G-convergence of monotone operators

Abstract

A general notion of $G$-convergence for sequences of maximal monotone operators of the form ${\mathcal{A}}_{h}u=-\mathrm{div}(a_h(x,Du))$ is introduced in terms of the asymptotic behavior, as $h\to +\infty$, of the solutions $u_h$ to the equations ${\mathcal{A}}_h{u}_h=f_h$ and of their momenta $a_h(x,D{u}_h)$. The main results of the paper are the local character of the $G$-convergence and the $G$-compactness of some classes of nonlinear monotone operators.

Résumé

On présente une notion générale de $G$-convergence pour des opérateurs maximaux monotones sous forme divergence. On démontre le caractère local de la $G$-convergence et de la $G$-compacité pour certaines classes d’opérateurs de ce type.