Homogenization and Convergence of Monotone Operators with Respect to Measures

Abstract

In this work, monotone operators are considered which satisfy suitable continuity and monotonicity assumptions. We are dealing with two cases when these operators depend or not on the thickness of thin structures which occur in junctions of structures with a small parameter. This parameter caracterizes the microscopic lenght-scale of the whole construction. We establish some results in convergences with respect to singular measures, inspiring essentially by works of V.V. Zhikov, D. Lukkassen and P. Wall, but without using two-scale convergence. Then we compare operators obtained for the singular structures and the classical ones obtained for thin constructions with vanishing thickness, and show the commutativity of the corresponding diagram under some assumptions.