New conditions for averaging of nonlinear dirichlet problems in perforated domains

We study the problem of averaging Dirichlet problems for nonlinear elliptic second-order equations in domains with fine-grained boundary. We consider a class of equations admitting degeneration with respect to the gradients of solutions. We prove a pointwise estimate for solutions of the model nonlinear boundary-value problem and construct an averaged boundary-value problem under new structural assumptions concerning perforated domains. In particular, it is not assumed that the diameters of cavities are small as compared to the distances between them.     

具有转向点曲线的椭圆型方程奇摄动边值问题

The singularly perturbed elliptic equation boundary value problem with a curve of turning point is considered. Using the method of multiple scales and the comparison theorem,the asymptotic behavior of solution for the boundary value problem is studied.     

Homogenization of low-cost control problems on perforated domains

The aim of this paper is to study the asymptotic behaviour of some low-cost control problems in periodically perforated domains with Neumann condition on the boundary of the holes. The optimal control problems considered here are governed by a second order elliptic boundary value problem with oscillating coefficients. It is assumed that the cost of the control is of the same order as that describing the oscillations of the coefficients. The asymptotic analysis of small cost problem is more delicate and need the H-convergence result for weak data. In this connection, an H-convergence result for weak data under some hypotheses is also proved.     

Some homogenization problems for the system of elasticity with nonlinear boundary conditions in perforated domains

The system of linear elasticity is considered in a perforated domain with an ε-periodic structure. External forces nonlinearly depending on the displacements are applied to the surface of the cavities (or channels), while the body is fixed along the outer portion of its boundary. We investigate the asymptotic behavior of solutions to such boundary value problems asε→0 and construct the limit problem, according to the external surface forces and their dependence on the parameter ε. In some cases, this dependence results in the homogenized problem having the form of a variational inequality over a certain closed convex cone in a Sobolev space. This cone is described in terms of the functions involved in the nonlinear boundary conditions on the perforated boundary. A homogenization theorem is also proved for some unilateral problems with boundary conditions of Signorini type for the system of elasticity in a perforated domain. We discuss some cases when the homogenized tensor may depend on the functions specifying the boundary conditions.     

The existence and concentration of weak solutions to a class of p-Laplacian type problems in unbounded domains

In this paper,we study the existence and concentration of weak solutions to the p-Laplacian type elliptic problem-εp△pu+V(z)|u|p-2u-f(u)=0 in Ω,u=0 on ■Ω,u0 in Ω,Np2,where Ω is a domain in RN,possibly unbounded,with empty or smooth boundary,εis a small positive parameter,f∈C1(R+,R)is of subcritical and V:RN→R is a locally Hlder continuous function which is bounded from below,away from zero,such that infΛVmin ■ΛV for some open bounded subset Λ of Ω.We prove that there is anε00 such that for anyε∈(0,ε0],the above mentioned problem possesses a weak solution uεwith exponential decay.Moreover,uεconcentrates around a minimum point of the potential V inΛ.Our result generalizes a similar result by del Pino and Felmer(1996)for semilinear elliptic equations to the p-Laplacian type problem.     

The existence of positive solution for singular boundary value problems of first order differential equation on unbounded domains

By constructing a special cone and using cone compression and expansion fixed point theorem, this paper presents some existence results of positive solutions of singular boundary value problem on unbounded domains for a class of first order differential equation. As applications of the main results, two examples are given at the end of this paper. Supported by the National Natural Science Foundation of China (No. 10671167) and the Natural Science Foundation of Liaocheng University (31805).     

Homogenization of eigenvalue problems in perforated domains

In this paper, we treat some eigenvalue problems in periodically perforated domains and study the asymptotic behaviour of the eigenvalues and the eigenvectors when the number of holes in the domain increases to infinity Using the method of asymptotic expansion, we give explicit formula for the homogenized coefficients and expansion for eigenvalues and eigenvectors. If we denote by ε the size of each hole in the domain, then we obtain the following aysmptotic expansion for the eigenvalues: Dirichlet: λ_ε = ε⁻² λ + λ₀ +O (ε), Stekloff: λ_ε = ελ₁ +O (ε²), Neumann: λ_ε = λ₀ + ελ₁ +O (ε²). Using the method of energy, we prove a theorem of convergence in each case considered here. We briefly study correctors in the case of Neumann eigenvalue problem.     

On output functionals of boundary value problems on stochastic domains

We consider the computation of output functionals of random solutions to elliptic boundary value problems in domains with random boundary perturbations. We use a second‐order shape calculus to linearize the problem around a fixed nominal domain. For known mean and two‐point correlation function of the boundary perturbation, we derive, with leading order, deterministic expressions for the mean and the variance of the random output functional. These expressions include the solution of the boundary value problem on the nominal domain and a further, deterministic solution of the so‐called adjoint equation. The theoretical findings are supported and quantified by numerical experiments. Copyright © 2009 John Wiley & Sons, Ltd.     

The asymptotic behaviour of weak solutions to the forward problem of electrical impedance tomography on unbounded three‐dimensional domains

The forward problem of electrical impedance tomography on unbounded domains can be studied by introducing appropriate function spaces for this setting. In this paper we derive the point‐wise asymptotic behaviour of weak solutions to this problem in the three‐dimensional case. Copyright © 2008 John Wiley & Sons, Ltd.     

Pullback attractors for the non-autonomous Benjamin-Bona-Mahony equation in unbounded domains

The existence of a pullback attractor is proven for the non-autonomous Benjamin-Bona-Mahony equation in unbounded domains.The asymptotic compactness of the solution operator is obtained by the uniform estimates on the tails of solutions.     

A singularly perturbed nonlinear traction problem in a periodically perforated domain: a functional analytic approach

We consider a periodically perforated domain obtained by making in a periodic set of holes, each of them of size proportional to ε. Then, we introduce a nonlinear boundary value problem for the Lamé equations in such a periodically perforated domain. The unknown of the problem is a vector‐valued function u, which represents the displacement attained in the equilibrium configuration by the points of a periodic linearly elastic matrix with a hole of size ε contained in each periodic cell. We assume that the traction exerted by the matrix on the boundary of each hole depends (nonlinearly) on the displacement attained by the points of the boundary of the hole. Then, our aim is to describe what happens to the displacement vector function u when ε tends to 0. Under suitable assumptions, we prove the existence of a family of solutions {u(ε, ? )}_{ε ∈ ]0,ε ′ [} with a prescribed limiting behavior when ε approaches 0. Moreover, the family {u(ε, ? )}_{ε ∈ ]0,ε ′ [} is in a sense locally unique and can be continued real analytically for negative values of ε. Copyright © 2013 John Wiley & Sons, Ltd.     

Asymptotic approximations for solutions to quasilinear and linear parabolic problems with different perturbed boundary conditions in perforated domains

We consider quasilinear and linear parabolic problems with rapidly oscillating coefficients in a domain Ω _ε that is ε-periodically perforated by small holes of order      

Solutions to the polynomial Dirac equations on unbounded domains in Clifford analysis

In this paper we study polynomial Dirac equation p(??)f = 0 including (?? ? λ)f = 0 with complex parameter λ and ??^kf = 0(k?1) as special cases over unbounded subdomains of ?^{n + 1}. Using the Clifford calculus, we obtain the integral representation theorems for solutions to the equations satisfying certain decay conditions at infinity over unbounded subdomains of ?^{n + 1}. Copyright © 2010 John Wiley & Sons, Ltd.     

Viscous two‐fluid flows in perturbed unbounded domains

Two stationary plane free boundary value problems for the Navier‐Stokes equations are studied. The first problem models the viscous two‐fluid flow down a perturbed or slightly distorted inclined plane. The second one describes the viscous two‐fluid flow in a perturbed or slightly distorted channel. For sufficiently small data and under certain conditions on parameters the solvability and uniqueness results are proved for both problems. The asymptotic behaviour of the solutions is investigated. For the second problem an example of nonuniqueness is constructed. Computational results of flow problems that are very close to the above problems are presented. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Nonconforming spline collocation methods in irregular domains

This article studies a class of nonconforming spline collocation methods for solving elliptic PDEs in an irregular region with either triangular or quadrilateral partition. In the methods, classical Gaussian points are used as matching points and the special quadrature points in a triangle or quadrilateral element are used as collocation points. The solution and its normal derivative are imposed to be continuous at the marching points. The authors present theoretically the existence and uniqueness of the numerical solution as well as the optimal error estimate in H¹‐norm for a spline collocation method with rectangular elements. Numerical results confirm the theoretical analysis and illustrate the high‐order accuracy and some superconvergence features of methods. Finally the authors apply the methods for solving two physical problems in compressible flow and linear elasticity, respectively. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

A singularly perturbed Dirichlet problem for the Laplace operator in a periodically perforated domain. A functional analytic approach

Let Ω be a sufficiently regular bounded connected open subset of such that 0 ∈ Ω and that is connected. Then we take q₁₁, … ,q_nn ∈ ]0,+ ∞ [and . If ε is a small positive number, then we define the periodically perforated domain , where {e₁, … ,e_n} is the canonical basis of . For ε small and positive, we introduce a particular Dirichlet problem for the Laplace operator in the set . Namely, we consider a Dirichlet condition on the boundary of the set p + εΩ, together with a periodicity condition. Then we show real analytic continuation properties of the solution and of the corresponding energy integral as functionals of the pair of ε and of the Dirichlet datum on p + ε?Ω, around a degenerate pair with ε = 0. Copyright © 2011 John Wiley & Sons, Ltd.     

Spectral analysis and numerical simulation for second order elliptic operator with highly oscillating coefficients in perforated domains with a periodic structure

This paper discusses the spectral properties and numerical simulation for the second order elliptic operators with rapidly oscillating coefficients in the domains which may contain small cavities distributed periodically with period ε. A multiscale asymptotic analysis formula for this problem is obtained by constructing properly the boundary layer. Finally, numerical results are given, which provide a strong support for the analytical estimates     

Corner boundary layer in nonlinear singularly perturbed elliptic problems

The Dirichlet problem in a rectangle is considered for the elliptic equation ?²Δu = F(u, x, y, ?), where F(u, x, y, ?) is a nonlinear function of u. The method of corner boundary functions is applied to the problem. Assuming that the leading term of the corner part of the asymptotics exists, an asymptotic expansion of the solution is constructed and the remainder is estimated.