Showing papers in "Asymptotic Analysis in 1989"

Homogenization of the stokes flow in a connected porous medium

Abstract: In this paper we prove the convergence of the homogenization process of the Stokes equations with Dirichlet boundary condition in a periodic porous medium. We consider here the case where the solid part of the porous medium is connected, and we generalize to this case the results obtained by Tartar (1980).

Existence and characterization of higher-order terms in an asymptotic expansion method for linearized elastic beams

Abstract: This work is a continuation of an earlier work by Bermudez and Viano (1984) on the same subject. In fact, using the same asymptotic expansion in linear elastic beams we give a complete characterization of displacements, bending moments and shear forces of orders 0, 1 and 2. These results include a characterization of the stress field of order 0 and of the axial and shear stresses of order 1. An appropriate physical interpretation of these results, which is considered elsewhere, will allow us to derive and to justify, from a mathematical point of view, the most well-known classical extension, bending and torsion theories for linear elastic beams, including the Bernoulli-Navier, Saint Venant, Timoshenko and Vlasov models.

The relaxation of functionals with surface energies

Abstract: The relaxation of a variational principle is determined when a superficial term is present. Lower semicontinuity of the functional may fail even when both integrands are convex. Generalized solutions, in terms of parametrized measures or Young measures, are introduced and analyzed. Several examples are given.

Remarks on the low-frequency asymptotics for the reduced wave equation and the schrodinger equation in low-dimensional spaces

Abstract: We study the low-frequency asymptotics of the solutions of the exterior Robin problem for the reduced wave equation in two dimensions and of the time-independent Schrodinger equation in one and two dimensions with a nonnegative potential with compact support, by employing integral equation methods. Applications to time-dependent problems are indicated. .

Asymptotic expansion for the optimal cost of an ill-posed problem with singular perturbations

Abstract: We give the development of j.: the optimal cost of the nonlinear problem with singular perturbations with state equation . , -Ilz"(t)-J(z(t))=vand z(O)=z(T)=O and with cost .fe(v, z) = 2\ IIz - Zd IIlho, T) + ~NII v 111>(0, T)· We make a formal expansion of the optimality system. In the case without constraints, we introduce boundary layer terms to approximate it to order O(llk) for any k> O. We show that the boundary layer terms decay exponentially. We deduce, from the approximate optimality system, the expansion ofj, to order O(1l2k) and the associated control.

Estimates for elliptic singular perturbations in Lp-spaces

Abstract: In this paper we study two-sided a-priori estimates in Lp-type spaces for a class of singularly perturbed elliptic problems with Dirichlet boundary conditions.

Reduction of linear differential systems to Arnold canonical form

Abstract: We consider systems of linear differential equations of the form eP dy/dx = A(k, e)y where A is an n X n matrix-valued function holomorphic in x possessing an asymptotic power series expansion in e, A(k, e) "" LkAk(x)ek uniformly valid on a neighborhood I x I 0+. We show that any such system can be reduced by a change of variable y --> Ty to a certain generic canonical form depending on the Jordan canonical form of A(O, 0). Here T(x, e) is an invertible matrix-valued function similarly possessing an asymptotic expansion T "" LkTk(x)ek uniformly valid on an e-dependent domain I x 1< r8(e) the size of which is determined by certain finer properties of the asymptotic expansion of A. We also obtain new results of classical type in which this domain is independent of e. The method of proof seems both new and of general purpose. It is based on an exacting analysis of finer termwise properties of divergent infinite formal series transformations which shows certain near-optimal finite sections to be powerful first approximations to actual transformations.