Gilles Francfort, LPMTM, Institut Galilee, University Paris Nord.
"Existence and homogenization for an x--dependent class of multi--valued operators"
This is joint work with F. Murat & L. Tartar.
We investigate the solutions to -div a(x,grad u)= f, together with appropriate boundary conditions, whenever a(x,e) is a maximal monotone graph in e, for every fixed x.
We propose an adequate setting for this problem which consists in looking at the graph after a 45circ rotation, for every fixed x. This "trick" --- due to Minty once again--- essentially does away with delicate measurability questions and permits to obtain an existence result.
In a second part, the stability of that class of operators under homogenization is established with the help of an abstract, but explicit maximal extension result for monotone graphs.
THURSDAY, November 21, 2002