Homogenization of Boundary-Value Problem in a Locally Periodic Perforated Domain

We consider a model homogenization problem for the Poisson equation in a locally periodic perforated domain with the smooth exterior boundary, the Fourier boundary condition being posed on the boundary of the holes. In the paper we construct the leading terms of formal asymptotic expansion. Then, we justify the asymptotics obtained and estimate the residual.     

Random homogenization of an obstacle problem

We study the homogenization of an obstacle problem in a perforated domain, when the holes are periodically distributed and have random shape and size. The main assumption concerns the capacity of the holes which is assumed to be stationary ergodic.     

Homogenization of Elliptic Problems with Quadratic Growth and Nonhomogenous Robin Conditions in Perforated Domains

This paper deals with the homogenization of a class of nonlinear elliptic problems with quadratic growth in a periodically perforated domain. The authors prescribe a Dirichlet condition on the exterior boundary and a nonhomogeneous nonlinear Robin condition on the boundary of the holes. The main difficulty, when passing to the limit, is that the solution of the problems converges neither strongly in L~2(Ω) nor almost everywhere in Ω. A new convergence result involving nonlinear functions provides suitable weak convergence results which permit passing to the limit without using any extension operator.Consequently, using a corrector result proved in [Chourabi, I. and Donato, P., Homogenization and correctors of a class of elliptic problems in perforated domains, Asymptotic Analysis, 92(1), 2015, 1–43, DOI: 10.3233/ASY-151288], the authors describe the limit problem, presenting a limit nonlinearity which is different for the two cases, that of a Neumann datum with a nonzero average and with a zero average.     

Homogenization of some degenerate pseudoparabolic variational inequalities

Multiscale analysis of a degenerate pseudoparabolic variational inequality, modelling the two-phase flow with dynamical capillary pressure in a perforated domain, is the main topic of this work. Regularisation and penalty operator methods are applied to show the existence of a solution of the nonlinear degenerate pseudoparabolic variational inequality defined in a domain with microscopic perforations, as well as to derive a priori estimates for solutions of the microscopic problem. The main challenge is the derivation of a priori estimates for solutions of the variational inequality, uniformly with respect to the regularisation parameter and to the small parameter defining the scale of the microstructure. The method of two-scale convergence is used to derive the corresponding macroscopic obstacle problem.     

一类具p-Laplacian算子的多点边值问题单调迭代正解的存在性

利用单调迭代的方法得到了一类具p-Laplacian算子的多点边值问题单调迭代正解的存在性,同时也得到了解的相应迭代序列.     

一类带p-Laplacian的Sturm-Liouville型四点边值问题多个正解的存在性

研究了一类带p-Laplacian的四点边值问题.通过运用锥上的不动点定理,我们得到了该类Sturm-Liouville型边值问题三个正解的存在性.     

Homogenization in perforated domains beyond the periodic setting

We study the homogenization of a second order linear elliptic differential operator in an open set in with isolated holes of size ε>0. The classical periodicity hypothesis on the coefficients of the operator is here substituted by an abstract assumption covering a variety of concrete behaviours such as the periodicity, the almost periodicity, and many more besides. Furthermore, instead of the usual “periodic perforation” we have here an abstract hypothesis characterizing the manner in which the holes are distributed. This is illustrated by practical examples ranging from the classical equidistribution of the holes to the more complex case in which the holes are concentrated in a neighbourhood of the hyperplane {x_N=0}. Our main tool is the recent theory of homogenization structures and our basic approach follows the direct line of two-scale convergence.     

Critical Homogenization of SDEs Driven by a Levy Process in Random Medium

We are concerned with homogenization of stochastic differential equations (SDE) with stationary coefficients driven by Poisson random measures and Brownian motions in the critical case, that is, when the limiting equation admits both a Brownian part as well as a pure jump part. We state an annealed convergence theorem. This problem is deeply connected with homogenization of integral partial differential equations.     

具有奇异系数的发展p-Laplace 方程

该文考虑Rⁿ内在球中和球面系数均有奇异的发展p-Laplac方程的初边值问题的可解性. 在一定条件下证明了轴对称整体解的存在性和不存在性.     

Very Weak Solutions of p-Laplacian Type Equations with VMO Coefficients

In this note we obtain a new a priori estimate for the very weak solutions of p-Laplacian type equations with VMO coefficients when p is close to 2, and then prove that the very weak solutions of such equations are the usual weak solutions. Our approath is based on the Hodge decomposition and the L^p-estimate for the corresponding linear equations. And this also provides a simpler proof for the results in [1].     

Homogenization in perforated domains with mixed conditions

In this paper we study the asymptotic behaviour of the Laplace equation in a periodically perforated domain of R ⁿ , where we assume that the period is ε and the size of the holes is of the same order of greatness. An homogeneous Dirichlet condition is given on the whole exterior boundary of the domain and on a flat portion of diameter if (, if n=2) of the boundary of every hole, while we take an homogeneous Neumann condition elsewhere.     

Homogenization of Elliptic Problems with Neumann Boundary Conditions in Non-smooth Domains

We consider a family of second-order elliptic operators {L_ε} in divergence form with rapidly oscillating and periodic coefficients in Lipschitz and convex domains in R~n. We are able to show that the uniform W~(1,p) estimate of second order elliptic systems holds for 2n/(n+1)-δ p 2n/(n-1)+ δ where δ 0 is independent of ε and the ranges are sharp for n = 2, 3. And for elliptic equations in Lipschitz domains, the W~(1,p) estimate is true for 3/2-δ p 3 + δ if n ≥ 4, similar estimate was extended to convex domains for 1 p ∞.     

一类带拒绝的多种类服务系统的平稳分布

本文在文献[2]的基础上考虑了一类有拒绝的可批量到达的服务系统的平稳分布。文章首先给出了Q过程的常返性以及遍历性的充要条件，最后得出了系统的不变测度。     

含非对称临界非线性项的p-Laplace方程的多解问题

本文考虑含非奇对称临界非线性项的p-Laplace方程Dirichlet问题。运用改进的集中列紧原理证明了在某些指数条件下非奇对称的临界非线性项仍能保证无穷多弱解的存在性。     

Remarks on Blow-Up Phenomena in p-Laplacian Heat Equation with Inhomogeneous Nonlinearity

We investigate the $p$-Laplace heat equation $u_t-Delta_p u=ζ(t)f(u)$ in a bounded smooth domain. Using differential-inequality arguments, we prove blow-up results under suitable conditions on $ζ, f$, and the initial datum $u_0$. We also give an upper bound for the blow-up time in each case.     

Global Existence and Uniqueness of Solutions to Evolution p-Laplacian Systems with Nonlinear Sources

This paper presents the global existence and uniqueness of the initial and boundary value problem to a system of evolution p-Laplacian equations coupled with general nonlinear terms. The authors use skills of inequality estimation and themethod of regularization to construct a sequence of approximation solutions, hence obtain the global existence of solutions to a regularized system. Then the global existence of solutions to the system of evolution p-Laplacian equations is obtained with the application of a standard limiting process. The uniqueness of the solution is proven when the nonlinear terms are local Lipschitz continuous.     

Homogenization of a parabolic equation in perforated domain with Dirichlet boundary condition

In this article, we study the homogenization of the family of parabolic equations over periodically perforated domains . Here, Ω_ɛ = ΩS _ε is a periodically perforated domain andd _ε is a sequence of positive numbers which goes to zero. We obtain the homogenized equation. The homogenization of the equations on a fixed domain and also the case of perforated domain with Neumann boundary condition was studied by the authors. The homogenization for a fixed domain and has been done by Jian. We also obtain certain corrector results to improve the weak convergence.     

Positive Solution of Multi-Point Boundary Value Problems for the One-Dimensional p-Laplacian with Singularities

In this paper, we obtain positive solution to the following multi-point singular boundary value problem with p-Laplacian operator,{（ φp（u＇））＇＋q（t）f（t,u,u＇）=0,0〈t〈1,u（0）=∑i=1^nαiu（ξi）,u＇（1）=∑i=1^nβiu＇（ξi）,whereφp（s）=｜s｜^p-2s,p≥2;ξi∈（0,1）（i=1,2,…,n）,0≤αi,βi〈1（i=1,2,…n）,0≤∑i=1^nαi,∑i=1^nβi〈1,and q（t） may be singular at t=0,1,f（t,u,u＇）may be singular at u＇=0