation of the calculus of variation in a bounded domain, perturbed by noise; we allow stochastic boundary conditions that depend on the time derivative of the solution on the boundary. This work provides the existence and uniqueness of the solution and it shows the existence of an ergodic invariant measure for the corresponding transition semigroup; furthermore, under suitable additional assumptions, uniqueness and strong asymptotic stability of the invariant measure are proved.