Homogenization of Monotone Operators

Abstract

Although Eduardo ZARANTONELLO first introduced monotone operators for solving a problem in continuum mechanics,¹ the theory of monotone operators quickly became taught as a part of functional analysis. In his course on nonlinear partial differential equations in the late 1960s, Jacques-Louis LIONS taught about a dichotomy, the compactness method, and the monotonicity method. During my stay in Madison in 1974-1975, I found that the div-curl lemma gave a natural framework to the monotonicity method for (stationary) diffusion equations, and that it was not so natural to classify the convexity method as being a part of the monotonicity method. A few years later, after developing the theory of compensated compactness with François MURAT, I unified all these methods in the compensated compactness method. I shall only discuss here the homogenization of monotone operators in the simple framework that I adopted in my Peccot lectures at the beginning of 1977.