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J. Mossino
France

 

Homogenization of some nonlinear problems with specific dependence upon coordinates



P. Courilleau$^{(1)}$, S. Fabre$^{(2)}$, J. Mossino$^{(2)}$

(1)Université de Cergy-Pontoise,
Département de mathématiques,
2, Avenue Adolphe Chauvin,
95302 Cergy Pontoise, France.


(2)Ecole Normale Supérieure de Cachan,
Centre de Mathématiques et Leurs Applications,
61, Avenue du Président Wilson,
94235 Cachan Cedex, France.

Abstract: The paper is concerned with a sequence of nonlinear partial differential equations in divergence form, of the type

\begin{displaymath}- div \left (Q^\varepsilon G(x, N^\varepsilon \nabla u) \right) = f^\varepsilon, \end{displaymath}

in a bounded domain $\Omega$ of the $n$-dimensional space. Here, $Q^\varepsilon = Q^\varepsilon (x)$ and $N^\varepsilon = N^\varepsilon (x)$ are matrices with bounded entries, $N^\varepsilon$ is invertible and the inverse matrix $R^\varepsilon$ also has bounded entries. The nonlinearity is due to thefunction $G = G(x,\xi)$; the growth condition and the monotonicity and coercivity assumptions are modeled on the $p$-Laplacian, $1<p<\infty$, and ensure existence of a solution $u^\varepsilon \in W^{1,p}_0 (\Omega)$ to each of these equations, for every fixed $f^\varepsilon \in W^{-1,p'}_0 (\Omega)$. A specific dependence on the coordinates is assumed for the coefficient matrices: $Q^\varepsilon (x) = \left(q^\varepsilon _{i,j} (x'_i)\right)$ and $R^\varepsilon (x) = \left(r^\varepsilon _{i,j} (x_i)\right)$, where thearbitrary point of $\Omega$ is denoted by $x = (x_i, x'_i)$, with $x_i$ real and $x'_i$ in the $(n-1)$-dimensional space. The main result (essentially) reads as follows. Assume the following convergence: for the coefficients, $Q^\varepsilon \rightharpoonup Q$, $R^\varepsilon \rightharpoonup R$, with respect to the weak* topology; for the source terms, $f^\varepsilon \rightarrow f$, with respect to the strong topology of $W^{-1,p'}_0 (\Omega)$; and for the solutions $u^\varepsilon \rightharpoonup u$, with respect to the weak topology of $W^{1,p}_0 (\Omega)$; then $u$ solves the limit equation

\begin{displaymath}- div \left (Q \, G(x, N^\varepsilon \, \nabla u )\right) = f. \end{displaymath}

A corrector type result is also proven and applications are given.

 

 

 

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Last modified: December 20, 1999