Scalar boundary value problems on junctions of thin rods and plates. I. Asymptotic analysis and error estimates

We derive asymptotic formulas for the solutions of the mixed boundary value problem for the Poisson equation on the union of a thin cylindrical plate and several thin cylindrical rods. One of the ends of each rod is set into a hole in the plate and the other one is supplied with the Dirichlet condition. The Neumann conditions are imposed on the whole remaining part of the boundary. Elements of the junction are assumed to have contrasting properties so that the small parameter, i.e. the relative thickness, appears in the differential equation, too, while the asymptotic structures crucially depend on the contrastness ratio. Asymptotic error estimates are derived in anisotropic weighted Sobolev norms.Comment: 34 pages, 4 figure

Bingham flow in porous media with obstacles of different size

By using the unfolding operators for periodic homogenization, we give a general compactness result for a class of functions defined on bounded domains presenting perforations of two different size. Then we apply this result to the homogenization of the flow of a Bingham fluid in a porous medium with solid obstacles of different size. Next, we give the interpretation of the limit problem in terms of a nonlinear Darcy law. Copyright (C) 2017 John Wiley & Sons, Ltd

Unfolding Method for the Homogenization of Bingham Flow

International audienceWe are interested in the homogenization of a stationary Bingham flow in a porous medium. The model and the formal expansion of this problem are introduced in Lions and Sanchez-Palencia (J. Math. Pures Appl. 60:341–360, 1981) and a rigorous justification of the convergence of the homogenization process is given in Bourgeat and Mikelic (J. Math. Pures Appl. 72:405–414, 1993), by using monotonicity methods coupled with the two-scale convergence method. In order to get the homogenized problem, we apply here the unfolding method in homogeniza-tion, method introduced in Cioranescu et al. (SIAM J. Math. Anal.40:1585–1620, 2008)

Unfolding method for the homogenization of Bingham flow

We are interested in the homogenization of a stationary Bingham ow in a porous medium.A rigorous justification ofthe convergence of the homogenization process is given by previous authors by using monotonicity methods coupled withthe two-scale convergence method. In order to get the homogenized problem, we apply here the unfoldingmethod in homogenization

On a waveguide with an infinite number of small windows

We consider a waveguide modeled by the Laplacian in a straight planar strip with the Dirichlet condition on the upper boundary, while on the lower one we impose periodically alternating boundary conditions with a small period. We study the case when the homogenization leads us to the Neumann boundary condition on the lower boundary. We establish the uniform resolvent convergence and provide the estimates for the rate of convergence. We construct the two-terms asymptotics for the first band functions of the perturbed operator and also the complete two-parametric asymptotic expansion for the bottom of its spectrum. (C) 2010 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved

SCALAR BOUNDARY VALUE PROBLEMS ON JUNCTIONS OF THIN RODS AND PLATES

We derive asymptotic formulas for the solutions of the mixed boundary value problem for the Poisson equation on the union of a thin cylindrical plate and several thin cylindrical rods. One of the ends of each rod is set into a hole in the plate and the other one is supplied with the Dirichlet condition. The Neumann conditions are imposed on the whole remaining part of the boundary. Elements of the junction are assumed to have contrasting properties so that the small parameter, i.e. the relative thickness, appears in the differential equation, too, while the asymptotic structures crucially depend on the contrastness ratio. Asymptotic error estimates are derived in anisotropic weighted Sobolev norms

Homogenization and asymptotics for a waveguide with an infinite number of closely located small windows

International audienceWe consider a planar waveguide modeled by the Laplacian in a straight infinite strip with the Dirichlet boundary condition on the upper boundary and with frequently alternating boundary conditions (Dirichlet and Neumann) on the lower boundary. The homogenized operator is the Laplacian subject to the Dirichlet boundary condition on the upper boundary and to the Dirichlet or Neumann condition on the lower one. We prove the uniform resolvent convergence for the perturbed operator in both cases and obtain the estimates for the rate of convergence. Moreover, we construct the leading terms of the asymptotic expansions for the first band functions and the complete asymptotic expansion for the bottom of the spectrum

On a Waveguide with Frequently Alternating Boundary Conditions: Homogenized Neumann Condition

We consider a waveguide modeled by the Laplacian in a straight planar strip. The Dirichlet boundary condition is taken on the upper boundary, while on the lower boundary we impose periodically alternating Dirichlet and Neumann condition assuming the period of alternation to be small. We study the case when the homogenization gives the Neumann condition instead of the alternating ones. We establish the uniform resolvent convergence and the estimates for the rate of convergence. It is shown that the rate of the convergence can be improved by employing a special boundary corrector. Other results are the uniform resolvent convergence for the operator on the cell of periodicity obtained by the Floquet-Bloch decomposition, the two terms asymptotics for the band functions, and the complete asymptotic expansion for the bottom of the spectrum with an exponentially small error term

Spectral approach to homogenization of an elliptic operator periodic in some directions

The operator A(epsilon) = D(1)g(1)(x(1)/epsilon,x(2))D(1) + D(2)g(2)(x(1)/epsilon,x(2))D(2) is considered in L(2)(R(2)), where g(j)(x(1),x(2)), j=1, 2, are periodic in x(1) with period 1, bounded and positive definite. Let function Q(x(1),x(2)) be bounded, positive definite and periodic in x(1) with period 1. Let Q(epsilon)(x(1),x(2))=Q(x(1)/epsilon,x(2)). The behavior of the operator (A epsilon+Q(epsilon))(-1) as epsilon -> 0 is studied. It is proved that the operator (A(epsilon)+Q(epsilon))(-1) tends to (A(0) + Q(0))(-1) in the operator norm in L(2)(R(2)). Here, A(0) is the effective operator whose coefficients depend only on x(2), Q(0) is the mean value of Q in x(1). A sharp order estimate for the norm of the difference (A(epsilon) + Q(epsilon))(-1) - (A(0) + Q(0))(-1) is obtained. The result is applied to homogenization of the Schrodinger operator with a singular potential periodic in one direction. Copyright (C) 2011 John Wiley & Sons, Ltd