Homogeneization at the boundary with singularities
by
Alexandre Demidov
Department of Mathematics, University of Moscow, Russia
Coauthors: Mohand Moussaoui (MAPLY, Ecole Centrale de Lyon, Ecully, France), Michelle Schatzman (MAPLY, CNRS et Université Claude Bernard -- Lyon 1, Villeurbanne, France)

Let L be the elliptic operator defined in R×R⁺ by

L =  \partial \partialx a(x, y)  \partial \partialx +  \partial \partialy b(x, y)  \partial \partialy .

Define singular and strongly oscillating data at the boundary with the help of a 2\pi-periodic infinitely differentiable function real f on R and of a function \sigma from R to itself which satisfies the following conditions: x (x)-x is 2\pi-periodic,
the restriction of \sigma to (0, 2\pi) is 2\pi-periodic,
for some C > 0, \tau in (0, 1) and all x in [-\pi/2, \pi/2], C |x|^-'(x) 1/C.

If u_\epsilon is the unique tempered solution of

L u\epsilon=0,     u\epsilon(x, 0)=f æ è \sigma(x)/\epsilon ö ø ,

we give an asymptotic expansion of u_\epsilon at all orders of the form

u\epsilon(x, y)= å j >= 0  \epsilonj vj(x, y, \xi, \eta),    \xi =  \sigma(x) \epsilon ,     \eta =  y \epsilon .

The term v^j includes exponentially decaying terms with respect to \eta, which was expected, but we have to deal with a number of algebraic and analytic difficulties due to the singular behavior of \sigma. The asymptotic expansion can be validated with the help of elliptic estimates.

Date received: June 13, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cahv-15.