Homogenization of some quasilinear problems for stratified media with low and high conductivities
Bernard Heron, Jacqueline Mossino, Colette Picard · Differential and Integral Equations · 1994
We are concerned with the homogenization of quasilinear elliptic problems (P") on structures which are stratified in some direction (say :q), when the coefficients belong to L 00 but are neither uniformly bounded from above or from below with respect to e:.The general framework of this study covers as a subcase the case of linear diffusion equations with conductivity matrices A"(:q) which are diagonal and, as well as their inverses, are not uniformly elliptic with respect to e:.We give assumptions on the coefficients -convergence in the sense of measures and a weakened uniform ellipticity condition-which imply the convergence of the problems (P") to a problem of the same form with L 00 -coefficients too. Introduction.Let rl =]0, 1(xrt' be a cylindrical domain in ffi.N representing a structure which is stratified in the x1-direction; we consider quasilinear elliptic problems (Pe:) -well-posed in the Sobolev space W 1 ,P(rt)of the form { a ( e( )au") '\'N e( )P-1 a ( 1 au") f • r. -axl 91 X, al Xl axl -L..Ji=2 ai Xl axi 9i X ' axi = Ill ~ 6' +boundary conditions L, ('Pe) where x = (x 1, x'), x' = (x2, ... , XN) and L stands for Dirichlet or mixed Dirichlet-Neumann boundary conditions.It is supposed that the functions gi have suitable growth and convexity properties, and that the coefficients a:[ which only depend on x 1, that is, the meaning of stratification in this paper, satisfy Cf ~ ai(x1) ~ c:[ > 0 almost everywhere for i = 1, ... , N. This paper is devoted to the homogenization of such problems in the case where the functions a:[ and I~ may not be bounded in L 00 (0, 1) ass tends to zero.Instead of fully des~ribing the class of problems ('P 8 ) that are concerned, let us introduce the chosen framework by examples of growing generality.The simplest